Bhakti movement

WEBVTT
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Language: en

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- [Instructor] In other
videos we have talked about

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the various empires of India

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as we exit the Vedic period.

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We talk about the Maurya Empire

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famous for the ruler Ashoka who converts

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and then spreads Buddhism.

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As we get into the Common Era

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we've talked about the Gupta Empire

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that once again unifies much of India,

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and is considered a golden
age in India's history.

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But after the fall of the
Gupta Empire India fragments

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and you could roughly consider this

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to be the Indian medieval period.

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This is what India looks
like in the year 1000

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where you have multiple Hindu kingdoms.

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Now as we get into the second
millennium of the Common Era,

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you start to have
significant Muslim influence.

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At first in the northwest

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but as we get into the late
12th and early 13th century

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you have the establishment
of the Delhi Sultanate

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and once again North India is now unified

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under Muslim rule which
will continue for some time

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under both the Delhi Sultanate
and the Mughal Empire

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until the British show up.

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And then eventually in 1947,

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India becomes an independent country.

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But as much transformation as there is

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depicted in this map in terms
of various empires and rulers,

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you also see a transformation

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in the nature of Hinduism.

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In other videos we have
talked about the Vedas

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composed over 3,000 years ago

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in the subset that is
focused on the metaphysical,

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on the spiritual known as the Upanishads.

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And this is an excerpt
from the Isha Upanishad

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which is considered one
of the most important

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that emphasizes this
notion of our inner Self,

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capitalized with an S, atma or atman

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and how it is of the
same substance as this

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formless, nameless ultimate reality

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sometimes referred to as Brahman.

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The Wise man, who realizes all beings

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as not distinct from his own Self, atman,

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and his own Self as
the Self of all beings,

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does not, by virtue of that
perception, hate anyone.

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What delusion, what sorrow
can there be for that wise man

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who realizes the unity of all existence

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by perceiving all beings as his own Self?

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And so you have this very abstract idea,

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some folks would call it monistic.

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All is one.

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Your true self is of the same
substance as my true self

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which is of the same substance
and is the same thing

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as the ultimate reality.

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But the Vedas also focus on rituals

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and there are also gods in the Vedas.

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In particular some of the gods

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that are mentioned most frequently.

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You have Indra who is the god of lightning

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and the god of storms

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also referred to in the
Vedas as the king of gods.

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You have Agni, the god of fire.

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It come from the same root word as ignite.

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You have Varuna, the god of water.

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But a question, how do we
bridge that to the gods

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and the practices of modern Hinduism?

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As we've talked about in
other videos on Hindu deities,

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most Hindus today view
themselves as devotees of Shiva

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or devotees of Vishnu.

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And even though Vishnu is
mentioned in the Vedas,

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he is by no means the focus.

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Modern Hindu practice is
really focused on devotion

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to Shiva and/or Vishnu or aspects of them

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or incarnations of them.

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For example, Rama or Krishna.

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How did Hinduism evolve in this way?

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And the answer is the Bhakti movement.

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And the word Bhakti can be translated as

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devotion or devotional love.

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And it's believed to have
come out of South India

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in the seventh century.

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Some people would call
it a reform of Hinduism

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that up until that point focused on

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the somewhat arcane rituals of the Vedas.

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And the Bhakti movement
provided an alternate path,

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a path of devotion

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Through devotion through a deity

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one can achieve that moksha.

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Some people would consider
it a reform movement,

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some people would say it was always there.

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Some historians think that

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it might have even been in
reaction to Muslim influence

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where there was this devotion

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or this surrender to a notion of god.

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Some historians think that it co-developed

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with Sufi Islam which is all about

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devotional love for God.

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But needless to say,

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as we go through the Indian middle ages,

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the Bhakti movement gains
more and more momentum

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and it's really the defining
movement for Hinduism today.

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And just to get an appreciation
for some Bhakti text

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here is an excerpt from the Bhagavad Gita

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which is considered by many modern Hindus

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to be one of their most
central scriptures.

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It was likely written over 2,000 years ago

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and really came into its final form

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during the Gupta Empire.

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But the Gita is considered one
of the central Bhakti text.

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And even though the Gita does focus a lot

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on notions of realizing the self

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and the ultimate reality and
the importance of meditation,

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it also makes clear that there's a path

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to self-realization through
love, through devotion.

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This is Krishna, the eighth
incarnation of Vishnu

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talking to the archer Arjun
as he goes into battle.

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It's a bit of a repudiation

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of what at the time in medieval India

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was becoming a more and
more rigid caste structure.

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I look upon all creatures equally

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none are less dear to
me and none more dear.

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And it is also very inclusive.

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It is not saying that one
needs to only worship Vishnu

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or only worship Shiva.

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Those who worship other
gods with faith and devotion

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also worship me.

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Not a fixation on being absolutely true

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to all of the rituals in the Vedas.

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Whatever I am offered in
devotion with a pure heart,

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even if it's a leaf, a
flower, a fruit or water,

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I accept with joy.

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With that, I'll leave you with a bhajan,

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a big part of the Bhakti movement

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of this notion of devotional
love to God is singing.

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(singing in foreign language)


Why were the Mongols so effective?

WEBVTT
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Language: en

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- [Instructor] The question
before us today is:

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why were Mongols so effective?

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How did they manage to take
an area, starting around here,

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and over the course of
20 years during the reign

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of Genghis Khan, from about 1206 to 1227,

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expand from this little part of Siberia,

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just south of Lake Baikal,
all the way into Central Asia?

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This is what the Mongol
Empire looked like in 1206,

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and this is what it looked like in 1279.

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This here is Genghis-Khan time,

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and this here is around
the reign of Kublai Khan.

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What enables a spread of territory

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on this scale in such a short time?

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The Mongols were not a centralized state.

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They weren't a centralized empire.

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They were a band of traveling horsemen.

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How did they end up taking this much land?

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The story of the Mongols is a story

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of a non-state actor
overcoming state actors,

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of a decentralized power
becoming centralized eventually.

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What I wanna talk about in this video

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is three reasons why the Mongols were

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so effective in beating
established empires.

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And then we're gonna talk about some

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of those follow-on effects.

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Reason one why the
Mongols were so effective

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was planning, what we
might call logistics.

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When Genghis Kahn,

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lemme pull up a picture
of good ol' Genghis,

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when Genghis was declared Emperor,

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or Great Khan, of the Mongols in 1206,

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one of the first things he
did was reformat the army.

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He swept away tribal
affiliations and favoritism.

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If you were good, you got
to have a higher position;

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if you were bad, you didn't.

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This meant that ordinary
people within the Mongol army

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felt that they could have a
measure of social mobility

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if they performed
especially well in the army.

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This made them very
committed and very loyal.

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Reason number two: the Mongols
were extremely adaptable.

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We're talking about a
group of horsemen here.

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These are the Mongols attacking
a group of Chinese soldiers.

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What they were used to was mounted combat

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against mounted opponents.

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What they weren't used to was attacking

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major cities and holding them,

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or even trying to get
major cities to break.

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But what the adaptability
of the Mongol army meant

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was that as they conquered new territories

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and as they fought and won and lost

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in various regions of
Asia, so, for example,

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as they started conquering
the Western Xia here,

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they learned siegecraft,

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and that's the art of attacking
walled cities and castles.

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It's a completely
different style of combat.

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What it involves is
sitting outside of a city,

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and waiting to starve out the inhabitants.

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Part of it involves using these
things called siege engines.

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This here is a kind of catapult.

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You can see here the rocks.

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The Mongols are waiting
with their archers here.

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The opponents have arrows, as well,

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but they're flinging rocks into the city

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in order to force a surrender.

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The Mongols didn't really
have this technology

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or these tactics when they
began their military campaigns;

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but as they rolled through
Central Asia they picked it up,

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and then they continued
to use those tactics

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as they swept westward.

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Now, the third reason that
the Mongols were so effective

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is because they were terrifying.

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Even though they mastered siege warfare,

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in the beginning at least,
the Mongols were not prepared

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to hold a city once they had conquered it.

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There are stories,
especially from the conquest

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of the Western Xia and the Northern Song

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and the Khwarazmian Empire over here,

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that tell of the Mongols winning a battle

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and then sweeping in
and murdering everyone.

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Just pulling plunder from corpses,

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and just looting the entire city,

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and leaving nothing standing.

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Because the Mongols weren't
especially interested,

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at least early, we're
talking about the period

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of Genghis Khan's life
here, so, 1200s to 1220s,

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they weren't especially
interested in governing.

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They wanted to hold
territory, and get tribute,

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and get money, but they weren't especially

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interested in running an empire.

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It actually made more sense to them

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to slaughter everybody
and take your things.

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Now, it got to the point
that this reputation

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for fearsomeness got so
powerful that some cities would

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just give up and surrender the
moment the Mongols arrived.

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Those are the three main
reasons that comprise

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Mongol effectiveness
against settled states.

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They were very good at planning.

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Everybody rode in their armies.

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They could move very quickly.

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The Mongols were extremely adaptable.

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Every time they lost,
they learned something

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from the people they lost
to, and that meant that

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they picked up new technologies
like siege warfare.

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And they were absolutely terrifying.

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Since, initially, the
Mongols didn't know what

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they were gonna do with a
conquered city besides loot it,

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they just killed everybody,
and that reputation

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really allowed them to conquer
a lot of land very quickly.

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And so we see out of the Mongol conquest,

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there are many follow-on effects,
but one of the chief ones

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is something called the Pax
Mongolica, the Mongolian Peace.

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If that phrasing looks familiar to you,

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it's because it's similar
to the Roman Peace,

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the Pax Romana; and the Chinese Peace,

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the Pax Sinica of the First Century CE.

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This is 1100 years later, but
the same principle applies.

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What happened was, with all
of this territory conquered

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under one ruling power,
it became safe to travel

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from one end of the empire to the other.

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Indeed, this is exactly what
Marco Polo did when he set off

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from Venice, and made his
way, eventually, to Beijing.

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That would've been around this time.

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Marco Polo's book, The Book
of the Marvels of the World,

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was published in 1300.

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Marco Polo was this Venetian
trader and adventurer

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who traveled extensively throughout Asia.

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That's one result of the Mongol conquests,

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is this Mongolian Peace.

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I should say that this Mongolian Peace

00:06:10.626 --> 00:06:12.534
came at a terrible price.

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When we were talking about
that terror of the Mongols,

00:06:14.241 --> 00:06:15.907
that's a for-real thing.

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They murdered millions of people.

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But what would come
after the Mongol Conquest

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as a result of this peace
would end up killing

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even more people than the
Mongol Conquest itself.

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This pacified region,
this Mongolian Peace,

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led to increased trade
throughout East Asia

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and reconnecting it to
Eurasia and the Arab States.

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That would result in the
exchanges of new ideas

00:06:41.974 --> 00:06:45.710
and new technologies and
new agricultural techniques.

00:06:45.710 --> 00:06:47.668
But, as trade always does,

00:06:47.668 --> 00:06:50.916
it also results in the
exchange of diseases.

00:06:50.916 --> 00:06:53.762
And so, out of the
Mongol homeland, in fact,

00:06:53.762 --> 00:06:57.221
comes this illness
called the Black Plague,

00:06:57.221 --> 00:06:58.845
or Bubonic Plague.

00:06:58.845 --> 00:07:01.242
Because of the stability of trade

00:07:01.242 --> 00:07:04.338
that had been established
by the Mongol Peace,

00:07:04.338 --> 00:07:07.301
it makes it so much easier
for this illness to sweep

00:07:07.301 --> 00:07:10.635
across Asia and Europe,
killing even more people than

00:07:10.635 --> 00:07:14.441
even Genghis Khan could've
achieved during his lifetime.


Knights Templar

WEBVTT
Kind: captions
Language: en

00:00:00.389 --> 00:00:02.049
- [Instructor] We've
already done multiple videos

00:00:02.049 --> 00:00:04.041
on the crusades but what
we're going to focus on

00:00:04.041 --> 00:00:07.374
in this video is how the
crusades help catalyze

00:00:07.374 --> 00:00:10.122
the start of what many historians consider

00:00:10.122 --> 00:00:14.620
to be the first international
financial institution

00:00:14.620 --> 00:00:17.503
and that is the Knights Templar.

00:00:17.503 --> 00:00:21.217
So let's just make sure we have
a good hold on the context.

00:00:21.217 --> 00:00:23.390
So over here we have our timeline.

00:00:23.390 --> 00:00:25.700
It's occurring during
the high middle ages.

00:00:25.700 --> 00:00:28.688
In other videos we saw
the first crusades start

00:00:28.688 --> 00:00:32.099
at the end of the 11th
century, started really

00:00:32.099 --> 00:00:36.189
by Pope Urban the second
with the purported reason

00:00:36.189 --> 00:00:38.710
to help the Byzantines take land

00:00:38.710 --> 00:00:42.150
from the Muslims,
especially the Holy Land.

00:00:42.150 --> 00:00:45.668
The first crusade is for
the most part successful

00:00:45.668 --> 00:00:48.671
but they don't return that
land to the Byzantines.

00:00:48.671 --> 00:00:51.340
They set up what are
known as Crusader States

00:00:51.340 --> 00:00:55.327
or Crusader Kingdoms including
the Kingdom of Jerusalem

00:00:55.327 --> 00:00:59.236
and just to read this
timeline that I drew up here,

00:00:59.236 --> 00:01:02.025
green shows a time period when at least

00:01:02.025 --> 00:01:05.168
Jerusalem is controlled by the Muslims.

00:01:05.168 --> 00:01:08.232
Red shows when Jerusalem is controlled

00:01:08.232 --> 00:01:11.280
by Western European powers or by these

00:01:11.280 --> 00:01:15.355
Crusader Kingdoms which were
essentially Western European.

00:01:15.355 --> 00:01:18.434
These red parts on our timeline show

00:01:18.434 --> 00:01:21.480
when the actual crusades occurred.

00:01:21.480 --> 00:01:25.573
So you see the first crusade
causes the change of control

00:01:25.573 --> 00:01:28.092
of Jerusalem from the Muslims to

00:01:28.092 --> 00:01:31.007
the Western European Christians

00:01:31.007 --> 00:01:33.241
and once Jerusalem was under their control

00:01:33.241 --> 00:01:35.248
and it would continue to
be under their control

00:01:35.248 --> 00:01:37.979
for roughly 90 years, you could imagine

00:01:37.979 --> 00:01:42.462
that many pilgrims from
Europe went to the holy sites

00:01:42.462 --> 00:01:45.916
in the Holy Land especially in Jerusalem,

00:01:45.916 --> 00:01:48.375
but this was not an easy journey.

00:01:48.375 --> 00:01:51.473
Not only was it a long
journey, these are the paths

00:01:51.473 --> 00:01:54.159
that I showed in previous
videos that the crusaders took,

00:01:54.159 --> 00:01:55.593
but this could also be the paths

00:01:55.593 --> 00:01:59.108
that many of the pilgrims
took in order to go to

00:01:59.108 --> 00:02:01.857
the Holy Land and you
could imagine a journey

00:02:01.857 --> 00:02:05.175
of this distance going
through multiple kingdoms

00:02:05.175 --> 00:02:09.162
in this time period could
be incredibly dangerous

00:02:09.162 --> 00:02:12.090
and many of the pilgrims
actually were robbed

00:02:12.090 --> 00:02:16.173
or sometimes worse on
their way to the Holy Land.

00:02:17.237 --> 00:02:20.421
So in 1119 a small group of Knights

00:02:20.421 --> 00:02:22.421
led by Hughes de Payens,

00:02:23.395 --> 00:02:28.270
they go to King Baldwin the
second, King of Jerusalem.

00:02:28.270 --> 00:02:30.066
Remember they set up this crusader state,

00:02:30.066 --> 00:02:32.615
the Kingdom of Jerusalem headquartered

00:02:32.615 --> 00:02:36.193
on the temple mount and these
knights tell King Baldwin

00:02:36.193 --> 00:02:39.332
the second and Warmund,
or Patriarch Warmund

00:02:39.332 --> 00:02:41.778
the Patriarch of Jerusalem, they say

00:02:41.778 --> 00:02:43.813
we would like to protect these pilgrims

00:02:43.813 --> 00:02:46.140
on their way to Jerusalem.

00:02:46.140 --> 00:02:48.764
These guys think it is a good idea

00:02:48.764 --> 00:02:52.387
and so in 1120 they give them some space,

00:02:52.387 --> 00:02:54.833
a headquarters on the temple mount

00:02:54.833 --> 00:02:57.333
in order to start their group.

00:02:58.923 --> 00:03:01.882
If you translate the Latin
version of their name

00:03:01.882 --> 00:03:05.820
it translates to Poor
Fellow-Soldiers of Christ

00:03:05.820 --> 00:03:08.568
and the Temple of Solomon.

00:03:08.568 --> 00:03:12.567
Now the reason why it says
and the Temple of Solomon

00:03:12.567 --> 00:03:15.375
is because they were
headquartered on the temple mount

00:03:15.375 --> 00:03:18.529
and the Al-Aqsa Mosque is believed by many

00:03:18.529 --> 00:03:22.453
to be on the site of the
original Temple of Solomon,

00:03:22.453 --> 00:03:27.268
but this is often shortened
to the Knights Templar.

00:03:27.268 --> 00:03:30.947
This is their symbol, two
knights riding on a horse

00:03:30.947 --> 00:03:35.371
to show their poverty, but
they don't stay poor for long.

00:03:35.371 --> 00:03:38.872
They are shortly thereafter
recognized by the Pope

00:03:38.872 --> 00:03:43.081
and many people start
donating money to this order.

00:03:43.081 --> 00:03:47.219
Many more people volunteer
to be Knights Templar

00:03:47.219 --> 00:03:50.373
and so over the remainder of the crusades

00:03:50.373 --> 00:03:54.540
they become an important
institution in Medieval Europe.

00:03:59.293 --> 00:04:02.626
Now most folks often
associate the Knights Templar

00:04:02.626 --> 00:04:03.862
with being Knights.

00:04:03.862 --> 00:04:07.374
You can see here one of
the Knights on horseback.

00:04:07.374 --> 00:04:10.300
Here you see the Knights
protecting pilgrims

00:04:10.300 --> 00:04:12.591
on their way to Jerusalem
and they did do that.

00:04:12.591 --> 00:04:15.485
They did protect the pilgrims
and they were also active

00:04:15.485 --> 00:04:18.979
in battle during the crusades thereafter,

00:04:18.979 --> 00:04:21.889
but it turns out that even
though on the peak there were

00:04:21.889 --> 00:04:24.814
on the order of 20,000 Knights Templar,

00:04:24.814 --> 00:04:28.584
only about 10% of them
were actual soldiers,

00:04:28.584 --> 00:04:30.893
were actually these warrior monks

00:04:30.893 --> 00:04:32.649
as they're sometimes called.

00:04:32.649 --> 00:04:35.412
The rest of the Templars
created this fairly significant

00:04:35.412 --> 00:04:38.125
complex infrastructure
to do multiple things

00:04:38.125 --> 00:04:41.979
to support the goal of the Knights Templar

00:04:41.979 --> 00:04:45.603
and perhaps most
significantly they developed

00:04:45.603 --> 00:04:49.878
a sprawling financial
enterprise of the time.

00:04:49.878 --> 00:04:53.308
If you were in London and
you wanted to do a pilgrimage

00:04:53.308 --> 00:04:55.908
to the Holy Land you would need resources

00:04:55.908 --> 00:04:59.013
in order to do that and you
would want to have access

00:04:59.013 --> 00:05:02.052
to your wealth once you got to Jerusalem

00:05:02.052 --> 00:05:03.921
and so you could imagine
if you were traveling

00:05:03.921 --> 00:05:06.441
over a long journey to
take gold coins with you,

00:05:06.441 --> 00:05:07.562
not only would they be heavy

00:05:07.562 --> 00:05:09.399
but more importantly
it would be dangerous.

00:05:09.399 --> 00:05:12.585
A lot of people might want
to take that gold from you

00:05:12.585 --> 00:05:15.884
and so one of the services
that the Knights Templar

00:05:15.884 --> 00:05:20.517
developed was that you
could go to their temple

00:05:20.517 --> 00:05:23.912
in London essentially deposit your money,

00:05:23.912 --> 00:05:26.235
they'll give you a note
of credit saying that,

00:05:26.235 --> 00:05:28.936
we the Knights owe you a
certain amount of money,

00:05:28.936 --> 00:05:30.543
and then you make your journey

00:05:30.543 --> 00:05:34.007
and when you get to Jerusalem
you can give that note

00:05:34.007 --> 00:05:38.038
to the Knights there and they
will give you your money.

00:05:38.038 --> 00:05:40.590
Now this is something that
we take for granted today.

00:05:40.590 --> 00:05:42.768
I can go to an ATM machine
anywhere in the world

00:05:42.768 --> 00:05:45.465
and for the most part I
can put in my ATM card

00:05:45.465 --> 00:05:48.701
and get access to money, but there's a lot

00:05:48.701 --> 00:05:50.993
of infrastructure that makes that happen.

00:05:50.993 --> 00:05:54.309
Today we have telecommunications,
we have satellites

00:05:54.309 --> 00:05:56.194
that help information go back and forth,

00:05:56.194 --> 00:05:59.834
we have servers that keep
track of all of that data.

00:05:59.834 --> 00:06:01.721
They didn't have that in the middle ages

00:06:01.721 --> 00:06:04.850
and so you can imagine this
is a very powerful idea

00:06:04.850 --> 00:06:07.693
that I could deposit money one place

00:06:07.693 --> 00:06:10.506
and then I could withdraw
it someplace else

00:06:10.506 --> 00:06:12.822
and we still don't fully understand

00:06:12.822 --> 00:06:15.472
how the knights were
able to pull this off.

00:06:15.472 --> 00:06:17.910
It requires a lot of trust
and how were they able

00:06:17.910 --> 00:06:22.152
to authenticate the notes of
credit at any of these points?

00:06:22.152 --> 00:06:25.272
Not only did they do this type of deposit

00:06:25.272 --> 00:06:27.707
and withdrawal in multiple locations,

00:06:27.707 --> 00:06:30.278
they also because they
had all of these deposits

00:06:30.278 --> 00:06:32.586
and people were also
donating money to them

00:06:32.586 --> 00:06:34.114
just to support their cause,

00:06:34.114 --> 00:06:36.308
they were able to provide loans.

00:06:36.308 --> 00:06:38.047
Often times loans to very powerful

00:06:38.047 --> 00:06:41.450
individuals like kings and nobility.

00:06:41.450 --> 00:06:44.815
They were often brokers
in major transactions.

00:06:44.815 --> 00:06:47.364
This is the reason why
historians really do consider

00:06:47.364 --> 00:06:51.281
them perhaps the first
true international bank.

00:06:52.617 --> 00:06:56.729
In other videos we talk
about how in 1187 Saladin

00:06:56.729 --> 00:06:59.927
is able to retake Jerusalem
but then in the Sixth Crusade

00:06:59.927 --> 00:07:02.999
and the Barons' Crusades it
goes back to Christian hands

00:07:02.999 --> 00:07:06.266
but then 1244 it's retaken
again by the Muslims

00:07:06.266 --> 00:07:08.314
and it stays under Muslim
control all the way

00:07:08.314 --> 00:07:10.785
until the end of World War One

00:07:10.785 --> 00:07:14.621
and so with Jerusalem taken
in 1244 the whole goal

00:07:14.621 --> 00:07:17.238
of the Knights Templar, the whole purpose

00:07:17.238 --> 00:07:20.200
of their existence starts
to become less important.

00:07:20.200 --> 00:07:22.443
Although at this point
they are a significant

00:07:22.443 --> 00:07:25.385
and influential financial institution.

00:07:25.385 --> 00:07:28.132
As we go to the end of the 13th century

00:07:28.132 --> 00:07:31.171
the whole crusader mentality
starts to lose steam.

00:07:31.171 --> 00:07:33.268
You have the Eighth and Ninth Crusades

00:07:33.268 --> 00:07:36.242
but they aren't successful
and as we get into the early

00:07:36.242 --> 00:07:40.409
14th century you have this
character Philip the Fourth,

00:07:42.336 --> 00:07:46.515
King of France and he has
incurred a lot of debt

00:07:46.515 --> 00:07:50.058
in order to have wars
especially with England

00:07:50.058 --> 00:07:51.732
and a good bit of this financing

00:07:51.732 --> 00:07:54.479
comes from the Knights Templar.

00:07:54.479 --> 00:07:56.966
He is a powerful man looking for solutions

00:07:56.966 --> 00:08:00.542
to his debt situation and
he realizes one solution

00:08:00.542 --> 00:08:02.708
is maybe just to arrest the people

00:08:02.708 --> 00:08:05.261
who you owe money to and then not only

00:08:05.261 --> 00:08:06.819
do you not owe any money to them,

00:08:06.819 --> 00:08:08.608
but you can seize their assets

00:08:08.608 --> 00:08:11.714
and Philip the fourth does exactly that.

00:08:11.714 --> 00:08:14.797
Famously on a Friday the 13th in 1307

00:08:16.975 --> 00:08:21.142
he arrests many Knights Templars
including the Grand Master,

00:08:22.718 --> 00:08:26.553
the leader of the Knights
Templar, Jacques de Molay.

00:08:26.553 --> 00:08:27.692
He tortures them.

00:08:27.692 --> 00:08:30.227
They eventually confess
to a whole bunch of things

00:08:30.227 --> 00:08:32.537
but then they retract that confession

00:08:32.537 --> 00:08:35.511
and they're not willing
to retract the retraction

00:08:35.511 --> 00:08:36.678
and so in 1314

00:08:39.626 --> 00:08:41.934
Jacques de Molay as well as

00:08:41.934 --> 00:08:45.071
other Knights Templar,
especially in the leadership,

00:08:45.071 --> 00:08:48.127
who do not agree to those
confessions are burned

00:08:48.127 --> 00:08:51.816
at the stake in Paris
and this is considered

00:08:51.816 --> 00:08:54.352
the end of the Knights Templar

00:08:54.352 --> 00:08:56.447
although there are many legends of how

00:08:56.447 --> 00:09:00.281
they have continued on perhaps in secrecy.

00:09:00.281 --> 00:09:03.259
So the Knights Templar are to this day

00:09:03.259 --> 00:09:05.681
are a subject of fascination.

00:09:05.681 --> 00:09:08.005
There's many legends associated with them,

00:09:08.005 --> 00:09:10.651
but one of their biggest
historical contributions

00:09:10.651 --> 00:09:13.759
that isn't often talked
about but we know happened,

00:09:13.759 --> 00:09:16.652
is that they really laid
the foundation for what

00:09:16.652 --> 00:09:20.569
we today consider modern
international banking.


Example estimating from regression line

WEBVTT
Kind: captions
Language: en

00:00:00.182 --> 00:00:02.993
- [Instructor] Liz's math test
included a survey question

00:00:02.993 --> 00:00:06.621
asking how many hours students
spent studying for the test.

00:00:06.621 --> 00:00:09.508
The scatter plot below
shows the relationship

00:00:09.508 --> 00:00:12.238
between how many hours
students spent studying

00:00:12.238 --> 00:00:14.358
and their score on the test.

00:00:14.358 --> 00:00:17.821
A line was fit to the data
to model the relationship.

00:00:17.821 --> 00:00:19.344
They don't tell us how the line was fit,

00:00:19.344 --> 00:00:21.173
but this actually looks
like a pretty good fit

00:00:21.173 --> 00:00:22.842
if I just eyeball it.

00:00:22.842 --> 00:00:24.384
Which of these linear equations

00:00:24.384 --> 00:00:26.510
best describes the given model?

00:00:26.510 --> 00:00:28.540
So this, you know, this
point right over here,

00:00:28.540 --> 00:00:30.746
this shows that some student
at least self-reported

00:00:30.746 --> 00:00:32.849
they studied a little bit
more than half an hour,

00:00:32.849 --> 00:00:34.813
and they didn't actually
do that well on the test,

00:00:34.813 --> 00:00:37.663
looks like they scored a
43 or a 44 on the test.

00:00:37.663 --> 00:00:39.146
This right over here shows,

00:00:39.146 --> 00:00:40.778
or like this one over here is a student

00:00:40.778 --> 00:00:42.314
who says they studied two hours,

00:00:42.314 --> 00:00:46.874
and it looks like they scored
about a 64, 65 on the test.

00:00:46.874 --> 00:00:50.560
And this over here or
this over here looks like

00:00:50.560 --> 00:00:52.554
a student who studied over four hours,

00:00:52.554 --> 00:00:53.497
or they reported that,

00:00:53.497 --> 00:00:56.714
and they got, looks like
a 95 or a 96 on the exam.

00:00:56.714 --> 00:00:58.722
And so then, and these are
all the different students,

00:00:58.722 --> 00:01:01.074
each of these points represents a student,

00:01:01.074 --> 00:01:02.605
and then they fit a line.

00:01:02.605 --> 00:01:04.439
And when they say which
of these linear equations

00:01:04.439 --> 00:01:06.832
best describes the given
model, they're really saying

00:01:06.832 --> 00:01:10.544
which of these linear equations describes

00:01:10.544 --> 00:01:14.260
or is being plotted right
over here by this line

00:01:14.260 --> 00:01:16.369
that's trying to fit to the,

00:01:16.369 --> 00:01:18.862
that's trying to fit to the data.

00:01:18.862 --> 00:01:20.307
So essentially, we just want to figure out

00:01:20.307 --> 00:01:21.940
what is the equation of this line?

00:01:21.940 --> 00:01:25.669
Well, it looks like the
y-intercept right over here is 20.

00:01:25.669 --> 00:01:27.681
And it looks like all
of these choices here

00:01:27.681 --> 00:01:30.437
have a y-intercept of 20, so
that doesn't help us much.

00:01:30.437 --> 00:01:32.198
But let's think about what the slope is.

00:01:32.198 --> 00:01:34.526
When we increase by one,

00:01:34.526 --> 00:01:38.166
when we increase along our x-axis by one,

00:01:38.166 --> 00:01:39.895
so change in x is one,

00:01:39.895 --> 00:01:41.517
what is our change in y?

00:01:41.517 --> 00:01:43.487
Our change in y looks like,

00:01:43.487 --> 00:01:45.261
let's see, we went from 20 to 40.

00:01:45.261 --> 00:01:47.700
It looks like we went up by 20.

00:01:47.700 --> 00:01:51.386
So our change in y over
change in x for this model,

00:01:51.386 --> 00:01:53.929
for this line that's
trying to fit to the data,

00:01:53.929 --> 00:01:55.441
is 20 over one.

00:01:55.441 --> 00:01:57.194
So this is going to be our slope.

00:01:57.194 --> 00:01:58.486
And if we look at all of these choices,

00:01:58.486 --> 00:02:01.497
only this one has a slope of 20.

00:02:01.497 --> 00:02:04.940
So it would be this
choice right over here.

00:02:04.940 --> 00:02:06.580
Based on this equation,

00:02:06.580 --> 00:02:08.792
estimate the score for a student

00:02:08.792 --> 00:02:11.375
that spent 3.8 hours studying.

00:02:11.375 --> 00:02:14.558
So we would go to 3.8,
which is right around,

00:02:14.558 --> 00:02:17.928
let's see, this would be, 3.8
would be right around here.

00:02:17.928 --> 00:02:19.788
So let's estimate that score.

00:02:19.788 --> 00:02:23.104
So if I go straight up, where
do we intersect our model?

00:02:23.104 --> 00:02:25.492
Where do we intersect our line?

00:02:25.492 --> 00:02:27.389
So it looks like they would
get a pretty high score.

00:02:27.389 --> 00:02:30.795
Let's see, if I were to take
it to the vertical axis,

00:02:30.795 --> 00:02:33.722
it looks like they would get about a 97.

00:02:33.722 --> 00:02:35.982
So I would write that my estimate is

00:02:35.982 --> 00:02:39.255
that they would get a
97 based on this model.

00:02:39.255 --> 00:02:40.637
And once again, this is only a model.

00:02:40.637 --> 00:02:43.665
It's not a guarantee that if
someone studies 3.8 hours,

00:02:43.665 --> 00:02:45.128
they're gonna get a 97,

00:02:45.128 --> 00:02:48.223
but it could give an
indication of what maybe,

00:02:48.223 --> 00:02:50.567
might be reasonable to expect,

00:02:50.567 --> 00:02:52.968
assuming that the time studying

00:02:52.968 --> 00:02:54.607
is the variable that matters.

00:02:54.607 --> 00:02:56.196
But you also have to be
careful with these models

00:02:56.196 --> 00:02:58.075
because it might imply if you kept going

00:02:58.075 --> 00:02:59.847
that if you get,

00:02:59.847 --> 00:03:01.310
if you study for nine hours,

00:03:01.310 --> 00:03:02.774
you're gonna get a 200 on the exam,

00:03:02.774 --> 00:03:04.803
even though something
like that is impossible.

00:03:04.803 --> 00:03:07.665
So you always have to be careful
extrapolating with models,

00:03:07.665 --> 00:03:09.211
and take it with a grain of salt.

00:03:09.211 --> 00:03:12.033
This is just a model that's
trying to fit to this data.

00:03:12.033 --> 00:03:15.795
And you might be able to
use it to estimate things

00:03:15.795 --> 00:03:18.842
or to maybe set some
form of an expectation,

00:03:18.842 --> 00:03:21.870
but take it all with a grain of salt.


Introduction to residuals and least squares regression

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:02.036
- [Narrator] Though I'm
interested in finding

00:00:02.036 --> 00:00:05.856
the relationship between
people's height in inches

00:00:05.856 --> 00:00:08.241
and their weight in pounds.

00:00:08.241 --> 00:00:10.332
And so I'm randomly
sampling a bunch of people

00:00:10.332 --> 00:00:12.207
measuring their heights,
measuring their weight

00:00:12.207 --> 00:00:14.161
and then for each person
I'm plotting a point

00:00:14.161 --> 00:00:17.709
that represents their height
and weight combination.

00:00:17.709 --> 00:00:19.687
So for example let's say I measure someone

00:00:19.687 --> 00:00:22.559
who is 60 inches tall, that'll
be about five feet tall

00:00:22.559 --> 00:00:24.686
and they weigh 100 pounds.

00:00:24.686 --> 00:00:28.680
And so I'd go to 60 inches
and then 100 pounds.

00:00:28.680 --> 00:00:32.162
Right over there so that
point right over there

00:00:32.162 --> 00:00:34.912
is the point 60 comma, 60 comma, 100.

00:00:38.097 --> 00:00:39.831
One way to think about
it, height we could say

00:00:39.831 --> 00:00:43.078
is being measured on our
X axis or plotted along

00:00:43.078 --> 00:00:47.500
our X axis and then
weight along our Y axis.

00:00:47.500 --> 00:00:51.544
And though this point from
this person is the 0.60, 100

00:00:51.544 --> 00:00:53.400
representing 60 inches, 100 pounds.

00:00:53.400 --> 00:00:56.596
And so so I've done it
for one, two, three, four

00:00:56.596 --> 00:00:59.977
five, six, seven, eight, nine
people and I could keep going

00:00:59.977 --> 00:01:01.890
but even with this I could say, well look,

00:01:01.890 --> 00:01:05.435
it looks like there's a roughly
linear relationship here.

00:01:05.435 --> 00:01:08.200
It looks like it's positive,
that generally speaking

00:01:08.200 --> 00:01:10.621
as height increases so does weight.

00:01:10.621 --> 00:01:14.461
Maybe I could try to put a line that can

00:01:14.461 --> 00:01:16.753
approximate this trend.

00:01:16.753 --> 00:01:19.491
Let me try to do that
so this is my line tool.

00:01:19.491 --> 00:01:21.121
I could think about a bunch of lines.

00:01:21.121 --> 00:01:23.379
Something like this
seems like it would be,

00:01:23.379 --> 00:01:26.974
you'd be, most of the
data is below the line

00:01:26.974 --> 00:01:29.250
so that seems like it's not right.

00:01:29.250 --> 00:01:31.417
I could do something like,

00:01:32.314 --> 00:01:33.710
I could do something like this

00:01:33.710 --> 00:01:35.263
but that doesn't seem like a good fit.

00:01:35.263 --> 00:01:38.339
Most the data seems to be above the line.

00:01:38.339 --> 00:01:40.873
And so once again I'm
just eyeballing it here,

00:01:40.873 --> 00:01:43.844
in the future you will learn
better methods of finding

00:01:43.844 --> 00:01:44.927
a better fit.

00:01:46.833 --> 00:01:49.493
But that's something like this
and I'm just eyeballing it

00:01:49.493 --> 00:01:51.500
looks about right.

00:01:51.500 --> 00:01:56.347
So that line, you could view
this as a regression line.

00:01:56.347 --> 00:01:59.870
We could view this as y equals mx plus b.

00:01:59.870 --> 00:02:01.696
Where we would have to
figure out the slope

00:02:01.696 --> 00:02:03.324
and the Y intercept and
we could figure it out

00:02:03.324 --> 00:02:06.343
based on what I just drew or
we could even think of this

00:02:06.343 --> 00:02:07.176
as weight.

00:02:08.569 --> 00:02:12.069
Weight is equal to our slope times height.

00:02:13.746 --> 00:02:17.472
Times height plus whatever
our Y intercept is,

00:02:17.472 --> 00:02:20.651
if you think of the vertical
axis as the weight axis

00:02:20.651 --> 00:02:22.949
you could think of it as
your weight intercept.

00:02:22.949 --> 00:02:24.958
But either way this is
the model that I'm just

00:02:24.958 --> 00:02:28.402
through eyeballing, this
is my regression line.

00:02:28.402 --> 00:02:30.291
Something that I'm trying
to fit to these points.

00:02:30.291 --> 00:02:32.067
But clearly it can't go through,

00:02:32.067 --> 00:02:34.698
one line won't be able to
go to all of these points.

00:02:34.698 --> 00:02:38.914
There's going to be for
each point some difference

00:02:38.914 --> 00:02:40.872
or not for all of them
but for many of them,

00:02:40.872 --> 00:02:44.309
some difference between the
actual and what would have been

00:02:44.309 --> 00:02:45.466
predicted by the line.

00:02:45.466 --> 00:02:49.278
And that idea, the difference
between the actual four point

00:02:49.278 --> 00:02:51.909
and what would have been
predicted, given say the height

00:02:51.909 --> 00:02:54.016
that is called a residual.

00:02:54.016 --> 00:02:56.204
Gonna write that down.

00:02:56.204 --> 00:03:00.136
The A residual for each
of these data points.

00:03:00.136 --> 00:03:02.974
And so for example if
I call this right here,

00:03:02.974 --> 00:03:06.765
if I call that point one,
the residual for point one.

00:03:06.765 --> 00:03:10.750
Is going to be well, for our variable,

00:03:10.750 --> 00:03:13.263
for our height variable 60 inches.

00:03:13.263 --> 00:03:16.411
The actual here is 100 pounds.

00:03:16.411 --> 00:03:20.578
From that we would subtract
what would be predicted.

00:03:22.051 --> 00:03:26.407
So what would be predicted
is right over here.

00:03:26.407 --> 00:03:30.563
I could just substitute
60 into this equation

00:03:30.563 --> 00:03:33.558
so it would be M times 60 plus b.

00:03:33.558 --> 00:03:35.675
So I could write it as M,

00:03:35.675 --> 00:03:39.056
maybe let me write it
this way, 60 M plus B.

00:03:39.056 --> 00:03:40.970
Once again I would just take the 60 pounds

00:03:40.970 --> 00:03:43.204
and put it into my model here and say,

00:03:43.204 --> 00:03:44.922
well what weight would
that have predicted.

00:03:44.922 --> 00:03:48.236
And I can even, just for the
sake of having a number here.

00:03:48.236 --> 00:03:52.153
I can look, I can, let
me get my line tool out.

00:03:53.639 --> 00:03:57.304
And try to get a straight
line from that point.

00:03:57.304 --> 00:04:01.137
So from this point let
me get a straight line.

00:04:03.050 --> 00:04:04.553
So that doesn't look quite straight,

00:04:04.553 --> 00:04:05.892
okay a little bit, okay.

00:04:05.892 --> 00:04:09.309
Okay so looks like it's about 150 pounds.

00:04:12.107 --> 00:04:15.893
So my model would have
predicted 150 pounds.

00:04:15.893 --> 00:04:20.060
So the residual here is going
to be equal to negative 50.

00:04:21.124 --> 00:04:23.572
So a negative residual is when your actual

00:04:23.572 --> 00:04:26.107
is below your predicted.

00:04:26.107 --> 00:04:27.696
So this right over here.

00:04:27.696 --> 00:04:30.797
This is our one, it is
a negative residual.

00:04:30.797 --> 00:04:33.676
If you had, if you tried to find,

00:04:33.676 --> 00:04:36.755
let's say this residual right
over here, for this point.

00:04:36.755 --> 00:04:40.338
This r2, this would be a positive residual

00:04:41.631 --> 00:04:44.791
because the actual is larger

00:04:44.791 --> 00:04:49.046
than what would have
actually been predicted.

00:04:49.046 --> 00:04:51.441
And so a residual is good for saying,

00:04:51.441 --> 00:04:54.598
well how good does your
line, does your regression,

00:04:54.598 --> 00:04:57.723
does your model fit a given data point

00:04:57.723 --> 00:04:59.946
or how does a given data
point compare to that.

00:04:59.946 --> 00:05:03.249
What you probably want
to do is think about

00:05:03.249 --> 00:05:05.476
some combination of all the residuals

00:05:05.476 --> 00:05:07.872
and try to minimize it.

00:05:07.872 --> 00:05:10.209
Now you might say, well
why don't I just add up

00:05:10.209 --> 00:05:11.866
all the residuals and
try to minimize that.

00:05:11.866 --> 00:05:14.143
But that gets tricky
because some are positive

00:05:14.143 --> 00:05:18.997
and some are negative and
so a big negative residual

00:05:18.997 --> 00:05:21.285
could counterbalance the
big positive residual

00:05:21.285 --> 00:05:23.192
and it would look, it would add up to zero

00:05:23.192 --> 00:05:24.974
and then it look like there's no residual

00:05:24.974 --> 00:05:27.750
so you could just add
up the absolute values.

00:05:27.750 --> 00:05:29.758
So you could say, well
let me just take the sum

00:05:29.758 --> 00:05:33.560
of all of the residual,
of the absolute value

00:05:33.560 --> 00:05:34.738
of all the residuals.

00:05:34.738 --> 00:05:38.238
And then let me change M and B for my line

00:05:39.132 --> 00:05:42.641
to minimize this and
that would be a technique

00:05:42.641 --> 00:05:46.284
of trying to create a regression line.

00:05:46.284 --> 00:05:47.939
But another way to do
it and this is actually

00:05:47.939 --> 00:05:50.009
the most typical way that
you will see in statistics

00:05:50.009 --> 00:05:53.509
is that people take the sum of the squares

00:05:55.081 --> 00:05:56.703
of the residuals.

00:05:56.703 --> 00:05:59.119
The sum of the squares and
when you square something

00:05:59.119 --> 00:06:00.612
whether it's negative or positive,

00:06:00.612 --> 00:06:03.229
it's going to be a positive
so it takes care of that issue

00:06:03.229 --> 00:06:06.164
of negatives and positives
canceling out with each other.

00:06:06.164 --> 00:06:09.707
And when you square a number,
things with large residuals

00:06:09.707 --> 00:06:12.543
are gonna become even
larger, relatively speaking.

00:06:12.543 --> 00:06:15.528
You know, if you square
a large, you know one is,

00:06:15.528 --> 00:06:18.138
if you think about this way,
let me put regular numbers,

00:06:18.138 --> 00:06:20.103
one, two, three, four.

00:06:20.103 --> 00:06:21.586
These are all one apart from each other

00:06:21.586 --> 00:06:25.076
but if I were to square
them, one, four, nine, 16,

00:06:25.076 --> 00:06:27.170
they get further and further apart

00:06:27.170 --> 00:06:29.732
and so something, the
larger the residual is

00:06:29.732 --> 00:06:32.568
when you square it,
when the sum of squares

00:06:32.568 --> 00:06:36.438
is going to represent a
bigger proportion of the sum.

00:06:36.438 --> 00:06:38.081
And so what we'll see in future videos

00:06:38.081 --> 00:06:39.724
is that there is a technique called

00:06:39.724 --> 00:06:41.807
least squares regression.

00:06:46.659 --> 00:06:49.022
Least squares regression.

00:06:49.022 --> 00:06:53.189
Where you can find an M and
a B for a given set of data

00:06:54.167 --> 00:06:58.334
so it minimizes the sum of
the squares of the residual.

00:06:59.699 --> 00:07:02.046
And that's valuable and the
reason why this is used most

00:07:02.046 --> 00:07:04.564
is it really tries to take in account

00:07:04.564 --> 00:07:06.249
things that are significant outliers.

00:07:06.249 --> 00:07:09.135
Things that sit from pretty
far away from the model,

00:07:09.135 --> 00:07:11.038
something like this is going to really,

00:07:11.038 --> 00:07:13.478
with a least squares regression.

00:07:13.478 --> 00:07:15.389
It's going to try to be minimized

00:07:15.389 --> 00:07:17.580
or it's going to be weighted
a little bit heavier

00:07:17.580 --> 00:07:18.963
because when you square it becomes

00:07:18.963 --> 00:07:20.844
even a bigger factor in this.

00:07:20.844 --> 00:07:23.029
But this is just a
conceptual introduction.

00:07:23.029 --> 00:07:26.416
In future videos we'll do
things like calculate residuals.

00:07:26.416 --> 00:07:27.848
And we'll actually derive the formula

00:07:27.848 --> 00:07:31.877
for how do you figure out
an M and a B for a line

00:07:31.877 --> 00:07:35.039
that actually minimizes
the sum of the squares

00:07:35.039 --> 00:07:36.456
of the residuals.


Calculating residual example

WEBVTT
Kind: captions
Language: en

00:00:00.008 --> 00:00:02.616
- [Instructor] Vera rents
bicycles to tourists.

00:00:02.616 --> 00:00:05.919
She recorded the height, in
centimeters, of each customer

00:00:05.919 --> 00:00:08.074
and the frame size, in centimeters,

00:00:08.074 --> 00:00:11.103
of the bicycle that customer rented.

00:00:11.103 --> 00:00:13.147
After plotting her results,

00:00:13.147 --> 00:00:15.494
Vera noticed that the relationship

00:00:15.494 --> 00:00:18.898
between the two variables
was fairly linear,

00:00:18.898 --> 00:00:20.906
so she used the data to calculate

00:00:20.906 --> 00:00:24.746
the following least
squares regression equation

00:00:24.746 --> 00:00:27.187
for predicting bicycle frame size

00:00:27.187 --> 00:00:28.344
from the height of the customer.

00:00:28.344 --> 00:00:29.641
And this is the equation.

00:00:29.641 --> 00:00:31.435
So before I even look at this question,

00:00:31.435 --> 00:00:33.417
let's just think about what she did.

00:00:33.417 --> 00:00:35.468
So she had a bunch of customers,

00:00:35.468 --> 00:00:38.598
and she recorded, given
the height of the customer,

00:00:38.598 --> 00:00:41.912
what size frame that person rented.

00:00:41.912 --> 00:00:45.421
And so she might've had
something like this,

00:00:45.421 --> 00:00:49.041
where in the horizontal
axis you have height

00:00:49.041 --> 00:00:51.171
measured in centimeters,

00:00:51.171 --> 00:00:53.423
and in the vertical
axis you have frame size

00:00:53.423 --> 00:00:55.496
that's also measured in centimeters.

00:00:55.496 --> 00:00:57.972
And so there might've been someone who

00:00:57.972 --> 00:00:59.966
measures 100 centimeters in height

00:00:59.966 --> 00:01:02.268
who gets a 25 centimeter frame.

00:01:02.268 --> 00:01:03.731
I don't know if that's reasonable or not,

00:01:03.731 --> 00:01:06.443
for you bicycle experts,
but let's just go with it.

00:01:06.443 --> 00:01:08.096
And so she would've plotted it there.

00:01:08.096 --> 00:01:11.135
Maybe there was another person
of 100 centimeters in height

00:01:11.135 --> 00:01:14.169
who got a frame that was slightly larger,

00:01:14.169 --> 00:01:15.890
and she plotted it there.

00:01:15.890 --> 00:01:19.538
And then, she did a
least squares regression.

00:01:19.538 --> 00:01:21.392
And a least squares regression

00:01:21.392 --> 00:01:24.237
is trying to fit a line to this data.

00:01:24.237 --> 00:01:27.416
Oftentimes, you would use a
spreadsheet or use a computer.

00:01:27.416 --> 00:01:29.976
And that line is trying
to minimize the square

00:01:29.976 --> 00:01:33.059
of the distance between these points.

00:01:34.245 --> 00:01:37.106
And so the least squares regression,

00:01:37.106 --> 00:01:39.267
maybe it would look something like this,

00:01:39.267 --> 00:01:41.698
and this is just a rough estimate of it.

00:01:41.698 --> 00:01:44.196
It might look something,

00:01:44.196 --> 00:01:46.062
let me get my ruler tool,

00:01:46.062 --> 00:01:49.208
it might look something like,

00:01:49.208 --> 00:01:51.763
it might look something

00:01:51.763 --> 00:01:52.691
like this.

00:01:52.691 --> 00:01:54.390
So let me plot it.

00:01:54.390 --> 00:01:55.223
So this,

00:01:56.240 --> 00:01:58.095
that would be the line.

00:01:58.095 --> 00:02:00.163
So our regression line, y-hat,

00:02:00.163 --> 00:02:04.750
is equal to 1/3 plus 1/3 x.

00:02:04.750 --> 00:02:07.467
And so this, you could view
this as a way of predicting,

00:02:07.467 --> 00:02:09.662
or either modeling the relationship

00:02:09.662 --> 00:02:12.377
or predicting that, hey,
if I get a new person,

00:02:12.377 --> 00:02:14.241
I could take their height and put as x

00:02:14.241 --> 00:02:17.839
and figure out what frame
size they're likely to rent.

00:02:17.839 --> 00:02:22.466
But they ask us, what is
the residual of a customer

00:02:22.466 --> 00:02:25.696
with a height of 155 centimeters

00:02:25.696 --> 00:02:27.282
who rents a bike

00:02:27.282 --> 00:02:30.249
with a 51 centimeter frame?

00:02:30.249 --> 00:02:32.194
So how do we think about this?

00:02:32.194 --> 00:02:34.741
Well, the residual is
going to be the difference

00:02:34.741 --> 00:02:38.254
between what they actually
produce and what the line,

00:02:38.254 --> 00:02:41.792
what our regression line
would have predicted.

00:02:41.792 --> 00:02:45.357
So we could say residual,
let me write it this way,

00:02:45.357 --> 00:02:46.190
residual

00:02:47.785 --> 00:02:49.618
is going to be actual,

00:02:50.857 --> 00:02:52.685
actual minus

00:02:52.685 --> 00:02:53.874
predicted.

00:02:53.874 --> 00:02:55.997
So if predicted is larger than actual,

00:02:55.997 --> 00:02:57.735
this is actually going
to be a negative number.

00:02:57.735 --> 00:03:00.099
If predicted as smaller than actual,

00:03:00.099 --> 00:03:01.880
this is gonna be a positive number.

00:03:01.880 --> 00:03:03.747
Well, we know the actual.

00:03:03.747 --> 00:03:05.016
They tell us that.

00:03:05.016 --> 00:03:07.355
They tell us that they rent,

00:03:07.355 --> 00:03:08.222
it's a,

00:03:08.222 --> 00:03:10.962
the 155 centimeter person rents a bike

00:03:10.962 --> 00:03:12.548
with a 51 centimeter frame,

00:03:12.548 --> 00:03:14.336
so this is 51 centimeters.

00:03:14.336 --> 00:03:16.057
But what is the predicted?

00:03:16.057 --> 00:03:18.581
Well, that's where we can
use our regression equation

00:03:18.581 --> 00:03:20.245
that Vera came up with.

00:03:20.245 --> 00:03:22.828
The predicted, I'll do that in orange,

00:03:22.828 --> 00:03:27.404
the predicted is going to be equal to 1/3

00:03:27.404 --> 00:03:30.511
plus 1/3 times the person's height.

00:03:30.511 --> 00:03:32.758
Their height is 155.

00:03:32.758 --> 00:03:33.791
That's the predicted.

00:03:33.791 --> 00:03:36.412
Y-hat is what our linear
regression predicts

00:03:36.412 --> 00:03:38.377
or our line predicts.

00:03:38.377 --> 00:03:39.410
So what is this going to be?

00:03:39.410 --> 00:03:44.282
This is going to be equal
to 1/3 plus 155 over three,

00:03:44.282 --> 00:03:47.326
which is equal to 156 over three,

00:03:47.326 --> 00:03:49.743
which comes out nicely to 52.

00:03:51.181 --> 00:03:54.366
So the predicted on our line is 52.

00:03:54.366 --> 00:03:55.366
And so here,

00:03:56.365 --> 00:03:57.763
so this person is 155,

00:03:57.763 --> 00:04:00.931
we can plot 'em right over here, 155.

00:04:00.931 --> 00:04:04.372
They're coming in slightly below the line.

00:04:04.372 --> 00:04:06.748
So they're coming in slightly
below the line right there,

00:04:06.748 --> 00:04:09.022
and that distance, which is,

00:04:09.022 --> 00:04:11.038
and we can see that
they are below the line,

00:04:11.038 --> 00:04:12.774
so the distance is going to be,

00:04:12.774 --> 00:04:15.385
or in this case, the residual
is going to be negative.

00:04:15.385 --> 00:04:17.267
So this is going to be negative one.

00:04:17.267 --> 00:04:20.240
And so if we were to
zoom in right over here,

00:04:20.240 --> 00:04:21.894
you can't see it that well,

00:04:21.894 --> 00:04:23.228
but let me draw it.

00:04:23.228 --> 00:04:26.431
So if we zoom in, let's say
we were to zoom in the line,

00:04:26.431 --> 00:04:27.961
and it looks like this.

00:04:27.961 --> 00:04:30.358
And our data point is right,

00:04:30.358 --> 00:04:33.369
our data point is right over here.

00:04:33.369 --> 00:04:34.733
We know we're below the line,

00:04:34.733 --> 00:04:36.336
and it's just gonna be
a negative residual.

00:04:36.336 --> 00:04:37.972
And the magnitude of that residual is

00:04:37.972 --> 00:04:40.001
how far we are below the line.

00:04:40.001 --> 00:04:42.750
And in this case, it is negative one.

00:04:42.750 --> 00:04:44.347
And so that is our residual.

00:04:44.347 --> 00:04:47.672
This is what actual, the actual data minus

00:04:47.672 --> 00:04:50.872
what was predicted by our regression line.


How parameters change as data is shifted and scaled

WEBVTT
Kind: captions
Language: en

00:00:00.191 --> 00:00:02.558
- [Instructor] So I have some
data here in the spreadsheet

00:00:02.558 --> 00:00:05.051
you could use Microsoft
Excel or you could use

00:00:05.051 --> 00:00:06.488
Google Spreadsheet

00:00:06.488 --> 00:00:08.694
and we're gonna use the spreadsheet

00:00:08.694 --> 00:00:11.260
to quickly calculate some parameters.

00:00:11.260 --> 00:00:12.767
Let's say this is a population.

00:00:12.767 --> 00:00:15.153
Let's say this is, we're looking
at a population of students

00:00:15.153 --> 00:00:17.133
and we wanna calculate some parameters

00:00:17.133 --> 00:00:18.284
and this is their ages,

00:00:18.284 --> 00:00:20.990
and we wanna calculate
some parameters on that.

00:00:20.990 --> 00:00:24.558
And so first I'm gonna calculate
it using the spreadsheet,

00:00:24.558 --> 00:00:26.735
and then we're gonna think about

00:00:26.735 --> 00:00:29.359
how those parameters change
as we do things to the data.

00:00:29.359 --> 00:00:31.074
If we were to shift the data up or down

00:00:31.074 --> 00:00:33.202
or if we were to multiply
all the points by some value,

00:00:33.202 --> 00:00:36.026
what does that do to
the actual parameters?

00:00:36.026 --> 00:00:38.274
So the first parameter I'm
gonna calculate as the mean.

00:00:38.274 --> 00:00:41.788
Then I'm gonna calculate
the standard deviation.

00:00:41.788 --> 00:00:44.179
Then I'm gonna calculate the median,

00:00:44.179 --> 00:00:46.050
and then I wanna calculate,

00:00:46.050 --> 00:00:49.073
let's say, the inter quartile range.

00:00:49.073 --> 00:00:51.073
Inter, I'll call it IQR.

00:00:53.604 --> 00:00:55.025
So let's do this.

00:00:55.025 --> 00:00:57.512
Let's first look at the
measures of central tendencies.

00:00:57.512 --> 00:00:59.932
So the mean, the function
on most spreadsheets

00:00:59.932 --> 00:01:01.342
is the average function,

00:01:01.342 --> 00:01:03.624
and then I cold use my mouse
and select all of these,

00:01:03.624 --> 00:01:05.862
or I could press Shift
with my arrow button

00:01:05.862 --> 00:01:07.249
and select all those.

00:01:07.249 --> 00:01:10.054
Okay, that's the mean of that data.

00:01:10.054 --> 00:01:12.008
Now let's think about what happens

00:01:12.008 --> 00:01:13.967
if I take all of that data

00:01:13.967 --> 00:01:16.061
and if I were to add a fixed amount to it.

00:01:16.061 --> 00:01:17.374
So if I took all the data

00:01:17.374 --> 00:01:18.914
and if I were to add five to it.

00:01:18.914 --> 00:01:20.839
So an easy way to do that in a spreadsheet

00:01:20.839 --> 00:01:23.242
is you select that, you add five,

00:01:23.242 --> 00:01:25.428
and then I can scroll down.

00:01:25.428 --> 00:01:27.511
And notice for every
data point I have before,

00:01:27.511 --> 00:01:29.408
I now have five more than that.

00:01:29.408 --> 00:01:30.885
So this is my new dataset,

00:01:30.885 --> 00:01:32.551
or as I'm calling Data+5.

00:01:32.551 --> 00:01:35.299
Let's see what the mean of that is.

00:01:35.299 --> 00:01:37.181
So the mean of that, notice,

00:01:37.181 --> 00:01:39.405
is exactly five more,

00:01:39.405 --> 00:01:41.004
and the same would have been true

00:01:41.004 --> 00:01:43.039
if I added or subtracted any number.

00:01:43.039 --> 00:01:47.659
The mean would change by the
amount that I add or subtract.

00:01:47.659 --> 00:01:49.570
That shouldn't surprise you,

00:01:49.570 --> 00:01:52.052
because when you're calculating the mean,

00:01:52.052 --> 00:01:53.708
you're adding all the numbers up

00:01:53.708 --> 00:01:56.081
and you're dividing by
the numbers you have.

00:01:56.081 --> 00:01:58.235
If all the numbers are five more,

00:01:58.235 --> 00:01:59.890
you're gonna add five.

00:01:59.890 --> 00:02:01.487
In this case, how many numbers are there?

00:02:01.487 --> 00:02:04.249
One, two, three, four,
five, fix, seven, eight,

00:02:04.249 --> 00:02:06.352
nine, 10, 11, 12.

00:02:06.352 --> 00:02:09.001
You're gonna add 12 more fives
and then you're gonna divide

00:02:09.001 --> 00:02:13.168
by 12, and so it makes sense
that your mean goes up by five

00:02:14.063 --> 00:02:17.670
let's think about how the
mean changes if you multiply.

00:02:17.670 --> 00:02:19.585
So if you take your data

00:02:19.585 --> 00:02:23.464
and if I were to multiply
it times five, what happens?

00:02:23.464 --> 00:02:26.047
So this equals this times five.

00:02:28.352 --> 00:02:32.566
So now all the data points
are five times more.

00:02:32.566 --> 00:02:35.210
Now what happens to my mean?

00:02:35.210 --> 00:02:38.627
Notice my mean is now five times as much.

00:02:39.607 --> 00:02:41.784
So the measures of central tendency,

00:02:41.784 --> 00:02:43.550
if I add or subtract,

00:02:43.550 --> 00:02:46.832
well I'm gonna add or subtract
the mean by that amount,

00:02:46.832 --> 00:02:48.472
and if I scale it up by five

00:02:48.472 --> 00:02:50.097
or if I scaled it down by five,

00:02:50.097 --> 00:02:53.131
well my mean would scale up
or down by that same amount,

00:02:53.131 --> 00:02:56.170
and if you numerically looked
at how you calculate a mean,

00:02:56.170 --> 00:02:59.641
it would make sense that this
is happening mathematically.

00:02:59.641 --> 00:03:02.824
Let's look at the other typical measure

00:03:02.824 --> 00:03:05.285
of central tendency,
and that is the median.

00:03:05.285 --> 00:03:07.728
To see if that has the same properties.

00:03:07.728 --> 00:03:09.982
So let's calculate the median here.

00:03:09.982 --> 00:03:12.117
So once again you order these numbers

00:03:12.117 --> 00:03:13.968
and just find the middle number.

00:03:13.968 --> 00:03:15.173
Which isn't too hard,

00:03:15.173 --> 00:03:16.806
but a computer can do it awfully fast.

00:03:16.806 --> 00:03:18.474
So that's the median for that dataset.

00:03:18.474 --> 00:03:20.213
What do you think the medians gonna be

00:03:20.213 --> 00:03:22.069
if you take all of the data plus five?

00:03:22.069 --> 00:03:23.424
Well the middle number,

00:03:23.424 --> 00:03:25.187
if you ordered all of these numbers

00:03:25.187 --> 00:03:26.650
and made them all five more,

00:03:26.650 --> 00:03:28.248
the orders, you could think of it

00:03:28.248 --> 00:03:29.631
as being the same order,

00:03:29.631 --> 00:03:31.641
but now the one in the
middle is gonna be five more.

00:03:31.641 --> 00:03:33.933
So this should be 10.5,

00:03:33.933 --> 00:03:35.948
and yes, it is indeed 10.5,

00:03:35.948 --> 00:03:38.612
and what would happen if you
multiply everything by five?

00:03:38.612 --> 00:03:42.103
Well once again, you still
have the same ordering.

00:03:42.103 --> 00:03:45.962
It should just multiply that by five.

00:03:45.962 --> 00:03:48.626
Yup, the middle number's now
gonna be five time larger.

00:03:48.626 --> 00:03:51.199
So both of these measures
of central tendency,

00:03:51.199 --> 00:03:54.138
if you shift all the data points,

00:03:54.138 --> 00:03:55.542
or if you scale them up,

00:03:55.542 --> 00:03:58.577
you're going to similarly
shift or scale up

00:03:58.577 --> 00:04:00.809
these measures of central tendency.

00:04:00.809 --> 00:04:03.350
Now let's think about
these measures of spread.

00:04:03.350 --> 00:04:06.159
See if that's the same with
these measures of spread.

00:04:06.159 --> 00:04:08.099
So standard deviation.

00:04:08.099 --> 00:04:08.932
So STDEV.

00:04:10.669 --> 00:04:12.892
I'm gonna take the population
standard deviation.

00:04:12.892 --> 00:04:15.528
I'm assuming that this
is my entire population.

00:04:15.528 --> 00:04:17.278
So let me analyze it.

00:04:18.466 --> 00:04:20.517
So let me make sure I'm doing,

00:04:20.517 --> 00:04:22.517
so standard deviation of

00:04:25.440 --> 00:04:26.357
all of this

00:04:27.390 --> 00:04:29.057
is going to be 2.99.

00:04:30.065 --> 00:04:33.118
Let's see what happens when
I shift everything by five.

00:04:33.118 --> 00:04:34.551
Actually, pause the video.

00:04:34.551 --> 00:04:36.040
What do you think is going to happen?

00:04:36.040 --> 00:04:37.308
This is a measure of spread.

00:04:37.308 --> 00:04:38.905
So if you shift, I'll
tell you what I think.

00:04:38.905 --> 00:04:41.390
If I shift everything by the same amount,

00:04:41.390 --> 00:04:45.529
the mean shifts but the distance
of everything from the mean

00:04:45.529 --> 00:04:46.956
should not change.

00:04:46.956 --> 00:04:49.814
So the standard deviation
should not change,

00:04:49.814 --> 00:04:51.596
I don't think, in this example,

00:04:51.596 --> 00:04:53.357
and indeed, it does not change.

00:04:53.357 --> 00:04:54.864
So if we shift the datasets.

00:04:54.864 --> 00:04:56.516
In this case we shifted it up by five,

00:04:56.516 --> 00:04:58.042
or if we shifted it down by one.

00:04:58.042 --> 00:05:00.318
Your measure of spread, in
this case standard deviation

00:05:00.318 --> 00:05:01.473
should not change,

00:05:01.473 --> 00:05:03.836
or at least the standard
deviation measure of spread

00:05:03.836 --> 00:05:04.851
does not change,

00:05:04.851 --> 00:05:07.580
but if we scale it, well
I think it should change,

00:05:07.580 --> 00:05:09.805
because you could imagine
a very simple dataset

00:05:09.805 --> 00:05:11.791
that things that were a
certain amount of distance

00:05:11.791 --> 00:05:14.477
from the mean are now going
to be five times further

00:05:14.477 --> 00:05:15.476
from the mean.

00:05:15.476 --> 00:05:17.222
So I think this actually should,

00:05:17.222 --> 00:05:18.939
we should multiply by five here,

00:05:18.939 --> 00:05:20.571
and it does look like that is the case.

00:05:20.571 --> 00:05:22.172
If I multiplied this by five.

00:05:22.172 --> 00:05:26.104
So scaling the dataset will
scale the standard deviation

00:05:26.104 --> 00:05:28.158
is a similar way.

00:05:28.158 --> 00:05:29.912
What about inter quartile range?

00:05:29.912 --> 00:05:32.446
Where essentially we're
taking the third quartile

00:05:32.446 --> 00:05:34.958
and subtracting from
that the first quartile

00:05:34.958 --> 00:05:37.861
to figure out kind of the
range of the middle 50%.

00:05:37.861 --> 00:05:39.200
Let's do that.

00:05:39.200 --> 00:05:43.298
We can have the quartile
function equals quartile

00:05:43.298 --> 00:05:46.632
and then we want to look at our data,

00:05:46.632 --> 00:05:49.215
and we want the third quartile.

00:05:50.388 --> 00:05:52.842
So that's gonna calculate
the third quartile.

00:05:52.842 --> 00:05:55.342
Minus quartile, same data set.

00:05:57.878 --> 00:06:00.187
So now we wanna select it again.

00:06:00.187 --> 00:06:03.011
So same dataset, but
this is now going to be

00:06:03.011 --> 00:06:04.106
the first quartile.

00:06:04.106 --> 00:06:05.949
So this is gonna give us
our inter quartile range.

00:06:05.949 --> 00:06:08.659
This calculates the third
quartile in that dataset

00:06:08.659 --> 00:06:11.652
and this calculates the first
quartile in that dataset.

00:06:11.652 --> 00:06:12.985
And we get 2.75.

00:06:14.109 --> 00:06:17.378
Now let's think about whether
the inter quartile range

00:06:17.378 --> 00:06:18.466
should change.

00:06:18.466 --> 00:06:19.662
And I don't think it will.

00:06:19.662 --> 00:06:21.248
Because remember, everything shifts,

00:06:21.248 --> 00:06:24.084
and even though the first
quartile is gonna be five more,

00:06:24.084 --> 00:06:26.170
but the third quartile is
gonna be five more as well.

00:06:26.170 --> 00:06:28.020
So the difference shouldn't change.

00:06:28.020 --> 00:06:30.309
And indeed look, the
distance does not change,

00:06:30.309 --> 00:06:31.952
or the difference does not change.

00:06:31.952 --> 00:06:34.753
But similarly, if we scale everything up,

00:06:34.753 --> 00:06:37.120
if we were to scale up the first quartile

00:06:37.120 --> 00:06:38.750
and the third quartile by five,

00:06:38.750 --> 00:06:41.422
well then their difference
should scale up by five,

00:06:41.422 --> 00:06:43.387
and we see that right over there.

00:06:43.387 --> 00:06:45.005
So the big takeaway here.

00:06:45.005 --> 00:06:46.939
I just use the example
of shifting up by five

00:06:46.939 --> 00:06:48.420
and scaling up by five,

00:06:48.420 --> 00:06:50.820
but you could subtract by any number,

00:06:50.820 --> 00:06:53.342
and you could divide by a number as well.

00:06:53.342 --> 00:06:56.291
The typical measures of central tendency

00:06:56.291 --> 00:06:59.939
mean and median, they both shift and scale

00:06:59.939 --> 00:07:02.162
as you shift and scale the data,

00:07:02.162 --> 00:07:04.605
but your typical measures of spread,

00:07:04.605 --> 00:07:08.236
standard deviation and
inter quartile range,

00:07:08.236 --> 00:07:11.206
they don't change if you shift the data,

00:07:11.206 --> 00:07:15.373
but they do change and they
scale as you scale the data.


Comparing with z-scores

WEBVTT
Kind: captions
Language: en

00:00:00.153 --> 00:00:02.243
- [Instructor] Before applying
to law school in the US,

00:00:02.243 --> 00:00:05.254
students need to take
an exam called the LSAT.

00:00:05.254 --> 00:00:07.114
Before applying to medical school,

00:00:07.114 --> 00:00:09.923
students need to take
an exam called the MCAT.

00:00:09.923 --> 00:00:12.597
Here are some summary
statistics for each exam.

00:00:12.597 --> 00:00:15.571
So the LSAT, the mean score is 151

00:00:15.571 --> 00:00:17.673
with a standard deviation of 10.

00:00:17.673 --> 00:00:20.294
And the MCAT, the mean score is 25.1

00:00:20.294 --> 00:00:23.183
with a standard deviation of 6.4

00:00:23.183 --> 00:00:25.439
Juwan took both exams.

00:00:25.439 --> 00:00:29.944
He scored 172 on the
LSAT and 37 on the MCAT.

00:00:29.944 --> 00:00:33.119
Which exam did he do relatively better on?

00:00:33.119 --> 00:00:36.539
So pause this video, and see
if you can figure it out.

00:00:36.539 --> 00:00:38.501
So the way I would think about it is

00:00:38.501 --> 00:00:40.634
you can't just look at the absolute score

00:00:40.634 --> 00:00:42.395
because they are on different scales

00:00:42.395 --> 00:00:44.571
and they have different distributions.

00:00:44.571 --> 00:00:46.037
But we can use this information.

00:00:46.037 --> 00:00:48.002
If we assume it's a normal distribution

00:00:48.002 --> 00:00:50.720
or relatively close to
a normal distribution

00:00:50.720 --> 00:00:52.912
with a meet, centered at this mean,

00:00:52.912 --> 00:00:55.716
we can think about, well,
how many standard deviations

00:00:55.716 --> 00:00:59.004
from the mean did he score
in each of these situations?

00:00:59.004 --> 00:01:01.203
In both cases, he scored above the mean.

00:01:01.203 --> 00:01:03.483
But how many standard
deviations above the mean?

00:01:03.483 --> 00:01:05.289
So let's see if we can figure that out.

00:01:05.289 --> 00:01:07.937
So on the LSAT, let's see,
let me write this down,

00:01:07.937 --> 00:01:09.515
on the LSAT,

00:01:09.515 --> 00:01:10.682
he scored 172.

00:01:11.739 --> 00:01:14.306
So how many standard
deviations is that going to be?

00:01:14.306 --> 00:01:17.944
Well, let's take 172, his
score, minus the mean,

00:01:17.944 --> 00:01:19.399
so this is the absolute number

00:01:19.399 --> 00:01:21.275
that he scored above the mean,

00:01:21.275 --> 00:01:24.555
and now let's divide that
by the standard deviation.

00:01:24.555 --> 00:01:26.139
So on the LSAT, this is what?

00:01:26.139 --> 00:01:27.972
This is going to be 21

00:01:29.132 --> 00:01:30.563
divided by 10.

00:01:30.563 --> 00:01:34.281
So this is 2.1 standard deviations,

00:01:34.281 --> 00:01:35.872
deviations

00:01:35.872 --> 00:01:37.656
above the mean,

00:01:37.656 --> 00:01:38.986
above

00:01:38.986 --> 00:01:39.819
the mean.

00:01:39.819 --> 00:01:41.199
You could view this as a z-score.

00:01:41.199 --> 00:01:43.033
It's a z-score of 2.1.

00:01:43.033 --> 00:01:47.513
We are 2.1 above the
mean in this situation.

00:01:47.513 --> 00:01:50.499
Now, let's think about
how he did on the MCAT.

00:01:50.499 --> 00:01:51.499
On the MCAT,

00:01:52.553 --> 00:01:53.803
he scored a 37.

00:01:56.121 --> 00:01:57.704
The mean is a 25.1,

00:01:59.600 --> 00:02:03.017
and there is a standard deviation of 6.4.

00:02:05.033 --> 00:02:07.361
So let's see, 37.1

00:02:07.361 --> 00:02:08.778
minus 25 would be

00:02:09.924 --> 00:02:12.507
12, but now it's gonna be 11.9,

00:02:13.369 --> 00:02:15.001
11.9

00:02:15.001 --> 00:02:16.251
divided by 6.4.

00:02:18.354 --> 00:02:21.421
So without even looking at this,

00:02:21.421 --> 00:02:24.766
so this is going to be approximately,

00:02:24.766 --> 00:02:27.321
well, this is gonna be a
little bit less than two.

00:02:27.321 --> 00:02:30.137
This is going to be less than two.

00:02:30.137 --> 00:02:31.600
So based on this information,

00:02:31.600 --> 00:02:33.971
and we could figure out
the exact number here.

00:02:33.971 --> 00:02:36.356
In fact, let me get my calculator out.

00:02:36.356 --> 00:02:38.907
So you get the calculator.

00:02:38.907 --> 00:02:40.240
So if we do 11.9

00:02:42.943 --> 00:02:44.193
divided by 6.4,

00:02:45.657 --> 00:02:47.871
that's gonna get us to one
point, I'll just say one point,

00:02:47.871 --> 00:02:50.619
I'll just say approximately 1.86,

00:02:50.619 --> 00:02:53.137
so approximately 1.86.

00:02:53.137 --> 00:02:57.978
So relatively speaking, he did
slightly better on the LSAT.

00:02:57.978 --> 00:03:00.159
He did more standard deviations,
although this is close.

00:03:00.159 --> 00:03:01.927
I would say they're comparable.

00:03:01.927 --> 00:03:04.170
He did roughly two standard deviations

00:03:04.170 --> 00:03:06.730
if we were to round to the
nearest standard deviation.

00:03:06.730 --> 00:03:08.412
But if you wanted to get precise,

00:03:08.412 --> 00:03:09.656
he did a little bit better,

00:03:09.656 --> 00:03:11.526
relatively speaking, on the LSAT.

00:03:11.526 --> 00:03:13.665
He did 2.1 standard deviations here

00:03:13.665 --> 00:03:17.722
while over here he did 1.86
or 1.9 standard deviations.

00:03:17.722 --> 00:03:18.970
But in everyday language,

00:03:18.970 --> 00:03:21.245
you would probably say,
well, this is comparable.

00:03:21.245 --> 00:03:23.115
If this was three standard deviations

00:03:23.115 --> 00:03:25.259
and this is one standard
deviation, then you'd be like,

00:03:25.259 --> 00:03:28.676
oh, he definitely did better on the LSAT.


Calculating percentile

WEBVTT
Kind: captions
Language: en

00:00:00.557 --> 00:00:01.759
- [Instructor] The dot plot shows the

00:00:01.759 --> 00:00:04.088
number of hours of daily driving time

00:00:04.088 --> 00:00:06.255
for 14 school bus drivers.

00:00:07.326 --> 00:00:09.903
Each dot represents a driver.

00:00:09.903 --> 00:00:13.933
So for example, one driver
drives one hour a day.

00:00:13.933 --> 00:00:16.109
Two drivers drive two hours a day.

00:00:16.109 --> 00:00:17.537
One driver drives three hours a day.

00:00:17.537 --> 00:00:20.259
It looks like there's five drivers

00:00:20.259 --> 00:00:23.479
that drive seven hours a day.

00:00:23.479 --> 00:00:26.274
Which of the following is the closest

00:00:26.274 --> 00:00:28.564
estimate to the percentile rank

00:00:28.564 --> 00:00:32.731
for the driver with a daily
driving time of six hours?

00:00:33.669 --> 00:00:35.330
And then they give us some choices.

00:00:35.330 --> 00:00:37.699
Which of the following
is the closest estimate

00:00:37.699 --> 00:00:39.959
to the percentile rank for the driver

00:00:39.959 --> 00:00:42.880
with the daily driving time of six hours?

00:00:42.880 --> 00:00:45.356
So pause the video and
see if you can figure out

00:00:45.356 --> 00:00:48.197
which of these percentiles
is the closest estimate

00:00:48.197 --> 00:00:50.723
to the percentile rank of a driver

00:00:50.723 --> 00:00:53.889
with a daily driving time of six hours,

00:00:53.889 --> 00:00:56.042
looking at this data right over here.

00:00:56.042 --> 00:00:58.139
Alright, now let's work
through this together.

00:00:58.139 --> 00:00:59.524
So when you think about percentile

00:00:59.524 --> 00:01:00.936
you really want to think about,

00:01:00.936 --> 00:01:02.198
so let me write this down.

00:01:02.198 --> 00:01:04.362
When we're talking about percentile

00:01:04.362 --> 00:01:06.313
we're really saying the percentage

00:01:06.313 --> 00:01:07.230
of the data

00:01:10.608 --> 00:01:12.248
that,

00:01:12.248 --> 00:01:13.939
and there's actually two ways
that you could compute it.

00:01:13.939 --> 00:01:15.874
One is the percentage of the data

00:01:15.874 --> 00:01:16.957
that is below

00:01:19.344 --> 00:01:21.261
the amount in question,

00:01:22.141 --> 00:01:23.724
amount in question.

00:01:26.343 --> 00:01:29.343
The other possibility is the percent

00:01:30.630 --> 00:01:31.547
of the data

00:01:33.491 --> 00:01:35.503
that is at or below,

00:01:35.503 --> 00:01:38.086
that is at or below the amount,

00:01:41.764 --> 00:01:43.681
the amount in question.

00:01:44.945 --> 00:01:46.882
So if we look at this right over here,

00:01:46.882 --> 00:01:48.899
let's just figure out
how many data points,

00:01:48.899 --> 00:01:50.244
what percentage of the data points

00:01:50.244 --> 00:01:52.577
are below six hours per day.

00:01:54.468 --> 00:01:56.764
So let's see, there are,
I'm just gonna count 'em.

00:01:56.764 --> 00:02:00.097
One, two, three, four, five, six, seven.

00:02:02.572 --> 00:02:05.822
So seven of the 14 are below six hours.

00:02:07.037 --> 00:02:08.599
So we could just say seven,

00:02:08.599 --> 00:02:10.336
if we use this first technique

00:02:10.336 --> 00:02:13.587
we would have seven of the 14 are below

00:02:13.587 --> 00:02:16.129
six hours per day, and so that would

00:02:16.129 --> 00:02:19.596
get us a number of 50%, that six hours

00:02:19.596 --> 00:02:21.622
is at the 50th percentile.

00:02:21.622 --> 00:02:23.092
If we want to say what percentage

00:02:23.092 --> 00:02:24.801
is at that number or below then

00:02:24.801 --> 00:02:27.796
we would also count this
one, so we would say eight,

00:02:27.796 --> 00:02:29.379
or eight out of 14.

00:02:30.245 --> 00:02:33.126
Eight out of 14, which is the same thing

00:02:33.126 --> 00:02:35.292
as four out of seven, and if we wanted

00:02:35.292 --> 00:02:37.080
to write that as a decimal, let's see,

00:02:37.080 --> 00:02:40.588
seven goes into four point zero zero zero,

00:02:40.588 --> 00:02:42.190
we just need to estimate.

00:02:42.190 --> 00:02:45.028
So seven goes into 40 five times.

00:02:45.028 --> 00:02:47.611
35, we subtract, we get a five,

00:02:48.620 --> 00:02:50.295
bring down a zero, it goes five times.

00:02:50.295 --> 00:02:52.642
Look, it's just gonna be 0.5 repeating.

00:02:52.642 --> 00:02:54.059
So 55.5555%.

00:02:55.447 --> 00:02:57.463
So either of these would actually be

00:02:57.463 --> 00:03:00.827
a legitimate response
to the percentile rank

00:03:00.827 --> 00:03:03.273
for the driver with the daily
driving time of six hours.

00:03:03.273 --> 00:03:06.073
It depends on whether you
include the six hours or not.

00:03:06.073 --> 00:03:07.844
So you could say either
the 50th percentile

00:03:07.844 --> 00:03:10.577
or roughly the 55th, or actually

00:03:10.577 --> 00:03:12.700
the 56th percentile if you wanted

00:03:12.700 --> 00:03:15.077
to round to the nearest percentile.

00:03:15.077 --> 00:03:16.995
Now if you look at these choices here,

00:03:16.995 --> 00:03:18.313
lucky for us there's only one choice

00:03:18.313 --> 00:03:20.033
that's even, that's reasonably close

00:03:20.033 --> 00:03:23.055
to either one of those, and
that's the 55th percentile,

00:03:23.055 --> 00:03:25.184
and it looks like the people
who wrote this question

00:03:25.184 --> 00:03:27.604
went with the calculation of percentile

00:03:27.604 --> 00:03:30.242
where they include the
data point in question.

00:03:30.242 --> 00:03:32.875
So everything at six hours or less,

00:03:32.875 --> 00:03:36.375
what percentage of the total data is that?


Analyzing a cumulative relative frequency graph

WEBVTT
Kind: captions
Language: en

00:00:00.484 --> 00:00:02.896
- Nutritionists measured the sugar content

00:00:02.896 --> 00:00:05.962
in grams for 32 drinks at Starbucks.

00:00:05.962 --> 00:00:08.856
A cumulative relative
frequency graph, let me

00:00:08.856 --> 00:00:13.797
underline that, a cumulative
relative frequency graph

00:00:13.797 --> 00:00:16.394
for the data is shown below.

00:00:16.394 --> 00:00:18.551
So, they have different
on the horizontal axis,

00:00:18.551 --> 00:00:21.410
different amounts of
sugar in grams and then,

00:00:21.410 --> 00:00:23.631
we have the cumulative
relative frequencies.

00:00:23.631 --> 00:00:25.795
Let's just make sure we
understand how to read this.

00:00:25.795 --> 00:00:29.962
This is saying that zero or
zero percent of the drinks

00:00:31.479 --> 00:00:35.646
have a sugar content,
have no sugar content.

00:00:35.646 --> 00:00:37.675
This right over here, this data point,

00:00:37.675 --> 00:00:40.599
this looks like it's at the .5 grams

00:00:40.599 --> 00:00:44.515
and then this looks like it's at 0.1.

00:00:44.515 --> 00:00:48.814
This says that 0.1, or
I guess we could say

00:00:48.814 --> 00:00:52.687
10 percent of the drinks that Starbucks

00:00:52.687 --> 00:00:55.937
offers has five grams of sugar or less.

00:00:57.240 --> 00:01:00.378
This data point tells us that a hundred

00:01:00.378 --> 00:01:03.873
percent of drinks at
Starbucks has 50 grams

00:01:03.873 --> 00:01:05.290
of sugar or less.

00:01:06.830 --> 00:01:08.592
The cumulative relative
frequency, that's why

00:01:08.592 --> 00:01:12.233
at each of these points we
say this is the frequency

00:01:12.233 --> 00:01:15.051
that has that much sugar or less.

00:01:15.051 --> 00:01:16.786
And, that's why it just
keeps on increasing

00:01:16.786 --> 00:01:18.956
and increasing as we add more sugar

00:01:18.956 --> 00:01:21.873
we're going to see a larger portion

00:01:22.709 --> 00:01:24.637
or a larger relative frequency has that

00:01:24.637 --> 00:01:26.220
much sugar or less.

00:01:27.576 --> 00:01:29.987
So, let's read the first question.

00:01:29.987 --> 00:01:33.188
An iced coffee has 15 grams of sugar.

00:01:33.188 --> 00:01:35.421
Estimate the percentile of this drink

00:01:35.421 --> 00:01:37.358
to the nearest whole percent.

00:01:37.358 --> 00:01:40.441
So, iced coffee has 15 grams of sugar

00:01:41.310 --> 00:01:43.432
which would be right over here.

00:01:43.432 --> 00:01:45.283
And so, let's estimate the percentile.

00:01:45.283 --> 00:01:46.559
So, we can see they actually have

00:01:46.559 --> 00:01:48.529
a data point right over here and

00:01:48.529 --> 00:01:51.362
we can see that 20 percent or 0.2,

00:01:52.911 --> 00:01:56.375
20 percent of the drinks that Starbucks

00:01:56.375 --> 00:01:59.458
offers has 15 grams of sugar or less.

00:02:01.154 --> 00:02:03.150
So, the percentile of this drink,

00:02:03.150 --> 00:02:04.859
if I were to estimate it, looks like

00:02:04.859 --> 00:02:08.037
it's the relative frequency 0.2 has

00:02:08.037 --> 00:02:10.277
that much sugar or less so this percentile

00:02:10.277 --> 00:02:11.816
would be 20 percent.

00:02:11.816 --> 00:02:13.096
Once again, another way to think about it,

00:02:13.096 --> 00:02:15.525
to read this you could
convert these to percentages.

00:02:15.525 --> 00:02:18.897
You could say that 20 percent
has this much sugar or less.

00:02:18.897 --> 00:02:21.402
15 grams of sugar or
less, so an iced coffee

00:02:21.402 --> 00:02:23.569
is in the 20th percentile.

00:02:24.968 --> 00:02:27.135
Let's do another question.

00:02:29.438 --> 00:02:33.732
So here, we are asked
to estimate the median

00:02:33.732 --> 00:02:35.764
of the distribution of drinks.

00:02:35.764 --> 00:02:38.143
Hint to think about the 50th percentile.

00:02:38.143 --> 00:02:40.209
So, the median, if you were to line up all

00:02:40.209 --> 00:02:42.002
of the drinks, you would
take the middle drink.

00:02:42.002 --> 00:02:43.607
And so, you could view that as well, what

00:02:43.607 --> 00:02:46.556
drink is exactly at the 50th percentile?

00:02:46.556 --> 00:02:48.357
So, now let's look at the 50th percentile

00:02:48.357 --> 00:02:52.097
would be a cumulative
relative frequency of 0.5,

00:02:52.097 --> 00:02:56.242
which would be right over
here on our vertical axis.

00:02:56.242 --> 00:02:59.956
Another way to think about
it is 0.5, or 50 percent

00:02:59.956 --> 00:03:02.888
of the drinks are going,
if we go to this point

00:03:02.888 --> 00:03:05.580
right over here, what
has a cumulative relative

00:03:05.580 --> 00:03:06.997
frequency of 0.5.

00:03:07.978 --> 00:03:11.445
We see that we are right at
looks like this is 25 grams.

00:03:11.445 --> 00:03:15.591
So, one way to interpret
this is 50 percent

00:03:15.591 --> 00:03:18.841
of the drinks have less than or have 25

00:03:19.752 --> 00:03:21.669
grams of sugar or less.

00:03:22.592 --> 00:03:24.724
So, this looks like a pretty good estimate

00:03:24.724 --> 00:03:27.785
for the median, for the middle data point.

00:03:27.785 --> 00:03:31.118
So, the median is approximately 25 grams

00:03:32.348 --> 00:03:36.515
that half of the drinks have
25 grams or less of sugar.

00:03:39.115 --> 00:03:43.088
Let's do one more based
on the same data set.

00:03:43.088 --> 00:03:46.652
So, here we're asked,
what is the best estimate

00:03:46.652 --> 00:03:50.870
for the interquartile range
of the distribution of drinks?

00:03:50.870 --> 00:03:53.071
So, the interquartile
range, we wanna figure out

00:03:53.071 --> 00:03:56.139
well, what's sitting
at the 25th percentile?

00:03:56.139 --> 00:03:57.466
And we wanna think about what's

00:03:57.466 --> 00:03:59.322
at the 75th percentile, and then we

00:03:59.322 --> 00:04:01.521
want to take the difference.

00:04:01.521 --> 00:04:04.552
That's what the interquartile range is.

00:04:04.552 --> 00:04:05.507
So, let's do that.

00:04:05.507 --> 00:04:08.092
So, first the 25th
percentile, we wanna look

00:04:08.092 --> 00:04:12.049
at the cumulative relative
frequency, so 25th

00:04:12.049 --> 00:04:13.975
this would be 30th, so
the 25th would be right

00:04:13.975 --> 00:04:17.826
around here and so, it looks like the 25th

00:04:17.826 --> 00:04:21.993
percentile is that looks
like about, I don't know,

00:04:23.377 --> 00:04:25.637
and we're estimating here,
so that looks like it's

00:04:25.637 --> 00:04:29.804
about this would be 15, I
would say maybe 18 grams.

00:04:32.106 --> 00:04:33.686
So, approximately 18 grams.

00:04:33.686 --> 00:04:35.682
Once again, one way to think about it is,

00:04:35.682 --> 00:04:39.849
25 percent of the drinks have
18 grams of sugar or less.

00:04:40.929 --> 00:04:42.923
Now, let's look at the 75th percentile.

00:04:42.923 --> 00:04:47.294
So it's the 70th, 75th
would be right over there.

00:04:47.294 --> 00:04:48.637
Actually, I can draw a
straighter line than that.

00:04:48.637 --> 00:04:51.127
I have a line tool here.

00:04:51.127 --> 00:04:54.960
75th percentile would
put me right over there.

00:04:57.262 --> 00:04:59.302
I don't know, that
looks like, I'll go with

00:04:59.302 --> 00:05:01.552
39 grams, roughly 39 grams.

00:05:04.898 --> 00:05:07.727
And so, what's the
difference between these two?

00:05:07.727 --> 00:05:09.714
Well, the difference
between these two, it looks

00:05:09.714 --> 00:05:12.421
like it's about 21 grams.

00:05:12.421 --> 00:05:15.834
So, our interquartile range, our estimate

00:05:15.834 --> 00:05:17.556
of our interquartile
range, looking at this

00:05:17.556 --> 00:05:20.172
cumulative relative
frequency distribution,

00:05:20.172 --> 00:05:22.356
'cause we're sayin',
hey look, it looks like

00:05:22.356 --> 00:05:24.752
the 25th percentile, looks like 25 percent

00:05:24.752 --> 00:05:26.801
of the drinks have 18 grams or less.

00:05:26.801 --> 00:05:30.055
75 percent of the drinks
have 39 grams or less.

00:05:30.055 --> 00:05:32.813
If we take the difference
between these two

00:05:32.813 --> 00:05:35.850
quartiles, this is the first quartile,

00:05:35.850 --> 00:05:37.634
this is our third quartile.

00:05:37.634 --> 00:05:39.602
We're gonna get 21 grams.

00:05:39.602 --> 00:05:41.442
Now, if we look at this
choice, the choices

00:05:41.442 --> 00:05:43.563
right over here, 20 grams definitely

00:05:43.563 --> 00:05:45.294
seems like the best estimate,

00:05:45.294 --> 00:05:47.722
closest to what we were able to estimate

00:05:47.722 --> 00:05:49.503
based on looking at this cumulative

00:05:49.503 --> 00:05:51.586
relative frequency graph.