Misleading line graphs

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.725
Thrill Soda hired
a marketing company

00:00:02.725 --> 00:00:05.455
to help them promote their
brand against Yummy Cola.

00:00:05.455 --> 00:00:07.530
The company gathered
the following data

00:00:07.530 --> 00:00:10.650
about consumers'
preference of soda.

00:00:10.650 --> 00:00:13.840
So they have, year by year,
percentage of respondents

00:00:13.840 --> 00:00:17.810
who preferred Yummy Cola,
percentage of respondents who

00:00:17.810 --> 00:00:19.730
preferred Thrill
Cola, and then these

00:00:19.730 --> 00:00:21.350
are people who
had no preference.

00:00:21.350 --> 00:00:26.310
So in 2006, 80% liked Yummy,
only 12% liked Thrill,

00:00:26.310 --> 00:00:30.200
and 8% didn't like either one
or didn't have any preference.

00:00:30.200 --> 00:00:31.790
And so actually
just from here you

00:00:31.790 --> 00:00:34.010
see that many, many more
people liked Yummy Cola

00:00:34.010 --> 00:00:36.360
than Thrill Cola, actually
every year over year.

00:00:36.360 --> 00:00:38.310
So, Thrill Cola
definitely has something.

00:00:38.310 --> 00:00:39.630
They have an uphill battle.

00:00:39.630 --> 00:00:41.713
But then they said the
advertising company created

00:00:41.713 --> 00:00:44.640
the following two graphs
to promote Thrill Soda.

00:00:44.640 --> 00:00:47.350
And so let's see what's
happening over here.

00:00:47.350 --> 00:00:50.790
And let's think about whether
this is misleading or not.

00:00:50.790 --> 00:00:55.060
So if we look at this graph
over here, in 2006, sure enough,

00:00:55.060 --> 00:00:57.096
80% liked Yummy Cola.

00:00:57.096 --> 00:00:59.080
Then in 2007, 76%.

00:00:59.080 --> 00:01:05.032
Then it keeps going to then
77%, then 73%, then 73% to 68%.

00:01:05.032 --> 00:01:07.016
So this is accurate data.

00:01:07.016 --> 00:01:10.984
It actually represents the data
that's given right over here.

00:01:10.984 --> 00:01:11.976
I'll do it in the same.

00:01:11.976 --> 00:01:14.456
It actually represents
this data very faithfully.

00:01:14.456 --> 00:01:16.936
Then right over here, if we
look at this chart, Percentage

00:01:16.936 --> 00:01:21.400
of People who Prefer Thrill
Soda, so over here in 2006, 12%

00:01:21.400 --> 00:01:24.376
preferred Thrill Soda.

00:01:24.376 --> 00:01:26.856
2007, 19%.

00:01:26.856 --> 00:01:29.336
2008, 19%.

00:01:29.336 --> 00:01:32.312
Then we go up to
20%, 21%, and 25%.

00:01:32.312 --> 00:01:34.296
So the graphs are
actually accurate.

00:01:34.296 --> 00:01:35.288
They're not lying.

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These are actually
the data points

00:01:37.070 --> 00:01:39.302
of the percentage who
prefer Thrill Soda.

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Now what's misleading
is if someone

00:01:41.286 --> 00:01:43.270
were to just look
at these two graphs

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without actually looking
at the scales over here,

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they'll see two things.

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They'll say, oh, look,
you see a declining trend.

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And that's what line graphs are
good for, for seeing trends.

00:01:52.694 --> 00:01:54.678
They say, look, I
see a declining trend

00:01:54.678 --> 00:01:58.150
in the percentage of people
who prefer Yummy Cola.

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And I see this increasing
trend in the percentage

00:02:01.622 --> 00:02:03.436
of people who
prefer Thrill Cola.

00:02:03.436 --> 00:02:04.102
And that's true.

00:02:04.102 --> 00:02:05.435
You have a declining trend here.

00:02:05.435 --> 00:02:07.078
And you have an
increasing trend here.

00:02:07.078 --> 00:02:09.062
But what's misleading
here is the way

00:02:09.062 --> 00:02:10.550
that they've plotted the scales.

00:02:10.550 --> 00:02:13.030
These scales are not the same.

00:02:13.030 --> 00:02:15.070
So when you look at
this, you say not only

00:02:15.070 --> 00:02:18.046
is there an increasing trend of
people who prefer Thrill Soda,

00:02:18.046 --> 00:02:20.030
but the way they
set up the scale,

00:02:20.030 --> 00:02:22.510
it looks like the
trend is above.

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The human brain is
tempted to compare these

00:02:25.486 --> 00:02:27.966
and to say, look, not only
is this an upward trend,

00:02:27.966 --> 00:02:30.446
but it's above this
trend right over here.

00:02:30.446 --> 00:02:32.182
Even in 2006, this
data point looks

00:02:32.182 --> 00:02:35.406
higher than these data
points right over here.

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But the reality
is that it's only

00:02:37.142 --> 00:02:38.878
because the scale is distorted.

00:02:38.878 --> 00:02:40.862
Now this is the oldest
trick in the book

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when plotting line graphs.

00:02:42.350 --> 00:02:43.838
It all depends on the scale.

00:02:43.838 --> 00:02:46.814
So this just looks good because
they used this scale that

00:02:46.814 --> 00:02:49.294
went from 0 to 30 as
opposed to 0 to 100.

00:02:49.294 --> 00:02:54.159
The better thing to do, or
the more genuine thing to do,

00:02:54.159 --> 00:02:56.284
or the more honest thing
to do, would have actually

00:02:56.284 --> 00:02:57.276
been to plot them
on the same graph.

00:02:57.276 --> 00:02:58.268
Although if they did
that, that wouldn't

00:02:58.268 --> 00:03:00.748
have painted a very good
picture for Thrill Soda.

00:03:00.748 --> 00:03:03.228
So if we plotted on the
same graph Thrill Soda,

00:03:03.228 --> 00:03:04.716
let's try that out.

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And actually, this
is even worse.

00:03:06.438 --> 00:03:08.188
You actually wouldn't
even be able to plot

00:03:08.188 --> 00:03:10.668
Thrill Soda on this graph
because they started this graph

00:03:10.668 --> 00:03:12.156
right over here at 50%.

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They didn't even start it at 0%.

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So you actually
would not even be

00:03:15.132 --> 00:03:17.116
able to plot Thrill
Soda on this graph.

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If you did, you would
have to extend this graph

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all the way down.

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So you would have to extend
this graph all the way down

00:03:23.812 --> 00:03:26.540
to-- so this would
have to be 40%.

00:03:26.540 --> 00:03:28.028
This would be 30%.

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This would be 20%.

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This would be 10%.

00:03:33.530 --> 00:03:36.403
And then down all the way
over here would be 0%.

00:03:36.403 --> 00:03:38.986
And then the Thrill Soda graph
would be all the way down here.

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So it was like 12%.

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And it goes all the
way up to like 25%.

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So the Thrill Soda, it
would have looked something

00:03:46.422 --> 00:03:46.922
like this.

00:03:46.922 --> 00:03:48.575
The graph would have
looked something

00:03:48.575 --> 00:03:50.394
like this, which
is nowhere near.

00:03:50.394 --> 00:03:54.362
If you plotted these on the
same scale, on the same graph,

00:03:54.362 --> 00:03:56.842
then it would have still
been pretty obvious

00:03:56.842 --> 00:03:59.322
that-- even though the trend
is downward-- a lot more

00:03:59.322 --> 00:04:03.786
people prefer Yummy
Cola to Thrill Cola.

00:04:03.786 --> 00:04:07.258
So there's two very disingenuous
things going on over here.

00:04:07.258 --> 00:04:08.746
One is the actual scale.

00:04:08.746 --> 00:04:12.237
For this amount of distance on
this scale, they represent 10%.

00:04:12.237 --> 00:04:14.570
So whatever the gain is, it
looks like it's a huge gain.

00:04:14.570 --> 00:04:16.554
And over here, on
that same amount,

00:04:16.554 --> 00:04:18.470
they're actually
representing a larger amount.

00:04:18.470 --> 00:04:20.870
They're representing
closer to 15% or 16%.

00:04:20.870 --> 00:04:24.470
And then the main thing is they
started the scale here at 50%.

00:04:24.470 --> 00:04:26.870
So they're not showing how
many people really prefer,

00:04:26.870 --> 00:04:30.170
how large 80% or
even 70% really is.

00:04:30.170 --> 00:04:32.046
And over here, they
start at 0% and they just

00:04:32.046 --> 00:04:32.878
have a larger scale.

00:04:32.878 --> 00:04:35.270
So it makes it look like out
the gate, a lot of people

00:04:35.270 --> 00:04:37.970
prefer, or a comparable amount
of people prefer, Thrill

00:04:37.970 --> 00:04:39.470
and that the trend is up.

00:04:39.470 --> 00:04:43.070
But the reality is still way
more people prefer Yummy Cola.

00:04:43.070 --> 00:04:44.870
So this was a little
bit-- or actually

00:04:44.870 --> 00:04:49.070
a very, very, not-so-honest
way of representing the data.

00:04:49.070 --> 00:04:51.820
Or I guess you could say they're
misrepresenting the data.


Example: All the ways you can flip a coin

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.580
Find the probability
of getting exactly two

00:00:03.580 --> 00:00:06.530
heads when flipping three coins.

00:00:06.530 --> 00:00:08.230
So let's think about
the sample space.

00:00:08.230 --> 00:00:10.910
Let's think about all of
the possible outcomes.

00:00:10.910 --> 00:00:12.230
So I could get all heads.

00:00:12.230 --> 00:00:14.470
So on flip one I get a
head, flip two I get a head,

00:00:14.470 --> 00:00:16.070
flip three I get a head.

00:00:16.070 --> 00:00:19.080
I could get two heads
and then a tail.

00:00:19.080 --> 00:00:23.190
I could get heads,
tails, heads, or I

00:00:23.190 --> 00:00:26.420
could get heads, tails, tails.

00:00:26.420 --> 00:00:30.760
I could get tails, heads, heads.

00:00:30.760 --> 00:00:33.420
I could get tails, heads, tails.

00:00:33.420 --> 00:00:36.040
I could get tails, tails, heads.

00:00:36.040 --> 00:00:38.992
Or I could get tails,
tails, and tails.

00:00:38.992 --> 00:00:40.450
These are all of
the different ways

00:00:40.450 --> 00:00:41.990
that I could flip three coins.

00:00:41.990 --> 00:00:43.406
And you can maybe
say that this is

00:00:43.406 --> 00:00:45.660
the first flip, the second
flip, and the third flip.

00:00:45.660 --> 00:00:47.930
Now, so this right over
here is the sample space.

00:00:47.930 --> 00:00:49.610
There's eight possible outcomes.

00:00:49.610 --> 00:00:52.340
Let me write this, the
probability of exactly two

00:00:52.340 --> 00:01:00.860
heads, I'll say H's
there for short.

00:01:00.860 --> 00:01:02.640
The probability of
exactly two heads, well

00:01:02.640 --> 00:01:04.959
what is the size of
our sample space?

00:01:04.959 --> 00:01:07.620
I have eight possible outcomes.

00:01:07.620 --> 00:01:10.510
So eight, this is
possible outcomes,

00:01:10.510 --> 00:01:15.150
or the size of our sample
space, possible outcomes.

00:01:15.150 --> 00:01:17.610
And how many of those
possible outcomes

00:01:17.610 --> 00:01:20.050
are associated with this event?

00:01:20.050 --> 00:01:21.620
You could call this
a compound event,

00:01:21.620 --> 00:01:23.610
because there's more
than one outcome that's

00:01:23.610 --> 00:01:25.560
associated with this.

00:01:25.560 --> 00:01:28.280
Let's think about
exactly two heads.

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This is three heads, so
it's not exactly two heads.

00:01:30.840 --> 00:01:33.390
This is exactly two
heads right over here.

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This is exactly two
heads right over here.

00:01:35.570 --> 00:01:36.990
There's only one head.

00:01:36.990 --> 00:01:38.690
This is exactly two heads.

00:01:38.690 --> 00:01:41.400
This is only one head,
only one head, no heads.

00:01:41.400 --> 00:01:46.400
So you have one, two, three
of the possible outcomes

00:01:46.400 --> 00:01:48.800
are associated with this event.

00:01:48.800 --> 00:01:51.410
So you have three
possible outcomes.

00:01:51.410 --> 00:01:53.620
Three outcomes
associated with event.

00:01:53.620 --> 00:01:59.870
Three outcomes
satisfy this event,

00:01:59.870 --> 00:02:01.570
are associated with this event.

00:02:01.570 --> 00:02:03.860
So the probability of
getting exactly two heads

00:02:03.860 --> 00:02:07.240
when flipping three
coins is three outcomes

00:02:07.240 --> 00:02:10.300
satisfying this event over
eight possible outcomes.

00:02:10.300 --> 00:02:12.410
So it is 3/8.


Finding probability example

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.680
Find the probability of pulling
a yellow marble from a bag

00:00:03.680 --> 00:00:07.900
with 3 yellow, 2 red,
2 green, and 1 blue--

00:00:07.900 --> 00:00:09.000
I'm assuming-- marbles.

00:00:09.000 --> 00:00:10.830
So they say the
probability-- I'll just

00:00:10.830 --> 00:00:12.460
say p for probability.

00:00:12.460 --> 00:00:17.340
The probability of
picking a yellow marble.

00:00:17.340 --> 00:00:19.720
And so this is sometimes
the event in question,

00:00:19.720 --> 00:00:21.880
right over here, is
picking the yellow marble.

00:00:21.880 --> 00:00:24.004
I'll even write down
the word "picking."

00:00:24.004 --> 00:00:25.420
And when you say
probability, it's

00:00:25.420 --> 00:00:27.650
really just a way of
measuring the likelihood

00:00:27.650 --> 00:00:29.570
that something is
going to happen.

00:00:29.570 --> 00:00:32.729
And the way we're going to
think about it is how many

00:00:32.729 --> 00:00:36.640
of the outcomes from this trial,
from this picking a marble out

00:00:36.640 --> 00:00:39.050
of a bag, how many
meet our constraints,

00:00:39.050 --> 00:00:40.600
satisfy this event?

00:00:40.600 --> 00:00:43.700
And how many possible
outcomes are there?

00:00:43.700 --> 00:00:46.280
So let me write the possible
outcomes right over here,

00:00:46.280 --> 00:00:52.280
so possible outcomes.

00:00:52.280 --> 00:00:54.680
And you'll see it's actually
a very straightforward idea.

00:00:54.680 --> 00:00:57.013
But I'll just make sure that
we understand all the words

00:00:57.013 --> 00:00:58.570
that people might
say, so the set

00:00:58.570 --> 00:01:00.552
of all the possible outcomes.

00:01:00.552 --> 00:01:02.010
Well, there's three
yellow marbles.

00:01:02.010 --> 00:01:05.379
So I could pick that yellow
marble, that yellow marble,

00:01:05.379 --> 00:01:08.400
or that yellow marble,
that yellow marble.

00:01:08.400 --> 00:01:10.020
These are clearly all yellow.

00:01:10.020 --> 00:01:11.840
There's two red
marbles in the bag.

00:01:11.840 --> 00:01:16.380
So I could pick that red
marble or that red marble.

00:01:16.380 --> 00:01:19.020
There's two green
marbles in the bag.

00:01:19.020 --> 00:01:23.140
So I could pick that green
marble or that green marble.

00:01:23.140 --> 00:01:25.720
And then there's one
blue marble in the bag.

00:01:25.720 --> 00:01:27.750
There's one blue marble.

00:01:27.750 --> 00:01:30.020
So this is all the
possible outcomes.

00:01:30.020 --> 00:01:34.370
And sometimes this is referred
to as the sample space,

00:01:34.370 --> 00:01:36.820
the set of all the
possible outcomes.

00:01:36.820 --> 00:01:38.880
Fancy word for
just a simple idea,

00:01:38.880 --> 00:01:41.960
that the sample space, when I
pick something out of the bag,

00:01:41.960 --> 00:01:44.540
and that picking out of
the bag is called a trial,

00:01:44.540 --> 00:01:46.549
there's 8 possible
things I can do.

00:01:46.549 --> 00:01:48.090
So when I think
about the probability

00:01:48.090 --> 00:01:50.840
of picking a yellow marble,
I want to think about, well,

00:01:50.840 --> 00:01:52.560
what are all of
the possibilities?

00:01:52.560 --> 00:01:57.030
Well, there's 8 possibilities,
8 possibilities for my trial.

00:01:57.030 --> 00:02:01.790
So the number of outcomes,
number of possible outcomes,

00:02:01.790 --> 00:02:04.896
you could view it as the size
of the sample space, number

00:02:04.896 --> 00:02:07.020
of possible outcomes, And
it's as simple as saying,

00:02:07.020 --> 00:02:08.362
look, I have 8 marbles.

00:02:08.362 --> 00:02:10.070
And then you say,
well, how many of those

00:02:10.070 --> 00:02:14.500
marbles meet my constraint,
that satisfy this event here?

00:02:14.500 --> 00:02:17.930
Well, there's 3 marbles
that satisfy my event.

00:02:17.930 --> 00:02:22.262
There's 3 outcomes that will
allow this event to occur,

00:02:22.262 --> 00:02:23.470
I guess is one way to say it.

00:02:23.470 --> 00:02:28.100
So there's 3 right over
here, so number that

00:02:28.100 --> 00:02:34.970
satisfy the event or the
constraint right over here.

00:02:34.970 --> 00:02:36.750
So it's very simple ideas.

00:02:36.750 --> 00:02:38.550
Many times the
words make them more

00:02:38.550 --> 00:02:40.250
complicated than they need to.

00:02:40.250 --> 00:02:43.010
If I say, what's the probability
of picking a yellow marble?

00:02:43.010 --> 00:02:45.690
Well, how many different
types of marbles can I pick?

00:02:45.690 --> 00:02:48.230
Well, there's 8 different
marbles I could pick.

00:02:48.230 --> 00:02:50.560
And then how many
of them are yellow?

00:02:50.560 --> 00:02:54.809
Well, there's 3 of them
that are actually yellow.


Box and whisker plot

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.010
An ecologist surveys the
age of about 100 trees

00:00:04.010 --> 00:00:05.450
in a local forest.

00:00:05.450 --> 00:00:10.070
He uses a box-and-whisker plot
to map his data shown below.

00:00:10.070 --> 00:00:13.540
What is the range of tree
ages that he surveyed?

00:00:13.540 --> 00:00:16.730
What is the median age
of a tree in the forest?

00:00:16.730 --> 00:00:19.040
So first of all, let's
make sure we understand

00:00:19.040 --> 00:00:21.760
what this box-and-whisker
plot is even about.

00:00:21.760 --> 00:00:24.310
This is really a way of
seeing the spread of all

00:00:24.310 --> 00:00:27.180
of the different data points,
which are the age of the trees,

00:00:27.180 --> 00:00:28.630
and to also give
other information

00:00:28.630 --> 00:00:29.780
like, what is the median?

00:00:29.780 --> 00:00:33.390
And where do most of the
ages of the trees sit?

00:00:33.390 --> 00:00:35.690
So this whisker part, so you
could see this black part

00:00:35.690 --> 00:00:37.440
is a whisker, this
is the box, and then

00:00:37.440 --> 00:00:39.670
this is another whisker
right over here.

00:00:39.670 --> 00:00:43.220
The whiskers tell us essentially
the spread of all of the data.

00:00:43.220 --> 00:00:46.190
So it says the lowest to
data point in this sample

00:00:46.190 --> 00:00:47.740
is an eight-year-old tree.

00:00:47.740 --> 00:00:51.580
I'm assuming that this axis
down here is in the years.

00:00:51.580 --> 00:00:54.800
And it says at the highest--
the oldest tree right over here

00:00:54.800 --> 00:00:56.310
is 50 years.

00:00:56.310 --> 00:00:58.172
So if we want the
range-- and when

00:00:58.172 --> 00:01:00.130
we think of range in a
statistics point of view

00:01:00.130 --> 00:01:01.980
we're thinking of
the highest data

00:01:01.980 --> 00:01:04.319
point minus the
lowest data point.

00:01:04.319 --> 00:01:06.630
So it's going to be 50 minus 8.

00:01:06.630 --> 00:01:09.411
So we have a range of 42.

00:01:09.411 --> 00:01:10.910
So that's what the
whiskers tell us.

00:01:10.910 --> 00:01:14.300
It tells us that everything
falls between 8 and 50 years,

00:01:14.300 --> 00:01:16.550
including 8 years and 50 years.

00:01:16.550 --> 00:01:19.400
Now what the box does,
the box starts at-- well,

00:01:19.400 --> 00:01:21.234
let me explain it
to you this way.

00:01:21.234 --> 00:01:23.150
This line right over
here, this is the median.

00:01:27.070 --> 00:01:29.490
And so half of
the ages are going

00:01:29.490 --> 00:01:30.740
to be less than this median.

00:01:30.740 --> 00:01:32.980
We see right over
here the median is 21.

00:01:32.980 --> 00:01:35.300
So this box-and-whiskers
plot tells us

00:01:35.300 --> 00:01:38.590
that half of the ages of
the trees are less than 21

00:01:38.590 --> 00:01:42.070
and half are older than 21.

00:01:42.070 --> 00:01:44.300
And then these endpoints
right over here,

00:01:44.300 --> 00:01:46.950
these are the medians for
each of those sections.

00:01:46.950 --> 00:01:48.960
So this is the median
for all the trees

00:01:48.960 --> 00:01:50.810
that are less than
the real median

00:01:50.810 --> 00:01:52.170
or less than the main median.

00:01:52.170 --> 00:01:55.360
So this is in the middle
of all of the ages of trees

00:01:55.360 --> 00:01:56.790
that are less than 21.

00:01:56.790 --> 00:01:59.280
This is the middle
age for all the trees

00:01:59.280 --> 00:02:02.610
that are greater than
21 or older than 21.

00:02:02.610 --> 00:02:06.780
And so we're actually
splitting all of the data

00:02:06.780 --> 00:02:08.070
into four groups.

00:02:08.070 --> 00:02:10.430
This we would call
the first quartile.

00:02:10.430 --> 00:02:12.680
So I'll call it Q1 for
our first quartile.

00:02:12.680 --> 00:02:15.170
Maybe I'll do 1Q.

00:02:15.170 --> 00:02:17.530
This is the first quartile.

00:02:17.530 --> 00:02:19.891
Roughly a fourth of the
tree, because the way

00:02:19.891 --> 00:02:21.640
you calculate it,
sometimes a tree ends up

00:02:21.640 --> 00:02:24.160
in one point or another,
about a fourth of the trees

00:02:24.160 --> 00:02:24.870
end up here.

00:02:24.870 --> 00:02:27.651
A fourth of the trees
are between 14 and 21.

00:02:27.651 --> 00:02:30.936
A fourth are between 21
and it looks like 33.

00:02:30.936 --> 00:02:32.560
And then a fourth
are in this quartile.

00:02:32.560 --> 00:02:36.310
So we call this the first
quartile, the second quartile,

00:02:36.310 --> 00:02:40.430
the third quartile, and
the fourth quartile.

00:02:40.430 --> 00:02:42.640
So to answer the question,
we already did the range.

00:02:42.640 --> 00:02:46.361
There's a 42-year spread between
the oldest and the youngest

00:02:46.361 --> 00:02:46.860
tree.

00:02:46.860 --> 00:02:51.440
And then the median age of a
tree in the forest is at 21.

00:02:51.440 --> 00:02:54.610
So even though you might have
trees that are as old as 50,

00:02:54.610 --> 00:02:56.360
the median of the
forest is actually

00:02:56.360 --> 00:03:01.387
closer to the lower end of
our entire spectrum of all

00:03:01.387 --> 00:03:02.370
of the ages.

00:03:02.370 --> 00:03:06.790
So if you view median as your
central tendency measurement,

00:03:06.790 --> 00:03:08.860
it's only at 21 years.

00:03:08.860 --> 00:03:09.910
And you can even see it.

00:03:09.910 --> 00:03:11.810
It's closer to the
left of the box

00:03:11.810 --> 00:03:14.930
and closer to the end
of the left whisker

00:03:14.930 --> 00:03:17.920
than the end of
the right whisker.


Finding mean, median, and mode

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.059
Find the mean, median,
and mode of the following

00:00:04.059 --> 00:00:05.190
sets of numbers.

00:00:05.190 --> 00:00:07.250
And they give us the
numbers right over here.

00:00:07.250 --> 00:00:09.312
So if someone just
says the mean,

00:00:09.312 --> 00:00:10.770
they're really
referring to what we

00:00:10.770 --> 00:00:13.280
typically, in everyday
language, call the average.

00:00:13.280 --> 00:00:14.780
Sometimes it's
called the arithmetic

00:00:14.780 --> 00:00:15.940
mean because you'll
learn that there's

00:00:15.940 --> 00:00:17.980
other ways of actually
calculating a mean.

00:00:17.980 --> 00:00:20.210
But it's really you just
sum up all the numbers

00:00:20.210 --> 00:00:22.210
and you divide by the
numbers there are.

00:00:22.210 --> 00:00:25.290
And so it's one way of
measuring the central tendency.

00:00:25.290 --> 00:00:27.270
The average, I
guess, we could say.

00:00:27.270 --> 00:00:28.160
So this is our mean.

00:00:28.160 --> 00:00:31.380
We want to average
23 plus 29-- or we're

00:00:31.380 --> 00:00:38.510
going to sum 23 plus 29 plus
20 plus 32 plus 23 plus 21

00:00:38.510 --> 00:00:44.710
plus 33 plus 25, and then divide
that by the number of numbers.

00:00:44.710 --> 00:00:49.480
So we have 1, 2, 3,
4, 5, 6, 7, 8 numbers.

00:00:49.480 --> 00:00:51.500
So you want to divide that by 8.

00:00:51.500 --> 00:00:53.330
So let's figure out
what that actually is.

00:00:53.330 --> 00:00:56.080
Actually, I'll just get the
calculator out for this part.

00:00:56.080 --> 00:00:59.310
I could do it by hand, but
we'll save some time over here.

00:00:59.310 --> 00:01:16.010
So we have 23 plus 29 plus
20 plus 32 plus 23 plus 21

00:01:16.010 --> 00:01:20.230
plus 33 plus 25.

00:01:20.230 --> 00:01:23.000
So the sum of all
the numbers is 206.

00:01:23.000 --> 00:01:25.940
And then we want
to divide 206 by 8.

00:01:25.940 --> 00:01:34.440
So if I say 206 divided
by 8 gets us 25.75.

00:01:34.440 --> 00:01:37.649
So the mean is equal to 25.75.

00:01:37.649 --> 00:01:39.190
So this is one way
to kind of measure

00:01:39.190 --> 00:01:41.430
the center, the
central tendency.

00:01:41.430 --> 00:01:43.600
Another way is with the median.

00:01:43.600 --> 00:01:47.001
And this is to pick out the
middle number, the median.

00:01:47.001 --> 00:01:49.000
And to figure out the
median, what we want to do

00:01:49.000 --> 00:01:51.790
is order these numbers
from least to greatest.

00:01:51.790 --> 00:01:56.030
So it looks like the
smallest number here is 20.

00:01:56.030 --> 00:01:57.180
Then, the next one is 21.

00:02:00.770 --> 00:02:03.580
There's no 22 here.

00:02:03.580 --> 00:02:05.410
Let's see, there's two 23's.

00:02:05.410 --> 00:02:07.010
23 and a 23.

00:02:07.010 --> 00:02:09.600
So 23 and a 23.

00:02:09.600 --> 00:02:10.889
And no 24's.

00:02:10.889 --> 00:02:12.770
There's a 25.

00:02:12.770 --> 00:02:13.580
25.

00:02:13.580 --> 00:02:15.710
There's no 26, 27, 28.

00:02:15.710 --> 00:02:17.460
There is a 29.

00:02:17.460 --> 00:02:18.860
29.

00:02:18.860 --> 00:02:20.690
Then you have your 32.

00:02:20.690 --> 00:02:21.470
32.

00:02:21.470 --> 00:02:23.420
And then you have your 33.

00:02:23.420 --> 00:02:24.170
33.

00:02:24.170 --> 00:02:26.420
So what's the middle number
now that we've ordered it?

00:02:26.420 --> 00:02:29.760
So we have 1, 2, 3,
4, 5, 6, 7, 8 numbers.

00:02:29.760 --> 00:02:30.751
We already knew that.

00:02:30.751 --> 00:02:32.750
And so there's actually
going to be two middles.

00:02:32.750 --> 00:02:35.080
If you have an even
number, there's

00:02:35.080 --> 00:02:37.680
actually two numbers
that qualify for close

00:02:37.680 --> 00:02:38.400
to the middle.

00:02:38.400 --> 00:02:40.980
And to actually get the median,
we're going to average them.

00:02:40.980 --> 00:02:42.940
So 23 will be one of them.

00:02:42.940 --> 00:02:44.440
That, by itself,
can't be the median

00:02:44.440 --> 00:02:45.856
because there's
three less than it

00:02:45.856 --> 00:02:47.330
and there's four
greater than it.

00:02:47.330 --> 00:02:49.590
And 25 by itself
can't be the median

00:02:49.590 --> 00:02:52.420
because there's three larger
than it and four less than it.

00:02:52.420 --> 00:02:55.480
So what we do is we take the
mean of these two numbers

00:02:55.480 --> 00:02:57.620
and we pick that as the median.

00:02:57.620 --> 00:03:03.840
So if you take 23 plus 25
divided by 2, that's 48 over 2,

00:03:03.840 --> 00:03:05.540
which is equal to 24.

00:03:05.540 --> 00:03:07.500
So even though 24 isn't
one of these numbers,

00:03:07.500 --> 00:03:09.260
the median is 24.

00:03:09.260 --> 00:03:10.920
So this is the middle number.

00:03:10.920 --> 00:03:13.550
So once again, this
is one way of thinking

00:03:13.550 --> 00:03:14.522
about central tendency.

00:03:14.522 --> 00:03:16.230
If you wanted a number
that could somehow

00:03:16.230 --> 00:03:17.105
represent the middle.

00:03:17.105 --> 00:03:19.396
And I want to be clear,
there's no one way of doing it.

00:03:19.396 --> 00:03:22.670
This is one way of
measuring the middle.

00:03:22.670 --> 00:03:24.660
Let me put that in quotes.

00:03:24.660 --> 00:03:25.450
The middle.

00:03:25.450 --> 00:03:28.510
If you had to represent
this data with one number.

00:03:28.510 --> 00:03:31.320
And this is another way of
representing the middle.

00:03:31.320 --> 00:03:33.430
Then finally, we can
think about the mode.

00:03:33.430 --> 00:03:35.060
And the mode is
just the number that

00:03:35.060 --> 00:03:37.590
shows up the most
in this data set.

00:03:37.590 --> 00:03:41.150
And all of these numbers show
up once, except we have the 23,

00:03:41.150 --> 00:03:42.610
it shows up twice.

00:03:42.610 --> 00:03:47.280
And since because 23 shows up
the most, it shows up twice.

00:03:47.280 --> 00:03:53.927
Every other number shows
up once, 23 is our mode.


Finding the range and mid-range

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.320
Find the range and the
mid-range of the following sets

00:00:04.320 --> 00:00:05.490
of numbers.

00:00:05.490 --> 00:00:08.172
So what the range tells us is
essentially how spread apart

00:00:08.172 --> 00:00:10.130
these numbers are, and
the way you calculate it

00:00:10.130 --> 00:00:11.630
is that you just
take the difference

00:00:11.630 --> 00:00:13.630
between the largest
of these numbers

00:00:13.630 --> 00:00:15.787
and the smallest
of these numbers.

00:00:15.787 --> 00:00:17.870
And so if we look at the
largest of these numbers,

00:00:17.870 --> 00:00:21.530
I'll circle it in magenta,
it looks like it is 94.

00:00:21.530 --> 00:00:23.200
94 is larger than
every other number

00:00:23.200 --> 00:00:25.990
here, so that's the
largest of the numbers.

00:00:25.990 --> 00:00:28.000
And from that, we
want to subtract

00:00:28.000 --> 00:00:29.490
the smallest of the numbers.

00:00:29.490 --> 00:00:32.100
And the smallest of the numbers
in our set right over here

00:00:32.100 --> 00:00:34.010
is 65.

00:00:34.010 --> 00:00:37.120
So you want to
subtract 65 from 94.

00:00:37.120 --> 00:00:40.420
And this is equal to-- Let's
see, if this was 95 minus 65,

00:00:40.420 --> 00:00:41.380
it would be 30.

00:00:41.380 --> 00:00:45.080
94 is one less than
that, so it is 29.

00:00:45.080 --> 00:00:47.810
So the larger this number is,
that means the more spread out.

00:00:47.810 --> 00:00:50.230
The larger the difference
between the largest

00:00:50.230 --> 00:00:51.310
and the smallest number.

00:00:51.310 --> 00:00:54.990
The smaller this is, that means
the tighter the range, just

00:00:54.990 --> 00:00:57.840
to use the word itself, of
the numbers actually are.

00:00:57.840 --> 00:00:59.110
So that's the range.

00:00:59.110 --> 00:01:01.820
The mid-range is
one way of thinking

00:01:01.820 --> 00:01:06.864
to some degree of kind of
central tendency, so mid-range.

00:01:06.864 --> 00:01:08.280
And what you do
with the mid-range

00:01:08.280 --> 00:01:12.690
is you take the average of the
largest number and the smallest

00:01:12.690 --> 00:01:13.570
number.

00:01:13.570 --> 00:01:14.940
So here we took the difference.

00:01:14.940 --> 00:01:15.730
That's the range.

00:01:15.730 --> 00:01:18.470
The mid-range would be the
average of these two numbers.

00:01:18.470 --> 00:01:21.364
So it would be 94 plus 65.

00:01:21.364 --> 00:01:22.780
And when I talk
about average, I'm

00:01:22.780 --> 00:01:26.330
talking about the
arithmetic mean over 2.

00:01:26.330 --> 00:01:27.540
So this is going to be what?

00:01:27.540 --> 00:01:30.750
90 plus 60 is 150.

00:01:30.750 --> 00:01:35.650
150, 4 plus 5 is 159.

00:01:35.650 --> 00:01:42.610
159 divided by 2 is equal
to-- 150 divided by 2 is 75.

00:01:42.610 --> 00:01:45.050
9 divided by 2 is 4 and 1/2.

00:01:45.050 --> 00:01:49.902
So this would be 79.5.

00:01:49.902 --> 00:01:51.360
So it's one kind
of way of thinking

00:01:51.360 --> 00:01:52.855
about the middle
of these numbers.

00:01:52.855 --> 00:01:54.730
Another way is obviously
the arithmetic mean,

00:01:54.730 --> 00:01:56.521
where you actually take
the arithmetic mean

00:01:56.521 --> 00:01:57.313
of everything here.

00:01:57.313 --> 00:01:58.979
Obviously, you could
also look at things

00:01:58.979 --> 00:02:00.280
like the median and the mode.

00:02:00.280 --> 00:02:02.810
So range and mid-range.


Stem-and-leaf plots

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.990
A statistician for
a basketball team

00:00:02.990 --> 00:00:04.800
tracked the number
of points that each

00:00:04.800 --> 00:00:07.600
of the 12 players on the
team had in one game.

00:00:07.600 --> 00:00:12.010
And then made a stem-and-leaf
plot to show the data.

00:00:12.010 --> 00:00:14.040
And sometimes it's
called a stem-plot.

00:00:14.040 --> 00:00:16.102
How many points
did the team score?

00:00:16.102 --> 00:00:18.310
And when you first look at
this plot right over here,

00:00:18.310 --> 00:00:19.950
it seems a little
hard to understand.

00:00:19.950 --> 00:00:22.444
Understand we have
0, 1, 2 under leaf

00:00:22.444 --> 00:00:23.860
you have all of
these digits here.

00:00:23.860 --> 00:00:26.190
How does this relate
to the number of points

00:00:26.190 --> 00:00:29.260
each student, or each
player, actually scored?

00:00:29.260 --> 00:00:31.750
And the way to interpret
a stem-and-leaf plot

00:00:31.750 --> 00:00:33.910
is the leafs contain--
at least the way

00:00:33.910 --> 00:00:38.470
that this statistician used it--
the leaf contains the smallest

00:00:38.470 --> 00:00:41.400
digit, or the ones digit,
in the number of points

00:00:41.400 --> 00:00:42.860
that each player scored.

00:00:42.860 --> 00:00:45.250
And the stem contains
the tens digits.

00:00:45.250 --> 00:00:48.870
And usually the leaf will
contain the rightmost digit,

00:00:48.870 --> 00:00:50.700
or the ones digit,
and then the stem

00:00:50.700 --> 00:00:52.840
will contain all of
the other digits.

00:00:52.840 --> 00:00:54.480
And what's useful
about this is it

00:00:54.480 --> 00:00:57.070
gives kind of a distribution
of where the players were.

00:00:57.070 --> 00:00:59.310
You see that most of
the players scored

00:00:59.310 --> 00:01:01.440
points that started with a 0.

00:01:01.440 --> 00:01:04.209
Then a few more scored
points that started with a 1.

00:01:04.209 --> 00:01:07.980
And then only one score scored
points to started with a 2,

00:01:07.980 --> 00:01:09.450
and it was actually 20 points.

00:01:09.450 --> 00:01:11.977
So I'm going to actually
write down all of this data

00:01:11.977 --> 00:01:13.810
in a way that maybe
you're a little bit more

00:01:13.810 --> 00:01:15.420
used to understanding it.

00:01:15.420 --> 00:01:18.920
So I'm going to write
the 0's in purple.

00:01:18.920 --> 00:01:23.020
So there's, let's see, 1,
2, 3, 4, 5, 6, 7 players

00:01:23.020 --> 00:01:25.070
had 0 as the first digit.

00:01:25.070 --> 00:01:28.462
So 1, 2, 3, 4, 5, 6, 7.

00:01:31.330 --> 00:01:33.040
I wrote seven 0's.

00:01:33.040 --> 00:01:37.090
And then this player also
had a 0 in his ones digit.

00:01:37.090 --> 00:01:39.800
This player, I'm going to
try to do all the colors,

00:01:39.800 --> 00:01:43.600
this player also had
a 0 in his ones digit.

00:01:43.600 --> 00:01:46.220
This player right here
had a 2 in his ones digit,

00:01:46.220 --> 00:01:48.710
so he scored a
total of 2 points.

00:01:48.710 --> 00:01:51.793
This player, let me
do orange, this player

00:01:51.793 --> 00:01:54.330
had 4 for his ones digit.

00:01:54.330 --> 00:01:58.580
This player had 7
for his ones digit.

00:01:58.580 --> 00:02:02.480
Then this player had
7 for his ones digit.

00:02:02.480 --> 00:02:05.960
And then, let me see, I'm
almost using all the colors,

00:02:05.960 --> 00:02:08.530
this player had 9
for his ones digit.

00:02:08.530 --> 00:02:11.460
So the way to read this is, you
had one player with 0 points.

00:02:11.460 --> 00:02:14.850
0, 2, 4, 7, 9 and 9.

00:02:14.850 --> 00:02:16.870
But you can see, and
it's kind of silly

00:02:16.870 --> 00:02:18.407
saying the zero
was a tens digit,

00:02:18.407 --> 00:02:19.990
you could have even
put a blank there.

00:02:19.990 --> 00:02:21.615
But the 0 lets us
know that they didn't

00:02:21.615 --> 00:02:24.700
score anything in
the tens place.

00:02:24.700 --> 00:02:27.940
But these are the actual
scores for those seven players.

00:02:27.940 --> 00:02:31.700
Now let's go to the next row
in the stem-and-leaf plot.

00:02:31.700 --> 00:02:34.060
So over here, all of
the digits start with,

00:02:34.060 --> 00:02:37.760
or all of the points start with
1, for each of the players.

00:02:37.760 --> 00:02:40.940
And there's four of them.

00:02:40.940 --> 00:02:44.710
So 1, 1, 1, and 1.

00:02:44.710 --> 00:02:46.600
And then we have this
player over here,

00:02:46.600 --> 00:02:51.250
his ones digit, or her
ones digit, is a 1.

00:02:51.250 --> 00:02:54.090
So this player,
this represents 11.

00:02:54.090 --> 00:02:57.150
1 in the tens place,
1 in the ones place.

00:02:57.150 --> 00:02:59.370
This player over
here also got 11.

00:02:59.370 --> 00:03:02.770
1 in the tens place,
1 in the ones place.

00:03:02.770 --> 00:03:06.080
This player, let me
do orange, this player

00:03:06.080 --> 00:03:08.350
has 3 in the ones place.

00:03:08.350 --> 00:03:10.250
So he or she scored 13 points.

00:03:10.250 --> 00:03:14.070
1 in the tens place,
3 in the ones place.

00:03:14.070 --> 00:03:15.770
13 points.

00:03:15.770 --> 00:03:18.020
And then I will
do this in purple.

00:03:18.020 --> 00:03:20.760
This player has 8
in their ones place.

00:03:20.760 --> 00:03:23.160
So he or she scored 18 points.

00:03:23.160 --> 00:03:25.710
1 in the tens place,
8 in the ones place.

00:03:25.710 --> 00:03:27.190
18 points.

00:03:27.190 --> 00:03:29.390
And then finally,
you have this player

00:03:29.390 --> 00:03:33.220
that has the tens digit is a 2.

00:03:33.220 --> 00:03:37.296
And then the ones digit is a 0.

00:03:37.296 --> 00:03:38.420
I'll circle that in yellow.

00:03:38.420 --> 00:03:40.160
It is a 0.

00:03:40.160 --> 00:03:43.030
So he or she scored 20 points.

00:03:43.030 --> 00:03:45.370
So looking at the
stem-and-leaf plot,

00:03:45.370 --> 00:03:48.179
we were able to extract out
all of the number of points

00:03:48.179 --> 00:03:49.470
that all of the players scored.

00:03:49.470 --> 00:03:51.261
And once again, what
was useful about this,

00:03:51.261 --> 00:03:54.110
is you see how many players
scored between 0 and 9 points,

00:03:54.110 --> 00:03:55.700
including 9 points.

00:03:55.700 --> 00:03:58.210
How many scored between
10 and 19 points,

00:03:58.210 --> 00:04:00.450
and then how many scored
20 points or over.

00:04:00.450 --> 00:04:02.620
And you see the distribution
right over here.

00:04:02.620 --> 00:04:04.360
But let's actually
answer the question

00:04:04.360 --> 00:04:06.360
that they're asking
us to answer.

00:04:06.360 --> 00:04:09.190
How many points
did the team score?

00:04:09.190 --> 00:04:10.920
So here we just
have to add up all

00:04:10.920 --> 00:04:12.530
of these numbers
right over here.

00:04:12.530 --> 00:04:16.230
So we're going to add up, I'll
start with the largest, so 20

00:04:16.230 --> 00:04:27.120
plus 18 plus 13 plus 11 plus
11-- 13, 11, 11-- plus 9 plus 7

00:04:27.120 --> 00:04:33.177
plus 7 again plus 4 plus 2.

00:04:33.177 --> 00:04:34.010
Did I do that right?

00:04:34.010 --> 00:04:38.570
We have two 11's, then a 9,
then two 7's, then a 4 then a 2,

00:04:38.570 --> 00:04:40.770
and then these two characters
didn't score anything.

00:04:40.770 --> 00:04:42.353
So let's add up all
of these together.

00:04:46.440 --> 00:04:53.110
So 0 plus 8 is 8, plus 3 is
11, plus 1 is 12, plus 1 is 13,

00:04:53.110 --> 00:05:03.490
plus 9 is 22, plus
7 is 27, 34, 38, 40.

00:05:03.490 --> 00:05:05.060
So that gets us to 40.

00:05:05.060 --> 00:05:06.590
Let me do that one more time.

00:05:06.590 --> 00:05:22.232
8, 11, 11, 12, 13, 22, 29, 29,
and then 29, 36, 40, and 42.

00:05:22.232 --> 00:05:23.940
Looks like I actually
might have messed--

00:05:23.940 --> 00:05:25.148
let me do that one more time.

00:05:25.148 --> 00:05:27.030
This is the hardest
part, adding these up.

00:05:27.030 --> 00:05:28.404
So let me try that
one last time.

00:05:28.404 --> 00:05:30.280
I'm just going to
state where my sum is.

00:05:30.280 --> 00:05:40.660
So 0, 8, add 3, 11, 12,
13, 22, 29, 36, 40, 42.

00:05:40.660 --> 00:05:42.660
So it's a good thing that
I double checked that.

00:05:42.660 --> 00:05:44.720
I made a mistake the first time.

00:05:44.720 --> 00:05:49.490
4 plus 2 is 6, 7, 8, 9, 10.

00:05:49.490 --> 00:05:52.200
So we get to 102 points.

00:05:52.200 --> 00:05:57.540
The team, in total,
scored 102 points.


Reading line graphs

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.050
An investment firm
creates a graph

00:00:03.050 --> 00:00:07.360
showing the performance of a
specific stock over 12 months.

00:00:07.360 --> 00:00:09.570
Over the course of
the year, is the price

00:00:09.570 --> 00:00:13.540
of the stock rising,
falling, or staying the same?

00:00:13.540 --> 00:00:16.480
So over this axis right over
here, the horizontal axis,

00:00:16.480 --> 00:00:17.680
they have month by month.

00:00:17.680 --> 00:00:20.210
And we move forward in time--
July, August, September,

00:00:20.210 --> 00:00:21.100
October.

00:00:21.100 --> 00:00:24.000
And in this axis, the vertical
axis, we have the price.

00:00:24.000 --> 00:00:29.410
So, for example, in July
the price of this stock

00:00:29.410 --> 00:00:31.620
was a little over $10.

00:00:31.620 --> 00:00:35.820
Then in August, it moved up
to-- it looks like around $11.

00:00:35.820 --> 00:00:37.930
And then we could keep
going month by month.

00:00:37.930 --> 00:00:39.560
And this type of
graph right over here

00:00:39.560 --> 00:00:41.380
is called a line
graph because you

00:00:41.380 --> 00:00:43.350
have the data points
for each month.

00:00:43.350 --> 00:00:45.200
And then we connected
them with a line.

00:00:45.200 --> 00:00:46.740
And the reason why we
connect them with a line

00:00:46.740 --> 00:00:48.739
is to really see if there's
some kind of a trend

00:00:48.739 --> 00:00:51.310
here to really show that
you have something that's

00:00:51.310 --> 00:00:53.770
moving from one
price to another.

00:00:53.770 --> 00:00:55.810
And so line graphs
tend to be used

00:00:55.810 --> 00:00:58.165
when you have something
that's changing over time.

00:00:58.165 --> 00:01:00.040
Now, with that out of
the way, let's actually

00:01:00.040 --> 00:01:01.410
answer their question.

00:01:01.410 --> 00:01:03.220
Over the course of
the year, is the price

00:01:03.220 --> 00:01:06.410
of the stock rising,
falling, or staying the same?

00:01:06.410 --> 00:01:08.640
So on a month-to-month
basis, you have, for example,

00:01:08.640 --> 00:01:10.680
from July to August,
the price went up.

00:01:10.680 --> 00:01:13.410
Then from August to September,
the price went down.

00:01:13.410 --> 00:01:14.940
Then it went up for two months.

00:01:14.940 --> 00:01:16.910
Then it went down for a month.

00:01:16.910 --> 00:01:18.640
Then it went up for
a couple more months.

00:01:18.640 --> 00:01:21.260
Then it went really up
from February to March,

00:01:21.260 --> 00:01:23.460
went all the way
up to almost $17.

00:01:23.460 --> 00:01:24.950
Then it went down again.

00:01:24.950 --> 00:01:27.590
And then it kept going up again.

00:01:27.590 --> 00:01:30.660
But they're asking us not
did it go up every month.

00:01:30.660 --> 00:01:34.400
They're saying over
the course of the year

00:01:34.400 --> 00:01:36.510
is the price of the
stock rising, falling,

00:01:36.510 --> 00:01:37.800
or staying the same?

00:01:37.800 --> 00:01:40.800
And if you go from July, which
is where our data starts right

00:01:40.800 --> 00:01:43.620
over here, our price
was around $10.

00:01:43.620 --> 00:01:47.070
And even though there were a
few months where it went down,

00:01:47.070 --> 00:01:50.250
the overall trend is that
the price is going up.

00:01:50.250 --> 00:01:52.410
The overall trend is
that the price going up.

00:01:52.410 --> 00:01:53.550
And you can even see that.

00:01:53.550 --> 00:01:55.080
In July, it was $10.

00:01:55.080 --> 00:01:57.800
And then by June
of the next year,

00:01:57.800 --> 00:01:59.440
it was approaching--
I don't know,

00:01:59.440 --> 00:02:01.800
it looks like it's
about a little

00:02:01.800 --> 00:02:04.440
over $16, maybe almost $17.

00:02:04.440 --> 00:02:05.880
So it actually
had gone up a lot.

00:02:05.880 --> 00:02:07.780
They don't give us
July of the next year.

00:02:07.780 --> 00:02:09.840
But the overall
trend is definitely

00:02:09.840 --> 00:02:11.986
the upwards direction
right over here.

00:02:11.986 --> 00:02:14.610
And you can see it just visually
by looking at this line graph.

00:02:14.610 --> 00:02:16.520
Even though there's a
few bumps that go down,

00:02:16.520 --> 00:02:19.538
the overall direction is upward.


Reading pie graphs (circle graphs)

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.500
For the past year,
a travel agency

00:00:02.500 --> 00:00:05.550
has collected data about the
number of individual tickets

00:00:05.550 --> 00:00:08.740
that it sells for its signature
product, a Mediterranean

00:00:08.740 --> 00:00:09.870
cruise.

00:00:09.870 --> 00:00:13.350
The monthly data on ticket
sales is shown below.

00:00:13.350 --> 00:00:16.700
What are the best and worst
months for cruise sales?

00:00:16.700 --> 00:00:18.550
So what they have
given us, this diagram.

00:00:18.550 --> 00:00:21.914
This is usually called
a pie chart or pie graph

00:00:21.914 --> 00:00:23.330
because it looks
like a pie that's

00:00:23.330 --> 00:00:24.980
sliced up into a
bunch of pieces.

00:00:24.980 --> 00:00:27.050
Sometimes this is
called a circle graph.

00:00:27.050 --> 00:00:29.117
But pie graph is
much more common.

00:00:29.117 --> 00:00:30.950
And then they say it's
monthly ticket sales.

00:00:30.950 --> 00:00:33.210
So each of these
slices represent

00:00:33.210 --> 00:00:34.940
the sales in a given month.

00:00:34.940 --> 00:00:37.420
So for example, this
blue slice over here

00:00:37.420 --> 00:00:40.060
represents the sales in January.

00:00:40.060 --> 00:00:41.980
And the way that a
pie chart is set up,

00:00:41.980 --> 00:00:45.100
each slice is bigger
or smaller, depending

00:00:45.100 --> 00:00:48.230
on what fraction of the
whole it represents.

00:00:48.230 --> 00:00:50.610
So for example, they're
telling us in January

00:00:50.610 --> 00:00:54.870
they sold 18% of the total
year's ticket sales in January.

00:00:54.870 --> 00:00:56.990
So if you add up all
of these percentages,

00:00:56.990 --> 00:00:59.300
it should add up to 100%.

00:00:59.300 --> 00:01:01.810
And not only do they tell
us that they sold 18%,

00:01:01.810 --> 00:01:03.910
but the slice of
this pie should be

00:01:03.910 --> 00:01:07.440
18% of the area
of the entire pie.

00:01:07.440 --> 00:01:09.480
It is literally 18% of the pie.

00:01:09.480 --> 00:01:11.060
If you were to eat
this slice, you

00:01:11.060 --> 00:01:13.382
would have eaten 18% of the pie.

00:01:13.382 --> 00:01:15.090
Now with that out of
the way, let's think

00:01:15.090 --> 00:01:16.480
about their questions.

00:01:16.480 --> 00:01:20.380
What are the best and the
worst months for cruise sales?

00:01:20.380 --> 00:01:23.270
So the best month is obviously
the month where they sell,

00:01:23.270 --> 00:01:26.240
where they have the largest
percentage of their tickets

00:01:26.240 --> 00:01:27.190
were sold.

00:01:27.190 --> 00:01:29.610
And actually, I started with
January, and if you just--

00:01:29.610 --> 00:01:31.660
and this is what's
neat about pie graphs.

00:01:31.660 --> 00:01:33.576
You wouldn't even have
to look at the numbers,

00:01:33.576 --> 00:01:35.750
January just jumps out as
the biggest slice of pie.

00:01:35.750 --> 00:01:36.920
If you didn't even
see the numbers,

00:01:36.920 --> 00:01:39.000
if you couldn't even read,
and you just looked at this

00:01:39.000 --> 00:01:41.070
and someone said what is
the largest slice of pie?

00:01:41.070 --> 00:01:42.778
You would immediately
say this is clearly

00:01:42.778 --> 00:01:44.640
the largest slice of
pie right over there.

00:01:44.640 --> 00:01:48.210
And so that is actually the
best month for cruise sales

00:01:48.210 --> 00:01:49.620
because they sold 18%.

00:01:49.620 --> 00:01:51.820
You see this 18%
is larger than all

00:01:51.820 --> 00:01:53.420
of the other
percentages over here.

00:01:53.420 --> 00:01:55.590
But it's clear, just by
looking at the graph,

00:01:55.590 --> 00:01:57.390
this is the largest slice.

00:01:57.390 --> 00:02:02.100
Now what's the worst month
for cruise ticket sales?

00:02:02.100 --> 00:02:05.560
Well, then we just have to
find the thinnest slice of pie.

00:02:05.560 --> 00:02:08.250
And if we look over
here, the slices of pie

00:02:08.250 --> 00:02:10.220
get pretty thin out down here.

00:02:10.220 --> 00:02:13.810
This is in the summer, in
June and July, and in May.

00:02:13.810 --> 00:02:16.450
But the smallest are
actually July and June.

00:02:16.450 --> 00:02:18.440
And this is where the
numbers become useful.

00:02:18.440 --> 00:02:22.527
Because when you just
eyeball it, you're not sure,

00:02:22.527 --> 00:02:23.860
hey, are these exactly the same?

00:02:23.860 --> 00:02:25.727
Or do they just look
exactly the same?

00:02:25.727 --> 00:02:27.560
And that's where these
numbers are valuable.

00:02:27.560 --> 00:02:29.184
And based on the data
they've given us,

00:02:29.184 --> 00:02:31.594
it looks like these are
tied for the worst in terms

00:02:31.594 --> 00:02:32.260
of ticket sales.

00:02:32.260 --> 00:02:36.350
In both of these months they
sell only 3% in each month.

00:02:36.350 --> 00:02:42.090
So the worst months for cruise
sales are July and June.

00:02:42.090 --> 00:02:44.470
July and June are
tied for the worst.

00:02:44.470 --> 00:02:48.640
The best is clearly January,
and then after January

00:02:48.640 --> 00:02:50.792
the next best-- they're
not really asking us that,

00:02:50.792 --> 00:02:52.750
but since we have the
pie chart in front of us,

00:02:52.750 --> 00:02:55.041
might as well ask ourselves
that, what's the next best?

00:02:55.041 --> 00:02:57.680
Well it looks like November is
slightly better than December.

00:02:57.680 --> 00:03:00.080
But in general,
it's pretty clear

00:03:00.080 --> 00:03:04.056
that the winter months do much
better in terms of cruise sales

00:03:04.056 --> 00:03:04.930
than these over here.

00:03:04.930 --> 00:03:08.324
And I guess people are looking
to get away from the cold.


Reading bar graphs

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:04.410
Based on the data below,
which student's score improved

00:00:04.410 --> 00:00:07.960
the most between the
midterm and final exams?

00:00:07.960 --> 00:00:10.000
And so they give us
this data in terms

00:00:10.000 --> 00:00:12.110
of a bar graph for each student.

00:00:12.110 --> 00:00:15.340
We actually have two bars
that show the midterm in blue

00:00:15.340 --> 00:00:16.790
and then the final exam in red.

00:00:16.790 --> 00:00:19.760
And they tell us that here--
midterm in blue and final exam.

00:00:19.760 --> 00:00:22.560
And sometimes this is called
a two-column bar graph

00:00:22.560 --> 00:00:25.850
because for each student here
you have two columns of data.

00:00:25.850 --> 00:00:27.990
So if you were to actually
look at the data itself,

00:00:27.990 --> 00:00:29.830
you have the midterm
data and then

00:00:29.830 --> 00:00:32.040
you have the final exam data.

00:00:32.040 --> 00:00:35.140
Now, they're asking us which
student's score improved

00:00:35.140 --> 00:00:38.980
the most between the
midterms and the final exams?

00:00:38.980 --> 00:00:40.944
So if we look at
Jasmine right over here,

00:00:40.944 --> 00:00:42.610
might as well start
with her since she's

00:00:42.610 --> 00:00:43.950
the most to the left.

00:00:43.950 --> 00:00:47.059
It looks like she definitely
did improve from the midterm

00:00:47.059 --> 00:00:47.600
to the final.

00:00:47.600 --> 00:00:49.600
It looks like in the
midterm, if I had to guess,

00:00:49.600 --> 00:00:51.220
this looks like about
a, I don't know,

00:00:51.220 --> 00:00:54.530
it looks like she got
maybe about a 72 or 73

00:00:54.530 --> 00:00:55.250
on the midterm.

00:00:55.250 --> 00:00:57.700
I'm just guessing because I
don't know the exact number.

00:00:57.700 --> 00:01:00.116
And then it looks like on the
final she got, I don't know,

00:01:00.116 --> 00:01:03.632
that looks like maybe a
77 or 78 approximately.

00:01:03.632 --> 00:01:05.590
So she improved a little
bit, about five points

00:01:05.590 --> 00:01:06.922
from the midterm to the final.

00:01:06.922 --> 00:01:08.880
The way that they've
given us this information,

00:01:08.880 --> 00:01:11.150
we don't know the exact
numbers because it's not

00:01:11.150 --> 00:01:13.194
super precise in terms
of marking off the bars,

00:01:13.194 --> 00:01:14.610
but hopefully it'll
become obvious

00:01:14.610 --> 00:01:17.000
when we look through
everyone's scores.

00:01:17.000 --> 00:01:18.030
Let's look at Jeff.

00:01:18.030 --> 00:01:22.960
So Jeff actually did worse from
the midterm to the final exam.

00:01:22.960 --> 00:01:25.560
He got looks like over
an 85 on the midterm

00:01:25.560 --> 00:01:29.150
and then on the final he
got about 84 or an 85.

00:01:29.150 --> 00:01:30.280
So he actually went down.

00:01:30.280 --> 00:01:31.520
So it's definitely not Jeff.

00:01:31.520 --> 00:01:33.300
He definitely did
not improve the most.

00:01:33.300 --> 00:01:35.000
He actually went down.

00:01:35.000 --> 00:01:36.790
Next, if you think
about Nevin, it

00:01:36.790 --> 00:01:40.990
looks like Nevin actually
improved about the same amount

00:01:40.990 --> 00:01:41.950
as Jasmine.

00:01:41.950 --> 00:01:45.850
On both scores, I don't know if
Nevin is a boy or girl's name,

00:01:45.850 --> 00:01:49.130
this person on both tests, this
person did better than Jasmine,

00:01:49.130 --> 00:01:51.680
but it looks like the actual
improvement is about the same.

00:01:51.680 --> 00:01:54.520
Looks like they went from
about 83 to, I don't know,

00:01:54.520 --> 00:01:55.850
about an 88.

00:01:55.850 --> 00:01:57.730
I'm just estimating
it, just trying

00:01:57.730 --> 00:02:00.640
to look at this axis
right over there

00:02:00.640 --> 00:02:02.997
and estimating what
those scores are.

00:02:02.997 --> 00:02:05.580
So Nevin and Jasmine right now,
based on the three we've seen,

00:02:05.580 --> 00:02:07.250
are tied for the lead.

00:02:07.250 --> 00:02:08.870
Now let's look at Alejandra.

00:02:08.870 --> 00:02:11.430
Now Alejandra-- OK,
so this jumps out.

00:02:11.430 --> 00:02:15.110
She definitely improved
dramatically from the midterm

00:02:15.110 --> 00:02:16.140
to the final exam.

00:02:16.140 --> 00:02:19.660
It looks like on the midterm
she maybe got an 81 or an 82.

00:02:19.660 --> 00:02:22.330
So maybe this was an 82
she got on the midterm,

00:02:22.330 --> 00:02:25.879
and then on the final, it
looks she got about a 95

00:02:25.879 --> 00:02:26.420
on the final.

00:02:26.420 --> 00:02:27.950
So there's dramatic improvement.

00:02:27.950 --> 00:02:30.900
So right now Alejandra
is the leading contender

00:02:30.900 --> 00:02:34.690
for most improved from
the midterm to the final.

00:02:34.690 --> 00:02:37.240
And then finally
Marta, right over here,

00:02:37.240 --> 00:02:41.120
it looks like she actually
got worse from the midterm

00:02:41.120 --> 00:02:41.750
to the final.

00:02:41.750 --> 00:02:44.630
She scored in the mid-90s
in the midterm and then low

00:02:44.630 --> 00:02:45.710
90s in the final.

00:02:45.710 --> 00:02:48.010
So she's definitely
not the most improved.

00:02:48.010 --> 00:02:51.050
So the winner here is Alejandra.

00:02:51.050 --> 00:02:55.750
Alejandra is the most
improved between the midterm

00:02:55.750 --> 00:02:57.991
and the final exam.


Reading pictographs

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.680
According to the
pictograph below,

00:00:03.680 --> 00:00:07.870
how many survey respondents
have type O positive blood?

00:00:07.870 --> 00:00:10.700
How many have O negative blood?

00:00:10.700 --> 00:00:12.750
So a pictograph is
really just a way

00:00:12.750 --> 00:00:14.990
of representing
data with pictures

00:00:14.990 --> 00:00:16.970
that are somehow
related to the data.

00:00:16.970 --> 00:00:18.970
So in this case, they
gave us little pictures

00:00:18.970 --> 00:00:21.870
of, I'm assuming, blood
drops right over here.

00:00:21.870 --> 00:00:26.470
And then they tell us that each
blood drop in this pictograph

00:00:26.470 --> 00:00:27.944
represents 8 people.

00:00:27.944 --> 00:00:30.360
So you could kind of view that
as a scale of these graphs.

00:00:30.360 --> 00:00:31.730
Each of these say 8 people.

00:00:31.730 --> 00:00:34.230
So, for example, if you say how
many people have A positive?

00:00:34.230 --> 00:00:39.570
It would be 1, 2, 3,
4, 5, 6, 7 blood drops.

00:00:39.570 --> 00:00:42.580
But each of those blood
drops represent 8 people,

00:00:42.580 --> 00:00:46.230
so it would be 56 people
have type A positive.

00:00:46.230 --> 00:00:47.730
But let's answer
the actual question

00:00:47.730 --> 00:00:48.920
that they're asking us.

00:00:48.920 --> 00:00:53.080
How many survey respondents
have type O positive?

00:00:53.080 --> 00:00:57.560
So this is O, and then
we care about O positive.

00:00:57.560 --> 00:01:00.532
So we have 1 blood drop, 2, 3.

00:01:00.532 --> 00:01:02.490
I'm going to do this in
a new, different color.

00:01:02.490 --> 00:01:09.000
We have 1, 2, 3, 4, 5, 6, 7, 8.

00:01:09.000 --> 00:01:11.880
So we have 8 drops.

00:01:11.880 --> 00:01:14.360
I'll put those in quotes because
they're pictures of drops.

00:01:14.360 --> 00:01:18.780
And then the scale is 8 people.

00:01:18.780 --> 00:01:20.170
Let me write it this way.

00:01:20.170 --> 00:01:21.940
Times 8 people per drop.

00:01:26.200 --> 00:01:28.370
And so 8 times 8-- and
actually even the drops,

00:01:28.370 --> 00:01:29.869
you could view them
as canceling out

00:01:29.869 --> 00:01:32.210
if you view them as
units, so drops, drops.

00:01:32.210 --> 00:01:36.412
8 times 8 is equal to 64 people.

00:01:36.412 --> 00:01:38.370
So they could have written
literally the number

00:01:38.370 --> 00:01:39.550
64 right over here.

00:01:39.550 --> 00:01:43.540
64 people have type
O positive blood.

00:01:43.540 --> 00:01:46.060
Now let's think about
the O negative case,

00:01:46.060 --> 00:01:47.260
O negative blood.

00:01:47.260 --> 00:01:50.480
Well, this is O. And then
within the blood group O,

00:01:50.480 --> 00:01:52.420
this is O negative.

00:01:52.420 --> 00:01:54.510
And we have 1 drops, 2 drops.

00:01:54.510 --> 00:02:00.220
So we have 2 drops
times 8 people per drop.

00:02:03.882 --> 00:02:06.420
And so 2 times 8, each
of these represent 8.

00:02:06.420 --> 00:02:12.030
So 8 and then 16, or 2 times
8 is equal to 16 people.

00:02:12.030 --> 00:02:17.060
So 16 of the survey respondents,
16 have type O negative.

00:02:17.060 --> 00:02:20.328
64 have type O positive.


Volume of a sphere

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:06.650
Find the volume of a sphere with
a diameter of 14 centimeters.

00:00:06.650 --> 00:00:09.520
So if I have a sphere-- so
this isn't just a circle,

00:00:09.520 --> 00:00:10.460
this is a sphere.

00:00:10.460 --> 00:00:12.350
You could view it as
a globe of some kind.

00:00:12.350 --> 00:00:14.183
So I'm going to shade
it a little bit so you

00:00:14.183 --> 00:00:15.830
can tell that it's
three-dimensional.

00:00:15.830 --> 00:00:17.320
They're giving us the diameter.

00:00:17.320 --> 00:00:19.120
So if we go from one
side of the sphere

00:00:19.120 --> 00:00:20.890
straight through
the center of it.

00:00:20.890 --> 00:00:23.430
So we're imagining that we
can see through the sphere.

00:00:23.430 --> 00:00:25.180
And we go straight
through the centimeter,

00:00:25.180 --> 00:00:28.960
that distance right over
there is 14 centimeters.

00:00:28.960 --> 00:00:32.810
Now, to find the volume of a
sphere-- and we've proved this,

00:00:32.810 --> 00:00:36.200
or you will see a proof for this
later when you learn calculus.

00:00:36.200 --> 00:00:38.480
But the formula for
the volume of a sphere

00:00:38.480 --> 00:00:43.880
is volume is equal
to 4/3 pi r cubed,

00:00:43.880 --> 00:00:47.411
where r is the
radius of the sphere.

00:00:47.411 --> 00:00:48.785
So they've given
us the diameter.

00:00:48.785 --> 00:00:54.280
And just like for circles,
the radius of the sphere

00:00:54.280 --> 00:00:56.020
is half of the diameter.

00:00:56.020 --> 00:01:00.440
So in this example, our radius
is going to be 7 centimeters.

00:01:00.440 --> 00:01:02.650
And in fact, the sphere
itself is the set

00:01:02.650 --> 00:01:05.670
of all points in three
dimensions that is exactly

00:01:05.670 --> 00:01:07.704
the radius away from the center.

00:01:07.704 --> 00:01:09.370
But with that out of
the way, let's just

00:01:09.370 --> 00:01:12.950
apply this radius being 7
centimeters to this formula

00:01:12.950 --> 00:01:14.140
right over here.

00:01:14.140 --> 00:01:16.540
So we're going to
have a volume is

00:01:16.540 --> 00:01:22.920
equal to 4/3 pi times 7
centimeters to the third power.

00:01:22.920 --> 00:01:25.280
So I'll do that in
that pink color.

00:01:25.280 --> 00:01:31.090
So times 7 centimeters
to the third power.

00:01:31.090 --> 00:01:33.430
And since it
already involves pi,

00:01:33.430 --> 00:01:35.880
and you could
approximate pi with 3.14.

00:01:35.880 --> 00:01:39.184
Some people even
approximate it with 22/7.

00:01:39.184 --> 00:01:41.100
But we'll actually just
get the calculator out

00:01:41.100 --> 00:01:43.590
to get the exact
value for this volume.

00:01:43.590 --> 00:01:47.870
So this is going to
be-- so my volume is

00:01:47.870 --> 00:01:51.940
going to be 4 divided by 3.

00:01:51.940 --> 00:01:53.900
And then I don't want
to just put a pi there,

00:01:53.900 --> 00:01:56.640
because that might interpret
it as 4 divided by 3 pi.

00:01:56.640 --> 00:02:07.420
So 4 divided by 3 times pi,
times 7 to the third power.

00:02:07.420 --> 00:02:09.370
In order of operations,
it'll do the exponent

00:02:09.370 --> 00:02:10.786
before it does the
multiplication,

00:02:10.786 --> 00:02:12.680
so this should work out.

00:02:12.680 --> 00:02:15.650
And the units are going to be
in centimeters cubed or cubic

00:02:15.650 --> 00:02:17.410
centimeters.

00:02:17.410 --> 00:02:19.340
So we get 1,436.

00:02:19.340 --> 00:02:21.010
They don't tell us
what to round it to.

00:02:21.010 --> 00:02:25.590
So I'll just round it to
the nearest 10th-- 1,436.8.

00:02:25.590 --> 00:02:32.750
So this is equal to
1,436.8 centimeters cubed.

00:02:32.750 --> 00:02:34.580
And we're done.


Perimeter and area of a non-standard polygon

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:04.280
Find the area and
perimeter of the polygon.

00:00:04.280 --> 00:00:06.770
So let's start with
the area first.

00:00:06.770 --> 00:00:08.580
So the area of this
polygon-- there's

00:00:08.580 --> 00:00:09.980
kind of two parts of this.

00:00:09.980 --> 00:00:12.580
First, you have this part
that's kind of rectangular,

00:00:12.580 --> 00:00:15.980
or it is rectangular,
this part right over here.

00:00:15.980 --> 00:00:17.710
And that area is
pretty straightforward.

00:00:17.710 --> 00:00:20.140
It's just going to
be base times height.

00:00:20.140 --> 00:00:25.490
So area's going to be 8 times
4 for the rectangular part.

00:00:25.490 --> 00:00:28.050
And then we have this
triangular part up here.

00:00:28.050 --> 00:00:30.410
So we have this area up here.

00:00:30.410 --> 00:00:35.065
And for a triangle, the area
is base times height times 1/2.

00:00:35.065 --> 00:00:36.690
And that actually
makes a lot of sense.

00:00:36.690 --> 00:00:38.890
Because if you just
multiplied base times height,

00:00:38.890 --> 00:00:41.020
you would get this entire area.

00:00:41.020 --> 00:00:43.190
You would get the area
of that entire rectangle.

00:00:43.190 --> 00:00:45.399
And you see that the triangle
is exactly 1/2 of it.

00:00:45.399 --> 00:00:46.940
If you took this
part of the triangle

00:00:46.940 --> 00:00:49.100
and you flipped it over,
you'd fill up that space.

00:00:49.100 --> 00:00:50.641
If you took this
part of the triangle

00:00:50.641 --> 00:00:52.770
and you flipped it over,
you'd fill up that space.

00:00:52.770 --> 00:00:56.280
So the triangle's area is
1/2 of the triangle's base

00:00:56.280 --> 00:00:57.940
times the triangle's height.

00:00:57.940 --> 00:01:01.400
So plus 1/2 times
the triangle's base,

00:01:01.400 --> 00:01:04.569
which is 8 inches, times
the triangle's height, which

00:01:04.569 --> 00:01:06.620
is 4 inches.

00:01:06.620 --> 00:01:08.440
And so let's just calculate it.

00:01:08.440 --> 00:01:13.710
This gives us 32
plus-- oh, sorry.

00:01:13.710 --> 00:01:14.784
That's not 8 times 4.

00:01:14.784 --> 00:01:15.950
I don't want to confuse you.

00:01:15.950 --> 00:01:18.260
The triangle's height is 3.

00:01:18.260 --> 00:01:21.160
8 times 3, right there.

00:01:21.160 --> 00:01:22.740
That's the triangle's height.

00:01:22.740 --> 00:01:25.270
So once again, let's go
back and calculate it.

00:01:25.270 --> 00:01:31.455
So this is going to be 32
plus-- 1/2 times 8 is 4.

00:01:31.455 --> 00:01:33.900
4 times 3 is 12.

00:01:33.900 --> 00:01:39.030
And so our area for our
shape is going to be 44.

00:01:39.030 --> 00:01:40.380
Now let's do the perimeter.

00:01:40.380 --> 00:01:42.560
The perimeter-- we just
have to figure out what's

00:01:42.560 --> 00:01:44.125
the sum of the sides.

00:01:44.125 --> 00:01:45.500
How long of a
fence would we have

00:01:45.500 --> 00:01:48.000
to build if we wanted to make
it around this shape, right

00:01:48.000 --> 00:01:50.280
along the sides of this shape?

00:01:50.280 --> 00:01:55.210
So the perimeter-- I'll
just write P for perimeter.

00:01:55.210 --> 00:02:05.260
It's going to be equal to 8 plus
4 plus 5 plus this 5, this edge

00:02:05.260 --> 00:02:08.350
right over here, plus--
I didn't write that down.

00:02:08.350 --> 00:02:12.300
So I have two 5's plus
this 4 right over here.

00:02:12.300 --> 00:02:15.540
So you have 8 plus 4 is 12.

00:02:15.540 --> 00:02:19.630
12 plus 10-- well, I'll
just go one step at a time.

00:02:19.630 --> 00:02:21.956
12 plus 5 is 17.

00:02:21.956 --> 00:02:23.684
17 plus 5 is 22.

00:02:23.684 --> 00:02:26.740
22 plus 4 is 26.

00:02:26.740 --> 00:02:28.940
So the perimeter is 26 inches.

00:02:28.940 --> 00:02:30.670
And let me get the
units right, too.

00:02:30.670 --> 00:02:33.440
Because over here, I'm
multiplying 8 inches

00:02:33.440 --> 00:02:34.530
by 4 inches.

00:02:34.530 --> 00:02:36.260
So you get square inches.

00:02:36.260 --> 00:02:39.240
8 inches by 3 inches, so
you get square inches again.

00:02:39.240 --> 00:02:41.980
So this is going to
be square inches.

00:02:41.980 --> 00:02:43.960
So area is 44 square inches.

00:02:43.960 --> 00:02:45.836
Perimeter is 26 inches.

00:02:45.836 --> 00:02:47.210
And that makes
sense because this

00:02:47.210 --> 00:02:48.584
is a two-dimensional
measurement.

00:02:48.584 --> 00:02:50.900
It's measuring something
in two-dimensional space,

00:02:50.900 --> 00:02:52.490
so you get a
two-dimensional unit.

00:02:52.490 --> 00:02:54.120
This is a one-dimensional
measurement.

00:02:54.120 --> 00:02:56.710
It's only asking you,
essentially, how long would

00:02:56.710 --> 00:02:59.520
a string have to be to
go around this thing.

00:02:59.520 --> 00:03:02.560
And so that's why you get
one-dimensional units.


Identifying rays

WEBVTT
Kind: captions
Language: en

00:00:02.067 --> 00:00:05.800
Identify all the rays shown in the image below.

00:00:05.800 --> 00:00:07.800
and this is a reminder what a ray is.

00:00:07.800 --> 00:00:13.000
A ray start at some point and then goes on forever in some direction.

00:00:13.000 --> 00:00:17.533
In order to find a ray you need that point that you're starting off on

00:00:17.533 --> 00:00:19.800
so let's that point over there is called X

00:00:19.800 --> 00:00:22.600
and then you need another point that sits on the ray

00:00:24.667 --> 00:00:26.533
and the ray is just keep going beyond, we

00:00:26.533 --> 00:00:35.533
will name that point as "Y" and we will call the ray "XY"

00:00:35.533 --> 00:00:41.600
It starts in "X" and keeps going in the direction of "y"

00:00:41.600 --> 00:00:45.533
for ever, it crosses "y" and keeps going further

00:00:45.533 --> 00:00:50.667
we need the second point to specify the direction

00:00:50.667 --> 00:00:56.200
in which the ray is going.

00:00:56.200 --> 00:01:00.867
So, lets identify all the rays shown in the image

00:01:00.867 --> 00:01:05.200
below, we can start anywhere, we will start at

00:01:05.200 --> 00:01:13.800
point J, the only line segment we have starting from J

00:01:13.800 --> 00:01:21.667
is JH going up, goes upto H and keeps on going

00:01:21.667 --> 00:01:27.467
in that direction beyond H, ray JH, starting from J

00:01:27.467 --> 00:01:31.267
going through H and going beyond it forever

00:01:31.267 --> 00:01:37.800
now if we go to H, there is no ray HJ as the line ends

00:01:37.800 --> 00:01:41.267
in J and does not keep going beyond J, there is no

00:01:41.267 --> 00:01:47.333
ray H as it is just one point, just usiing one point,

00:01:47.333 --> 00:01:51.600
we cannot say it as a ray. now looking at our diagram

00:01:51.600 --> 00:02:08.067
the only ray is JH. Now let us look at the other points.

00:02:08.067 --> 00:02:15.067
If we look at point C, once again there is no point

00:02:15.067 --> 00:02:21.533
after C to specify it as a ray, we can have a ray CE,

00:02:21.533 --> 00:02:27.600
starts at C goes through E and goes on for ever,

00:02:27.600 --> 00:02:32.600
you can also have a ray starting at C, going through F

00:02:32.600 --> 00:02:37.933
and going on forever, CE & CF are the same rays

00:02:37.933 --> 00:02:49.133
as F sits on ray CE and E sits of ray CF,

00:02:49.133 --> 00:03:02.933
so CE & CF are the same rays,

00:03:02.933 --> 00:03:08.067
now lets think about what we can do from point E,

00:03:08.067 --> 00:03:12.333
We can start at point E , go in the direction of C

00:03:12.333 --> 00:03:17.600
and go beyond C, so it is a ray, ray EC

00:03:17.600 --> 00:03:22.200
you could start in E and go in the direction of A and go

00:03:22.200 --> 00:03:26.067
beyond A, so EA is a ray, and we can start at E and

00:03:26.067 --> 00:03:30.200
go in the direction of F and go beyond F, that is ray EF

00:03:30.200 --> 00:03:39.267
ray EF and ray CF are different as the

00:03:39.267 --> 00:03:46.200
starting points are different

00:03:46.200 --> 00:03:53.667
Now lets go to point F. To the left of point F

00:03:53.667 --> 00:04:00.667
there is no other point. We can look at the right,

00:04:00.667 --> 00:04:02.533
we can have ray FE, start at F go through E and

00:04:02.533 --> 00:04:07.800
beyond E, you can have ray FC, start at F

00:04:07.800 --> 00:04:17.667
go through C and beyond C that is teh same thing as FE

00:04:17.667 --> 00:04:21.933
ray FE and ray FC are the same as the point E

00:04:21.933 --> 00:04:33.200
is on ray FC, then finally we have not focussed on point A

00:04:33.200 --> 00:04:39.133
you may think there is ray AE, but the line does not

00:04:39.133 --> 00:04:44.800
go beyond E, so it is not a ray, to the top of A

00:04:44.800 --> 00:04:48.000
there is no other point, so there is no ray there either

00:04:48.000 --> 00:04:55.933
that is all the rays based on the points specified.

00:04:55.933 --> 00:04:59.067
If they had given us a point over there, we could have

00:04:59.067 --> 00:05:32.133
had other rays,

00:05:32.133 --> 99:59:59.999
There are 6 rays in this problem


Perimeter of a parallelogram

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.307
Find the perimeter
of the parallelogram.

00:00:03.307 --> 00:00:05.140
And we have a parallelogram
right over here.

00:00:05.140 --> 00:00:07.230
Opposite sides are parallel.

00:00:07.230 --> 00:00:10.500
That side is parallel to
that side right over here.

00:00:10.500 --> 00:00:12.830
This side is parallel
to that side over here.

00:00:12.830 --> 00:00:14.760
And also in a parallelogram,
opposite sides

00:00:14.760 --> 00:00:15.990
have the same length.

00:00:15.990 --> 00:00:18.600
Now, we have to worry about
the perimeter here, which

00:00:18.600 --> 00:00:21.290
essentially just says that
if we had a field the size

00:00:21.290 --> 00:00:22.930
or the shape of
this parallelogram,

00:00:22.930 --> 00:00:26.010
and we wanted to make a
fence around the field that

00:00:26.010 --> 00:00:28.230
was the shape of this
parallelogram, how long would

00:00:28.230 --> 00:00:29.670
that fence have to be?

00:00:29.670 --> 00:00:32.200
And obviously, it
would be the sum

00:00:32.200 --> 00:00:34.040
of the lengths of the sides.

00:00:34.040 --> 00:00:35.510
So we wouldn't
even have to worry

00:00:35.510 --> 00:00:36.770
about this thing
right over here.

00:00:36.770 --> 00:00:38.770
We could ignore it for
the sake of that problem.

00:00:38.770 --> 00:00:40.756
We don't care about
the actual height,

00:00:40.756 --> 00:00:42.130
or this altitude
right over here.

00:00:42.130 --> 00:00:44.090
We just care about the
lengths of its sides.

00:00:44.090 --> 00:00:46.253
So the perimeter of
this parallelogram

00:00:46.253 --> 00:00:50.560
is going to be 12
inches plus 8 inches

00:00:50.560 --> 00:00:53.787
plus 12 inches plus 8 inches.

00:00:58.530 --> 00:00:59.680
12 plus 8 is 20.

00:00:59.680 --> 00:01:01.910
12 plus 8 is 20.

00:01:01.910 --> 00:01:06.850
So this is all going
to add up to 40 inches.

00:01:06.850 --> 00:01:07.990
And we're done.

00:01:07.990 --> 00:01:09.570
I guess this is a
very small field,

00:01:09.570 --> 00:01:10.970
because it's only 8 inches.

00:01:10.970 --> 00:01:13.530
I guess it would be a
little plot of land,

00:01:13.530 --> 00:01:15.600
a little parallelogram
plot of land.

00:01:15.600 --> 00:01:16.660
Anyway, we're done.

00:01:16.660 --> 00:01:19.370
Its perimeter is 40 inches.


Identifying Complementary and Supplementary Angles

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:02.760
Identify complementary and supplementary

00:00:03.120 --> 00:00:05.900
angles in the image below. So just as a reminder

00:00:06.780 --> 00:00:11.480
complementary angles are two angles
that add up to 90 degrees and

00:00:11.840 --> 00:00:15.120
supplementary angles, supplementary angles add up to

00:00:15.850 --> 00:00:20.430
180 degrees.  So let's look at this diagram.
First let's look for the complementary,

00:00:21.640 --> 00:00:28.709
complementary angles. So angles that add up to 90 degrees and so they drew this right angle right over here.

00:00:28.709 --> 00:00:30.500
So we know that angle FAH

00:00:30.500 --> 00:00:33.580
is a right angle, because they did this little
half box right over there,

00:00:33.630 --> 00:00:35.559
and it's made up of two adjacent angles

00:00:35.560 --> 00:00:40.900
and these two adjacent angles make up a 90 degree angle so they must add up to 90 degrees.

00:00:40.900 --> 00:00:44.140
so the two angles-- angle FAC

00:00:45.540 --> 00:00:48.660
Angle FAC

00:00:49.180 --> 00:00:51.060
and Angle

00:00:51.060 --> 00:00:52.660
CAH and

00:00:52.660 --> 00:00:58.220
Angle CAH, let me do it in a different color since I already used that) and angle CAH and

00:00:58.980 --> 00:01:01.820
Angle CAH are

00:01:01.920 --> 00:01:04.000
complementary because the measures of these two angles

00:01:04.180 --> 00:01:10.640
must add up to 180 degrees are-- sorry, must add up to this 90 degrees right over here.

00:01:10.640 --> 00:01:13.100
And I think that's all the complementary angles.

00:01:13.100 --> 00:01:16.520
Just these two, they add up to this
right angle, to this 90 degrees.

00:01:16.760 --> 00:01:20.600
Now let's think about the supplementary angles, so

00:01:21.549 --> 00:01:23.439
supplementary,

00:01:23.440 --> 00:01:26.940
supplementary angles.  They add up to 180 degrees.

00:01:27.540 --> 00:01:34.640
So you have angle FAC, or we could call it CAF.
It's the same angle. So angle CAF,

00:01:35.290 --> 00:01:40.559
and then this angle out here, FAB.

00:01:40.990 --> 00:01:45.547
When they're adjacent to each other
and they clearly form a straight angle

00:01:45.547 --> 00:01:48.750
which is 180 degrees
when you add their two measures.

00:01:48.820 --> 00:01:50.770
So these two are clearly

00:01:50.770 --> 00:01:52.360
supplementary and

00:01:52.360 --> 00:01:56.340
angle FAB.   FAB.

00:01:56.340 --> 00:02:01.760
CAF and FAB together definitely add up to 180 degrees

00:02:01.760 --> 00:02:05.060
so they're supplementary.  And then the other one--

00:02:05.060 --> 00:02:06.280
we can do angle CAH,

00:02:07.200 --> 00:02:09.480
angle CAH,

00:02:09.480 --> 00:02:15.100
which is this right over here,
that plus HAB clearly add up.

00:02:15.100 --> 00:02:16.500
They form-- They're adjacent.

00:02:16.500 --> 00:02:17.920
They form a straight angle.

00:02:18.100 --> 00:02:21.960
So they are clearly adding up to 180 degrees.
So they are also supplementary.

00:02:23.020 --> 00:02:24.520
CAH and

00:02:24.520 --> 00:02:27.600
angle HAB.

00:02:27.760 --> 00:02:30.800
And we're done.
These two angles add up to 90.

00:02:31.020 --> 00:02:34.780
These two add up to 180,
these two add up to 180.


Identifying parallel and perpendicular lines

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.470
Identify all sets of
parallel and perpendicular

00:00:03.470 --> 00:00:05.630
lines in the image below.

00:00:05.630 --> 00:00:08.160
So let's start with
the parallel lines.

00:00:08.160 --> 00:00:11.850
And just as a
reminder, two lines

00:00:11.850 --> 00:00:14.382
are parallel if they're
in the same plane,

00:00:14.382 --> 00:00:16.590
and all of these lines are
clearly in the same plane.

00:00:16.590 --> 00:00:18.006
They're in the
plane of the screen

00:00:18.006 --> 00:00:19.200
you're viewing right now.

00:00:19.200 --> 00:00:21.560
But they are two lines that
are in the same plane that

00:00:21.560 --> 00:00:23.400
never intersect.

00:00:23.400 --> 00:00:26.240
And one way to verify,
because you can sometimes--

00:00:26.240 --> 00:00:28.000
it looks like two
lines won't intersect,

00:00:28.000 --> 00:00:30.633
but you can't just always
assume based on how it looks.

00:00:30.633 --> 00:00:32.299
You really have to
have some information

00:00:32.299 --> 00:00:34.900
given in the diagram or
the problem that tells you

00:00:34.900 --> 00:00:36.660
that they are
definitely parallel,

00:00:36.660 --> 00:00:38.850
that they're definitely
never going to intersect.

00:00:38.850 --> 00:00:40.520
And one of those
pieces of information

00:00:40.520 --> 00:00:42.730
which they give
right over here is

00:00:42.730 --> 00:00:45.680
that they show that
line ST and line

00:00:45.680 --> 00:00:51.229
UV, they both intersect line
CD at the exact same angle,

00:00:51.229 --> 00:00:52.270
at this angle right here.

00:00:52.270 --> 00:00:54.370
And in particular,
it's at a right angle.

00:00:54.370 --> 00:00:57.340
And if you have two lines
that intersect a third line

00:00:57.340 --> 00:01:00.317
at the same angle--
so these are actually

00:01:00.317 --> 00:01:02.400
called corresponding angles
and they're the same--

00:01:02.400 --> 00:01:05.489
if you have two of these
corresponding angles the same,

00:01:05.489 --> 00:01:07.280
then these two
lines are parallel.

00:01:07.280 --> 00:01:10.220
So line ST is
parallel to line UV.

00:01:10.220 --> 00:01:11.830
And we can write it like this.

00:01:11.830 --> 00:01:17.650
Line ST, we put the arrows
on each end of that top bar

00:01:17.650 --> 00:01:20.470
to say that this is a line,
not just a line segment.

00:01:20.470 --> 00:01:26.850
Line ST is parallel to line UV.

00:01:26.850 --> 00:01:29.830
And I think that's the
only set of parallel lines

00:01:29.830 --> 00:01:31.260
in this diagram.

00:01:31.260 --> 00:01:33.760
Yep.

00:01:33.760 --> 00:01:36.180
Now let's think about
perpendicular lines.

00:01:36.180 --> 00:01:40.810
Perpendicular lines
are lines that

00:01:40.810 --> 00:01:43.830
intersect at a 90-degree angle.

00:01:43.830 --> 00:01:48.030
So, for example, line ST is
perpendicular to line CD.

00:01:48.030 --> 00:01:53.650
So line ST is
perpendicular to line CD.

00:01:53.650 --> 00:01:55.650
And we know that they
intersect at a right angle

00:01:55.650 --> 00:01:57.180
or at a 90-degree angle
because they gave us

00:01:57.180 --> 00:01:58.930
this little box here
which literally means

00:01:58.930 --> 00:02:01.910
that the measure of this
angle is 90 degrees.

00:02:01.910 --> 00:02:08.490
By the exact same argument, line
the UV is perpendicular to CD.

00:02:08.490 --> 00:02:11.710
Let me make sure I
specified these as lines.

00:02:11.710 --> 00:02:15.705
Line UV is perpendicular to CD.

00:02:15.705 --> 00:02:19.920
So I did UV, ST, they're
perpendicular to CD.

00:02:19.920 --> 00:02:22.739
And then after that, the
only other information

00:02:22.739 --> 00:02:25.280
where they definitely tell us
that two lines are intersecting

00:02:25.280 --> 00:02:28.730
at right angles
are line AB and WX.

00:02:28.730 --> 00:02:38.900
So AB is definitely
perpendicular to WX, line WX.

00:02:38.900 --> 00:02:41.760
And I think we are done.

00:02:41.760 --> 00:02:45.930
And one thing to think
about, AB and CD,

00:02:45.930 --> 00:02:47.930
well, they don't even
intersect in this diagram.

00:02:47.930 --> 00:02:49.620
So you can't make any
comment about perpendicular,

00:02:49.620 --> 00:02:51.080
but they're definitely
not parallel.

00:02:51.080 --> 00:02:52.440
You could even
imagine that it looks

00:02:52.440 --> 00:02:53.840
like they're about to intersect.

00:02:53.840 --> 00:02:56.200
And they give us no
information that they intersect

00:02:56.200 --> 00:02:58.930
the same lines at
the same angle.

00:02:58.930 --> 00:03:01.485
So if somehow they told us that
this is a right angle, even

00:03:01.485 --> 00:03:03.610
though it doesn't look
anything like a right angle,

00:03:03.610 --> 00:03:06.085
then we would have to
suspend our judgment based

00:03:06.085 --> 00:03:07.460
on how it actually
looks and say,

00:03:07.460 --> 00:03:13.475
oh, I guess maybe those
things are perpendicular,

00:03:13.475 --> 00:03:15.172
or maybe these two
things are parallel.

00:03:15.172 --> 00:03:16.380
But they didn't tell us that.

00:03:16.380 --> 00:03:17.796
And that would
actually be bizarre

00:03:17.796 --> 00:03:19.272
because it looks
so not parallel.

00:03:19.272 --> 00:03:20.730
And actually then
this would end up

00:03:20.730 --> 00:03:24.654
being parallel to other things
as well if that was done.

00:03:24.654 --> 00:03:26.320
It's a good thing
that wasn't because it

00:03:26.320 --> 00:03:27.550
would look very strange.

00:03:27.550 --> 00:03:29.710
But based on the
information they gave us,

00:03:29.710 --> 00:03:33.610
these are the parallel and
the perpendicular lines.


Interpreting linear graphs word problems example 1

WEBVTT
Kind: captions
Language: en

00:00:00.273 --> 00:00:07.464
what i want to do with this video is think about the relationship between variables and then think about what the graph of that relationship should look like.

00:00:07.464 --> 00:00:13.212
So let's say these two axis, the horizontal axis over here I plot

00:00:13.212 --> 00:00:20.065
the price of a product and lets say this vertical axis over here i plot the demand for the product, and

00:00:20.065 --> 00:00:24.498
i'm only plotting the first quadrant here because i'm assuming that the price can only be positive and

00:00:24.498 --> 00:00:30.320
i'm assuming that the demand can only be positive, that people aren't going to pay someone to take the product

00:00:30.320 --> 00:00:37.114
away from them. So let's think about what would happen for the price and demand for most normal products.

00:00:37.114 --> 00:00:43.231
So, if the price is low, you would expect that a lot of poeple are willing to buy that thing they're

00:00:43.231 --> 00:00:46.652
like "Oh it's a good price, i would like to buy it."

00:00:46.652 --> 00:00:51.163
So, if the price is low, then the demand would be high so maybe it would be somplace over here, all the

00:00:51.163 --> 00:00:54.697
way that you would have really high demand if the price was zero.

00:00:54.697 --> 00:00:58.581
so if the price was low the demand would be high.

00:00:58.581 --> 00:01:02.331
Now what happens is the price -- so right here the price is low, demand is high, if the price were to

00:01:02.331 --> 00:01:06.464
go up a little bit then maybe the demand goes down a little bit, right?

00:01:06.464 --> 00:01:09.280
price went up a little bit demand went down a little bit.

00:01:09.280 --> 00:01:13.165
if the price went up a little bit more then maybe demand goes down a little bit more.

00:01:13.165 --> 00:01:17.849
as the price went up a bunch then demand would go down a bunch and so the line that represents how the

00:01:17.849 --> 00:01:28.115
demand relates to price might look something like this, and i'm just going to assume it's

00:01:28.115 --> 00:01:33.099
it a line. It might not be a line, it might be a curve. It might look something like that.

00:01:33.099 --> 00:01:37.766
Or it might look something like that. But in general is someone were to ask you,

00:01:37.766 --> 00:01:43.030
if you saw this magenta curve that as price increases what happens to demand.

00:01:43.030 --> 00:01:47.931
You just say "Well look price increases, as price increases what happens to demand?"

00:01:47.931 --> 00:01:53.731
Well demand is decreasing. Now let's think about a different scenario.

00:01:53.731 --> 00:02:02.951
Let's talk about the demand for real estate. For actual property, and lets say that on this axis

00:02:02.951 --> 00:02:08.648
that we plot the population. The population in the area, and this right over here this is

00:02:08.648 --> 00:02:17.682
demand for land. So when the population is very low, you can imagine, if the population is

00:02:17.682 --> 00:02:23.447
zero there is no one there that would want to buy land. So if the population is very low

00:02:23.447 --> 00:02:26.914
the demand is going to be very low. And as population increases, demand should increase.

00:02:26.914 --> 00:02:31.280
If the population increases, more people are going to want to buy land. And if the

00:02:31.280 --> 00:02:36.564
population goes up a bunch then a lot of people are going to want to buy land. So you'll see a

00:02:36.564 --> 00:02:41.364
line that looks something like this. And once again I drew a line, it doesn't have to be a line

00:02:41.364 --> 00:02:47.581
it could be a curve of some kind. It could be a curve that looks something like that, or a curve

00:02:47.581 --> 00:02:50.397
that looks something like that. We don't know but the general idea is that if someone

00:02:50.397 --> 00:02:56.498
showed you a graph that looked like this. And as population increases what happens to demand.

00:02:56.498 --> 00:03:00.231
We'll you'd say "Look, this is population increasing, what happens to demand?" Demand

00:03:00.231 --> 00:03:07.049
is going up. Where as price increased the demand went down. Here as population is increasing

00:03:07.049 --> 00:03:12.180
demand went up. And you can just make that more general with variables. We're talking about

00:03:12.180 --> 00:03:18.298
specific cases here. But if I were to plot something like this, if you were to see a graph that

00:03:18.298 --> 00:03:26.479
looked like that and this is the variable x and this is the variable y. And someone were to ask you what

00:03:26.479 --> 00:03:38.465
happens to y as x increases. Well you take any x that's the y that we have for that x.

00:03:38.465 --> 00:03:42.713
And as you increase x, as you move in the positive horizontal directions. As you increase

00:03:42.713 --> 00:03:49.850
x what is happening to y? Y is going up. So for this example, as x increases

00:03:49.850 --> 00:04:00.098
y is increasing. If we had a graph that looked like this. Let's call that the

00:04:00.098 --> 00:04:05.530
a axis and this is the b axis and maybe our graph looked like this.

00:04:05.530 --> 00:04:14.281
What happens as A increases? If you pick an A right over here. We're at that A and that B.

00:04:14.281 --> 00:04:24.897
As A increases what's happening to B? Well as A increases our B is lower. As A

00:04:24.897 --> 00:04:32.931
increases here B is decreasing. So, just wanting to give you a general idea, when X and Y increased

00:04:32.931 --> 00:04:39.864
together the line goes from the bottom left to the top right, we would call this an upward sloping line.

00:04:39.864 --> 00:04:49.115
We would call this a positive slope. Everytime X is increased Y also increases is upwards sloping.

00:04:49.115 --> 00:04:54.857
When our independant variable increases and our dependent variable decreases.

00:04:54.857 --> 00:05:05.466
When the independent variable is increasing, then you say it has a downward slope, when you go from the top left to the bottom right.


Quadrilateral properties

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.870
Which of the
following names can be

00:00:02.870 --> 00:00:05.680
used to describe the
geometric shape below?

00:00:05.680 --> 00:00:08.516
So the first name in
question is a quadrilateral.

00:00:08.516 --> 00:00:11.570
And a quadrilateral is
literally any closed shape

00:00:11.570 --> 00:00:13.550
that has four sides.

00:00:13.550 --> 00:00:17.870
And this is definitely a closed
shape that has four sides.

00:00:17.870 --> 00:00:20.670
So it is definitely
a quadrilateral.

00:00:20.670 --> 00:00:23.940
Next, we have to think about
whether it is a parallelogram.

00:00:23.940 --> 00:00:27.060
A parallelogram
is a quadrilateral

00:00:27.060 --> 00:00:30.570
that has two pairs of parallel
sides, where in each pair

00:00:30.570 --> 00:00:32.180
they're opposite sides.

00:00:32.180 --> 00:00:35.390
And in this case, if you
look at this side over here,

00:00:35.390 --> 00:00:37.770
it forms a 90-degree
angle with this line.

00:00:37.770 --> 00:00:40.460
And this side over here
also forms a 90-degree angle

00:00:40.460 --> 00:00:42.040
with this line over here.

00:00:42.040 --> 00:00:45.452
So these two sides are parallel.

00:00:45.452 --> 00:00:47.410
And then you could make
the exact same argument

00:00:47.410 --> 00:00:48.830
for the other two sides.

00:00:48.830 --> 00:00:52.970
This line up here forms a
90-degree angle with this side.

00:00:52.970 --> 00:00:54.260
And so does this side.

00:00:54.260 --> 00:00:56.940
It forms a 90-degree angle
with this line right over here.

00:00:56.940 --> 00:00:59.590
They form the same
angle with this line.

00:00:59.590 --> 00:01:00.500
They're parallel.

00:01:00.500 --> 00:01:04.300
So this side is parallel to
that side right over there.

00:01:04.300 --> 00:01:08.260
So this is definitely
also a parallelogram.

00:01:08.260 --> 00:01:11.800
Next, we ask about a trapezoid.

00:01:11.800 --> 00:01:13.750
Now, trapezoid is interesting.

00:01:13.750 --> 00:01:18.580
Sometimes a trapezoid is defined
as any quadrilateral having

00:01:18.580 --> 00:01:21.720
at least one pair
of parallel sides.

00:01:21.720 --> 00:01:24.520
Sometimes it's defined
as having only one

00:01:24.520 --> 00:01:26.060
pair of parallel sides.

00:01:26.060 --> 00:01:28.000
So let me write this down.

00:01:28.000 --> 00:01:30.710
Trapezoid, there's
a debate here.

00:01:30.710 --> 00:01:31.970
It's not completely settled.

00:01:31.970 --> 00:01:42.870
Some people say at least
one pair of parallel sides.

00:01:42.870 --> 00:01:45.410
That's one definition,
one possible definition.

00:01:45.410 --> 00:01:55.955
The other one is at exactly
one pair of parallel sides.

00:02:01.270 --> 00:02:03.740
How we answer this question
depends on which definition

00:02:03.740 --> 00:02:05.410
for trapezoid we pick.

00:02:05.410 --> 00:02:07.480
Now, the one that
people most refer to

00:02:07.480 --> 00:02:09.620
is actually this
one right over here,

00:02:09.620 --> 00:02:12.070
exactly one pair
of parallel sides.

00:02:12.070 --> 00:02:17.040
So when you think
of a trapezoid,

00:02:17.040 --> 00:02:21.030
they think of something like
this, where this side over here

00:02:21.030 --> 00:02:23.110
is parallel to
that side over here

00:02:23.110 --> 00:02:25.340
and those two are not parallel.

00:02:25.340 --> 00:02:27.620
But sometimes you'll also
see this at least one

00:02:27.620 --> 00:02:29.610
pair of parallel sides.

00:02:29.610 --> 00:02:31.345
And so this would
include parallelograms.

00:02:31.345 --> 00:02:32.970
It would be inclusive
of parallelograms

00:02:32.970 --> 00:02:35.820
because parallelograms have
two pairs of parallel sides.

00:02:35.820 --> 00:02:38.300
But I'm going to go with this
definition right over here,

00:02:38.300 --> 00:02:40.720
exactly one pair
of parallel sides.

00:02:40.720 --> 00:02:43.270
This has two pairs
of parallel sides

00:02:43.270 --> 00:02:47.080
so I will not call
it a trapezoid.

00:02:47.080 --> 00:02:50.260
But it's always important to
clarify what people are talking

00:02:50.260 --> 00:02:52.080
about because some
people might say

00:02:52.080 --> 00:02:55.610
a trapezoid is at least
one pair of parallel sides.

00:02:55.610 --> 00:02:57.570
And if we used that
definition, then we

00:02:57.570 --> 00:02:58.930
would call it a trapezoid.

00:02:58.930 --> 00:03:01.900
So it really depends on the
definition that you're using.

00:03:01.900 --> 00:03:04.910
Now, let's go on to rhombus.

00:03:04.910 --> 00:03:10.020
So a rhombus is a quadrilateral
where four of the sides

00:03:10.020 --> 00:03:11.123
are congruent.

00:03:14.100 --> 00:03:18.000
So a rhombus will
look like this.

00:03:18.000 --> 00:03:21.700
All four sides have
the same length.

00:03:21.700 --> 00:03:25.010
They're not necessarily at
right angles to each other.

00:03:25.010 --> 00:03:27.180
This figure over here,
we have two pairs

00:03:27.180 --> 00:03:28.880
of a size that are
the same length,

00:03:28.880 --> 00:03:30.590
but there's no
information that tells us

00:03:30.590 --> 00:03:32.370
that this side is
equal to that side

00:03:32.370 --> 00:03:34.410
or that this side is
equal to that side.

00:03:34.410 --> 00:03:37.840
So we can't make the claim that
this is necessarily a rhombus.

00:03:37.840 --> 00:03:38.921
We don't know for sure.

00:03:38.921 --> 00:03:41.420
If someone told us that this
length is equal to that length,

00:03:41.420 --> 00:03:43.030
then things change.

00:03:43.030 --> 00:03:45.780
But for the sake
of this one, we're

00:03:45.780 --> 00:03:47.465
not going to go with a rhombus.

00:03:47.465 --> 00:03:51.070
A rectangle is essentially
a parallelogram

00:03:51.070 --> 00:03:53.780
that has four right angles.

00:03:53.780 --> 00:03:56.320
And we already established
this is a parallelogram,

00:03:56.320 --> 00:03:59.950
and it also has four right
angles-- one, two, three, four.

00:03:59.950 --> 00:04:01.597
So this is a rectangle.

00:04:01.597 --> 00:04:03.180
Another way to think
about a rectangle

00:04:03.180 --> 00:04:06.140
is opposite sides
have the same length,

00:04:06.140 --> 00:04:08.890
and you have four right angles.

00:04:08.890 --> 00:04:10.870
So this is definitely
a rectangle.

00:04:10.870 --> 00:04:14.070
A square, a couple of way
you can think about a square.

00:04:14.070 --> 00:04:17.969
You could view a square as a
rhombus with four right angles

00:04:17.969 --> 00:04:18.560
.

00:04:18.560 --> 00:04:21.540
So if were to straighten it out
a little bit, it's a rhombus

00:04:21.540 --> 00:04:23.050
so all the four
sides are the same.

00:04:23.050 --> 00:04:24.610
And you have four right angles.

00:04:24.610 --> 00:04:26.320
That's one way to
think about a square.

00:04:26.320 --> 00:04:27.940
Or you could view
it as a rectangle

00:04:27.940 --> 00:04:30.140
where all four
sides are congruent.

00:04:30.140 --> 00:04:33.360
But in either case, you have
to have all four sides be

00:04:33.360 --> 00:04:35.260
congruent in order
to be a square.

00:04:35.260 --> 00:04:38.720
And we already established we
ruled out this being a rhombus,

00:04:38.720 --> 00:04:41.290
that all four sides here are
not necessarily congruent.

00:04:41.290 --> 00:04:42.910
You have two pairs
of congruent sides,

00:04:42.910 --> 00:04:46.240
but we don't know whether
this side and this side

00:04:46.240 --> 00:04:47.400
are congruent.

00:04:47.400 --> 00:04:50.600
So we cannot call this a square.

00:04:50.600 --> 00:04:52.660
So it's not a square,
not a rhombus,

00:04:52.660 --> 00:04:55.130
not a trapezoid by the
definition we picked,

00:04:55.130 --> 00:04:58.790
which is the less
inclusive version where

00:04:58.790 --> 00:05:00.980
you say exactly one
pair of parallel sides.

00:05:00.980 --> 00:05:03.000
It is a quadrilateral.

00:05:03.000 --> 00:05:04.350
It is a parallelogram.

00:05:04.350 --> 00:05:06.257
It is a rectangle.