Statistics: Standard deviation

WEBVTT
Kind: captions
Language: en

00:00:00.424 --> 00:00:02.840
Let's review a little bit of
everything we learned so far.

00:00:02.840 --> 00:00:04.350
And hopefully make
everything fit together

00:00:04.350 --> 00:00:04.920
a little bit better.

00:00:04.920 --> 00:00:07.480
And then we'll do a bunch of
calculations with real numbers.

00:00:07.480 --> 00:00:09.590
And I think it'll really
hit the point home.

00:00:09.590 --> 00:00:12.760
So first of all, let
me make some columns.

00:00:12.760 --> 00:00:15.720
So if we're dealing
with-- let's see.

00:00:15.720 --> 00:00:17.310
We could call it the concept.

00:00:21.670 --> 00:00:24.810
And then we'll call
it the-- whether we're

00:00:24.810 --> 00:00:34.470
dealing with a
population or a sample.

00:00:34.470 --> 00:00:38.060
So the first statistical
concept we came up with

00:00:38.060 --> 00:00:41.310
was the notion of the mean
or the central tendency.

00:00:41.310 --> 00:00:43.010
And we learned that
that was one way

00:00:43.010 --> 00:00:47.630
to measure the average or
central tendency of a data set.

00:00:47.630 --> 00:00:50.160
The other ways were the
median and the mode.

00:00:50.160 --> 00:00:52.660
But the mean tends to show up
a lot more, especially when we

00:00:52.660 --> 00:00:54.470
start talking about
variances, and we'll

00:00:54.470 --> 00:00:56.440
do in this video, the
standard deviation.

00:00:56.440 --> 00:00:58.210
But the mean of a
population, we learned--

00:00:58.210 --> 00:01:03.630
we use the Greek letter mu--
is equal to the sum of each

00:01:03.630 --> 00:01:05.780
of the data points
in the population.

00:01:05.780 --> 00:01:10.290
That's an I. Let me make
sure it looks like an I.

00:01:10.290 --> 00:01:13.240
So you're going to sum up
each of those data points.

00:01:13.240 --> 00:01:15.647
You're going to start
with the first one.

00:01:15.647 --> 00:01:17.230
And you're going to
go to the Nth one.

00:01:17.230 --> 00:01:20.070
We're assuming that there are n
data points in the population.

00:01:20.070 --> 00:01:22.924
And then you divide by the
total number that you have.

00:01:22.924 --> 00:01:24.590
And this is like the
average that you're

00:01:24.590 --> 00:01:27.080
used to taking before you
learned any of the statistic

00:01:27.080 --> 00:01:27.580
stuff.

00:01:27.580 --> 00:01:28.871
You add up all the data points.

00:01:28.871 --> 00:01:30.780
And you divide by
the number there are.

00:01:30.780 --> 00:01:32.380
The sample is the same thing.

00:01:32.380 --> 00:01:35.170
We just use a slightly
different terminology.

00:01:35.170 --> 00:01:39.100
The mean of a sample-- and I'll
do it in a different color.

00:01:39.100 --> 00:01:42.940
Just write it as x
with a line on top.

00:01:42.940 --> 00:01:45.780
And that's equal to
the sum of all the data

00:01:45.780 --> 00:01:47.900
points in the sample.

00:01:47.900 --> 00:01:49.962
So each of the x
sine in the sample.

00:01:49.962 --> 00:01:51.920
But we're assuming the
sample is something less

00:01:51.920 --> 00:01:52.930
than a population.

00:01:52.930 --> 00:01:55.490
So you start with
the first one still.

00:01:55.490 --> 00:01:57.840
And then you go to
the lowercase n,

00:01:57.840 --> 00:02:01.040
where we assume that lowercase
n is less than the big N.

00:02:01.040 --> 00:02:03.960
If this was the same thing,
then we're just actually taking

00:02:03.960 --> 00:02:06.850
the average, or we're taking the
mean, of the entire population.

00:02:06.850 --> 00:02:08.850
And then you divide by
the number of data points

00:02:08.850 --> 00:02:09.710
you added.

00:02:09.710 --> 00:02:11.990
You get to n.

00:02:11.990 --> 00:02:15.572
Then we said, OK, this gives
us the central tendency.

00:02:15.572 --> 00:02:17.280
It's one measure of
the central tendency.

00:02:17.280 --> 00:02:19.630
But what if we wanted to
know how good of an indicator

00:02:19.630 --> 00:02:21.810
this is for the population
or for the sample.

00:02:21.810 --> 00:02:25.690
Or on average, how far are the
data points from this mean?

00:02:25.690 --> 00:02:28.490
And that's where we came up
with the concept of variance.

00:02:28.490 --> 00:02:30.390
And I'll arbitrarily
switch colors again.

00:02:35.940 --> 00:02:40.040
And in a population,
the variable,

00:02:40.040 --> 00:02:42.220
or the notation for variance,
is the sigma squared.

00:02:42.220 --> 00:02:44.510
This means variance.

00:02:44.510 --> 00:02:47.760
And that is equal to-- you
take each of the data points.

00:02:50.560 --> 00:02:53.040
You find the difference
between that and the mean

00:02:53.040 --> 00:02:55.180
that you calculate up there.

00:02:55.180 --> 00:02:56.060
You square it.

00:02:56.060 --> 00:02:57.820
So you get the
squared difference.

00:02:57.820 --> 00:03:00.400
And then you essentially take
the average of all of these.

00:03:00.400 --> 00:03:02.760
You take the average of all
of these squared distances.

00:03:02.760 --> 00:03:08.960
So that's if you take the sum
from i is equal to one to N.

00:03:08.960 --> 00:03:13.535
And you divide it by
N. That's the variance.

00:03:13.535 --> 00:03:14.910
And then the
variance of a sample

00:03:14.910 --> 00:03:17.180
mean, and this was a little
bit more interesting.

00:03:17.180 --> 00:03:19.500
And we talked a little bit
about it in the last video.

00:03:19.500 --> 00:03:21.680
You actually want
to provide a, you

00:03:21.680 --> 00:03:24.137
want to estimate the
variance of the population

00:03:24.137 --> 00:03:25.970
when you're taking the
variance of a sample.

00:03:25.970 --> 00:03:29.030
And in order to provide
an unbiased estimate,

00:03:29.030 --> 00:03:30.830
you do something
very similar to here.

00:03:30.830 --> 00:03:32.800
But you end up
dividing by n minus 1.

00:03:32.800 --> 00:03:34.460
So let me write that down.

00:03:34.460 --> 00:03:36.930
So the variance of a
population-- sorry.

00:03:36.930 --> 00:03:39.310
The variance of a sample,
or sample variance,

00:03:39.310 --> 00:03:41.150
or unbiased sample
variance-- and that's

00:03:41.150 --> 00:03:43.570
why we're going to
divide by n minus 1.

00:03:43.570 --> 00:03:46.315
That's denoted by s squared.

00:03:46.315 --> 00:03:48.440
What you do is you take
the difference between each

00:03:48.440 --> 00:03:53.810
of the data points in the
sample minus the sample mean.

00:03:53.810 --> 00:03:56.110
We assume that we don't
know the population mean.

00:03:56.110 --> 00:03:56.800
Maybe we did.

00:03:56.800 --> 00:03:59.369
If we knew the population
mean, we actually

00:03:59.369 --> 00:04:01.410
wouldn't have to do the
unbiased thing that we're

00:04:01.410 --> 00:04:02.910
going to do here
in the denominator.

00:04:02.910 --> 00:04:05.442
But when you have a sample,
the only way to kind of figure

00:04:05.442 --> 00:04:07.150
out the population
mean is to estimate it

00:04:07.150 --> 00:04:08.250
with the sample mean.

00:04:08.250 --> 00:04:11.060
So we assume that we only
have the sample mean.

00:04:11.060 --> 00:04:13.080
And you're going
to square those.

00:04:13.080 --> 00:04:15.140
And then you're
going to sum them up.

00:04:15.140 --> 00:04:20.420
Sum them up from i is equal
to one to i is equal to n.

00:04:20.420 --> 00:04:22.029
Because you have n data points.

00:04:22.029 --> 00:04:23.990
And if you want an
unbiased estimator,

00:04:23.990 --> 00:04:26.999
you divide by n minus 1.

00:04:26.999 --> 00:04:28.790
And we talked a little
bit before about why

00:04:28.790 --> 00:04:30.970
you want this to be n
minus 1 instead of an n.

00:04:30.970 --> 00:04:32.630
And actually, in a
couple of videos,

00:04:32.630 --> 00:04:34.320
I'll actually prove this to you.

00:04:34.320 --> 00:04:37.181
One I'll prove, maybe
experimentally, using Excel.

00:04:37.181 --> 00:04:38.930
And then I'll-- which
wouldn't be a proof.

00:04:38.930 --> 00:04:40.600
But it'll just give you a
little bit of intuition.

00:04:40.600 --> 00:04:42.230
And then I'll actually prove
it a little bit more formally

00:04:42.230 --> 00:04:43.000
later on.

00:04:43.000 --> 00:04:45.070
But you don't have to
worry about it right now.

00:04:45.070 --> 00:04:46.995
Now, the next thing
we'll learn is something

00:04:46.995 --> 00:04:49.274
that you've probably
heard a lot of.

00:04:49.274 --> 00:04:50.940
Especially, sometimes,
in class teachers

00:04:50.940 --> 00:04:53.800
talk about the standard
deviation of a test,

00:04:53.800 --> 00:04:57.220
or it's actually, probably,
one of the most used words

00:04:57.220 --> 00:04:58.410
in statistics.

00:04:58.410 --> 00:05:00.680
I think a lot of
people, unfortunately,

00:05:00.680 --> 00:05:06.410
maybe use it or maybe use it
without fully appreciating

00:05:06.410 --> 00:05:08.330
everything that it involves.

00:05:08.330 --> 00:05:11.050
But the goal-- well,
we'll eventually,

00:05:11.050 --> 00:05:15.140
hopefully, appreciate all
that it involves soon.

00:05:15.140 --> 00:05:19.620
But the standard deviation.

00:05:19.620 --> 00:05:23.430
And once you know variance, it's
actually quite straightforward.

00:05:23.430 --> 00:05:26.830
It's the square root
of the variance.

00:05:26.830 --> 00:05:30.170
So the standard
deviation of a population

00:05:30.170 --> 00:05:33.630
is written as sigma which
is equal to the square root

00:05:33.630 --> 00:05:35.690
of the variance.

00:05:35.690 --> 00:05:37.260
And now, I think,
you understand why

00:05:37.260 --> 00:05:40.100
variance is written
as sigma squared.

00:05:40.100 --> 00:05:42.720
And that is equal to just
the square root of all that.

00:05:42.720 --> 00:05:45.040
It's equal to the square
root-- I'll probably

00:05:45.040 --> 00:05:46.590
run out of space--
of all of that.

00:05:46.590 --> 00:05:49.497
So I won't write the
top at the bottom.

00:05:49.497 --> 00:05:50.330
That makes it messy.

00:05:50.330 --> 00:05:56.280
If xi minus mu squared.

00:05:56.280 --> 00:05:58.570
Everything over N.

00:05:58.570 --> 00:06:01.240
And then, if you wanted the
standard deviation of a sample,

00:06:01.240 --> 00:06:03.780
and it actually gets a
little bit interesting.

00:06:03.780 --> 00:06:06.180
Because the standard
deviation of a sample,

00:06:06.180 --> 00:06:08.100
which is equal to
the square root

00:06:08.100 --> 00:06:09.467
of the variance of a sample.

00:06:09.467 --> 00:06:11.050
It actually turns
out that this is not

00:06:11.050 --> 00:06:12.920
an unbiased estimator for this.

00:06:12.920 --> 00:06:15.220
And I don't want to get too
technical for it right now.

00:06:15.220 --> 00:06:18.930
But this is actually a
very good estimate of this.

00:06:18.930 --> 00:06:21.520
The expected value of
this is going to be this.

00:06:21.520 --> 00:06:26.540
And I'll go into more depth on
expected values in the future.

00:06:26.540 --> 00:06:28.860
But it turns out that
this is not quite

00:06:28.860 --> 00:06:30.990
the same expected value as this.

00:06:30.990 --> 00:06:33.220
But you don't have to
worry about it for now.

00:06:33.220 --> 00:06:35.870
So why even talk about
the standard deviation?

00:06:35.870 --> 00:06:37.870
Well, one, the units
work out a little better.

00:06:37.870 --> 00:06:39.860
If, let's say, all
of our data points

00:06:39.860 --> 00:06:41.230
we're measured in meters, right?

00:06:41.230 --> 00:06:44.720
If we were taking a bunch
of measurements of length,

00:06:44.720 --> 00:06:47.400
then the units of the variance
would be meter squared, right?

00:06:47.400 --> 00:06:49.350
Because we're taking
meters minus meters.

00:06:49.350 --> 00:06:49.870
This would be a meter.

00:06:49.870 --> 00:06:51.036
And then you're squaring it.

00:06:51.036 --> 00:06:52.340
You're getting meters squared.

00:06:52.340 --> 00:06:54.340
And that's kind of a
strange concept if you say,

00:06:54.340 --> 00:06:57.534
the average dispersion from the
center has been meter squared.

00:06:57.534 --> 00:06:59.700
So well, first, when you
take the square root of it,

00:06:59.700 --> 00:07:02.762
you get something
that's, again, in meters.

00:07:02.762 --> 00:07:04.220
So you're kind of
saying, oh, well,

00:07:04.220 --> 00:07:07.045
the standard deviation
is x or y meters.

00:07:07.045 --> 00:07:08.420
And then we'll
learn a little bit

00:07:08.420 --> 00:07:12.020
that If you can actually model
your data as a bell curve,

00:07:12.020 --> 00:07:14.727
or if you assume that your data
has a distribution of a bell

00:07:14.727 --> 00:07:16.810
curve, then this tells you
some interesting things

00:07:16.810 --> 00:07:20.930
about where all of the
probability of finding someone

00:07:20.930 --> 00:07:23.957
within one or two standard
deviations of the mean.

00:07:23.957 --> 00:07:26.290
But, anyway, I don't want to
go too technical right now.

00:07:26.290 --> 00:07:28.920
Let's just calculate a bunch.

00:07:28.920 --> 00:07:37.000
Let's calculate, let's say, if
I had numbers 1, 2, 3, 8, and 7.

00:07:37.000 --> 00:07:38.990
And let's say that
this is a population.

00:07:38.990 --> 00:07:40.870
So what would its mean be?

00:07:40.870 --> 00:07:44.340
So I have 1 plus 2 plus 3.

00:07:44.340 --> 00:07:46.570
So it's 3 plus 3 is 6.

00:07:46.570 --> 00:07:49.241
6 plus 8 is 14.

00:07:49.241 --> 00:07:51.640
14 plus 7 is 21.

00:07:51.640 --> 00:07:54.490
So the mean of this population,
you sum up all the data points.

00:07:54.490 --> 00:07:59.250
You get 21 divided by the
total number of data points.

00:07:59.250 --> 00:08:01.250
One, two, three, four, five.

00:08:01.250 --> 00:08:03.480
21 divided by 5 which
is equal to what?

00:08:03.480 --> 00:08:06.020
4.20.

00:08:06.020 --> 00:08:07.350
Fair enough.

00:08:07.350 --> 00:08:09.860
Now, we want to figure
out the variance.

00:08:09.860 --> 00:08:12.720
And we're assuming that this
is the entire population.

00:08:12.720 --> 00:08:15.390
So the variance of this
population is going to be equal

00:08:15.390 --> 00:08:18.800
to the sum of the squared
differences of each of these

00:08:18.800 --> 00:08:20.060
numbers from 4.20

00:08:20.060 --> 00:08:22.260
I'm going to have to
get my calculator out.

00:08:22.260 --> 00:08:27.180
So it's going to be
1 minus 4.20 squared,

00:08:27.180 --> 00:08:37.380
plus 2 minus 4.20 squared,
plus 3 minus 4.20 squared,

00:08:37.380 --> 00:08:47.370
plus 8 minus 4.20 squared,
plus 7 minus 4.20 squared.

00:08:47.370 --> 00:08:48.994
And it's going to
be all of that--

00:08:48.994 --> 00:08:50.660
and I know it looks
a little bit funny--

00:08:50.660 --> 00:08:55.680
divided by the number of data
points we have, divided by 5.

00:08:55.680 --> 00:08:57.470
So let me take the
calculator out.

00:09:00.070 --> 00:09:02.180
All right.

00:09:02.180 --> 00:09:03.084
Here we go.

00:09:03.084 --> 00:09:05.500
Actually, maybe I should have
used the graphing calculator

00:09:05.500 --> 00:09:08.300
that I have.

00:09:08.300 --> 00:09:11.920
Let me see if I
can get this thing,

00:09:11.920 --> 00:09:14.683
if I could get this-- Oh.

00:09:19.599 --> 00:09:20.140
There you go.

00:09:20.140 --> 00:09:22.890
Yeah, I think the graphing
one will be better.

00:09:22.890 --> 00:09:25.122
Because I can see
everything that I'm writing.

00:09:25.122 --> 00:09:27.170
OK, so let me clear this.

00:09:27.170 --> 00:09:35.360
So I want to take 1 minus
4.20 squared, plus--

00:09:35.360 --> 00:09:44.640
let me write it down--
plus 2 minus 4.20 squared,

00:09:44.640 --> 00:09:59.460
plus 3 minus 4.20 squared, plus
8 minus 4.20 squared, right?

00:09:59.460 --> 00:10:01.990
I'm just taking the sum
of the squared distances

00:10:01.990 --> 00:10:04.370
from the mean.

00:10:04.370 --> 00:10:10.640
One more, plus 7
minus 4.20 squared.

00:10:10.640 --> 00:10:12.330
So that's the sum.

00:10:12.330 --> 00:10:14.340
The sum is 38.80,
so the numerator.

00:10:14.340 --> 00:10:18.950
So this is going to be
equal to 38.80 divided by 5.

00:10:18.950 --> 00:10:21.530
So this is the sum of the
square distances, right?

00:10:21.530 --> 00:10:23.340
Each of these, just
so you can relate

00:10:23.340 --> 00:10:28.710
to the formula, each of that
is xi minus the mean squared.

00:10:28.710 --> 00:10:31.650
And so if we take the sum
of all of them, right?

00:10:31.650 --> 00:10:34.310
This numerator is
the sum of each

00:10:34.310 --> 00:10:42.850
of the xi minus the mean
squared from i equals 1 to N.

00:10:42.850 --> 00:10:45.222
And that ended up to be 38.80.

00:10:45.222 --> 00:10:46.680
And I just calculated
it like that.

00:10:46.680 --> 00:10:49.450
I just took each of the data
points minus the mean squared,

00:10:49.450 --> 00:10:50.300
added them all up.

00:10:50.300 --> 00:10:51.810
And I got 38.80.

00:10:51.810 --> 00:10:53.990
And I want to divide
it by N which is 5.

00:10:53.990 --> 00:10:57.440
So this N up here is
actually, also, 5.

00:10:57.440 --> 00:10:58.040
Right?

00:10:58.040 --> 00:11:08.480
And so 38.80 divided
by 5 is 7.76.

00:11:08.480 --> 00:11:14.160
So let me scroll
down a little bit.

00:11:14.160 --> 00:11:19.180
The variance is equal to 7.76.

00:11:19.180 --> 00:11:24.330
Now, if this was a sample
of a larger distribution,

00:11:24.330 --> 00:11:27.460
if the 1, 2, 3, 8, and 7
weren't the population,

00:11:27.460 --> 00:11:30.950
if it was a sample from
a larger population,

00:11:30.950 --> 00:11:33.510
instead of dividing by 5,
we would have divided by 4.

00:11:33.510 --> 00:11:37.080
And we would have gotten
the variance as 38.80.

00:11:37.080 --> 00:11:38.630
Sorry.

00:11:38.630 --> 00:11:45.190
38.80 divided by N minus
1 which is divided by 4.

00:11:45.190 --> 00:11:49.740
So then we would have gotten
the sample variance at 9.70.

00:11:49.740 --> 00:11:52.110
If you would have divided
by N minus 1 instead of n.

00:11:52.110 --> 00:11:53.410
But, anyway, don't worry
about that right now.

00:11:53.410 --> 00:11:54.960
That's just a change
of N. But once

00:11:54.960 --> 00:11:56.876
you have the variance,
very easy to figure out

00:11:56.876 --> 00:11:58.140
the standard deviation.

00:11:58.140 --> 00:11:59.890
You just take the
square root of it.

00:11:59.890 --> 00:12:09.750
The square root of 7.76, 2.78.

00:12:09.750 --> 00:12:13.060
Let's say 2.79 is the
standard deviation.

00:12:13.060 --> 00:12:15.830
So this gives us some
measure of, on average,

00:12:15.830 --> 00:12:21.130
how far the numbers are away
from the mean which was 4.20.

00:12:21.130 --> 00:12:23.200
And it gives it in
kind of the units

00:12:23.200 --> 00:12:24.520
of the original measurement.

00:12:24.520 --> 00:12:25.999
Anyway, I'm all out of time.

00:12:25.999 --> 00:12:27.290
I'll see you in the next video.

00:12:27.290 --> 00:12:28.665
Well, actually,
let's figure out.

00:12:28.665 --> 00:12:33.960
We said if this was a sample,
if those numbers were a sample

00:12:33.960 --> 00:12:38.100
and not the population, then
we figured out that the sample

00:12:38.100 --> 00:12:40.670
variance was 9.70.

00:12:40.670 --> 00:12:43.430
And so then the sample
standard deviation

00:12:43.430 --> 00:12:46.000
is just going to be the
square root of that.

00:12:46.000 --> 00:12:53.932
The square root of 9.70
which would be 3.13, 3.11.

00:12:53.932 --> 00:12:56.390
Anyway, hopefully that makes
it a little bit more concrete.

00:12:56.390 --> 00:12:58.590
We've been dealing with
these sigma notation

00:12:58.590 --> 00:13:00.090
variables and all that so far.

00:13:00.090 --> 00:13:01.715
So when you actually
do it with numbers

00:13:01.715 --> 00:13:03.920
you see it's, hopefully,
not that difficult.

00:13:03.920 --> 00:13:06.028
Anyway, see you
in the next video.


Statistics: Sample variance

WEBVTT
Kind: captions
Language: en

00:00:01.120 --> 00:00:03.390
This video, here, is a
groundbreaking video,

00:00:03.390 --> 00:00:05.500
for multiple reasons.

00:00:05.500 --> 00:00:09.280
One, I'm going to introduce
you to the variance

00:00:09.280 --> 00:00:11.960
of a sample, which is
interesting in its own right.

00:00:11.960 --> 00:00:14.300
And I'm attempting to
record this video in HD,

00:00:14.300 --> 00:00:16.730
and hopefully you can see it
bigger and clearer than ever

00:00:16.730 --> 00:00:17.230
before.

00:00:17.230 --> 00:00:19.330
But we'll see how
all of that goes.

00:00:19.330 --> 00:00:22.240
So this is a bit of an
experiment, so bear with me.

00:00:22.240 --> 00:00:25.170
But so, just before we go
into the variance of a sample,

00:00:25.170 --> 00:00:27.600
I think it's
instructive to review

00:00:27.600 --> 00:00:32.360
the variance the population, and
we can compare their formulas.

00:00:32.360 --> 00:00:34.100
The variance of a
population-- and it's

00:00:34.100 --> 00:00:36.150
this Greek letter, sigma.

00:00:36.150 --> 00:00:37.610
Lowercase sigma squared.

00:00:37.610 --> 00:00:38.570
That means variance.

00:00:38.570 --> 00:00:41.037
I know it's weird that
a variable already

00:00:41.037 --> 00:00:41.870
has a squared in it.

00:00:41.870 --> 00:00:43.244
You're not squaring
the variable.

00:00:43.244 --> 00:00:44.270
This is the variable.

00:00:44.270 --> 00:00:45.850
Sigma squared means variance.

00:00:45.850 --> 00:00:47.225
Actually, let me
write that down.

00:00:47.225 --> 00:00:48.130
That equals variance.

00:00:51.640 --> 00:00:55.400
And that is equal to--
you take each data point--

00:00:55.400 --> 00:00:58.745
and we'll call them x sub i.

00:00:58.745 --> 00:01:00.480
You take each data
point, find out

00:01:00.480 --> 00:01:04.379
how far it is from the
mean of the population--

00:01:04.379 --> 00:01:06.370
the mean of the population.

00:01:06.370 --> 00:01:11.051
You square it, and then you take
the average of all of those.

00:01:11.051 --> 00:01:12.050
So you take the average.

00:01:12.050 --> 00:01:12.930
You sum them all up.

00:01:12.930 --> 00:01:14.250
You go from i is equal to 1.

00:01:14.250 --> 00:01:17.720
So from the first point all
the way to the N-th point.

00:01:17.720 --> 00:01:19.780
And then to average,
you sum them all up,

00:01:19.780 --> 00:01:22.080
and then you divide by N.

00:01:22.080 --> 00:01:26.000
So the variance is the average
of the squared distances

00:01:26.000 --> 00:01:27.262
of each point from the mean.

00:01:27.262 --> 00:01:28.970
And just to give you
the intuition again,

00:01:28.970 --> 00:01:32.420
it essentially says, on
average, roughly how far away

00:01:32.420 --> 00:01:34.300
are each of the points
from the middle?

00:01:34.300 --> 00:01:36.300
That's the best way to
think about the variance.

00:01:36.300 --> 00:01:39.336
Now what if we're dealing-- this
was for a population, right?

00:01:39.336 --> 00:01:40.960
And we said, if we
wanted to figure out

00:01:40.960 --> 00:01:43.552
the variance of men's
heights in the country,

00:01:43.552 --> 00:01:45.010
it would be very
hard to figure out

00:01:45.010 --> 00:01:46.610
the variance for the population.

00:01:46.610 --> 00:01:49.010
You would have to go
and, essentially, measure

00:01:49.010 --> 00:01:51.330
everyone's height--
250 million people.

00:01:51.330 --> 00:01:54.915
Or what if it's for some
population where it's just

00:01:54.915 --> 00:01:56.540
completely impossible
to have the data,

00:01:56.540 --> 00:01:57.690
or some random variable?

00:01:57.690 --> 00:01:58.920
And we'll go more
into that later.

00:01:58.920 --> 00:02:00.419
So a lot of times,
you actually want

00:02:00.419 --> 00:02:03.280
to estimate this
variance by taking

00:02:03.280 --> 00:02:04.740
the variance of a sample.

00:02:04.740 --> 00:02:07.560
Same way that you could never
get the mean of a population,

00:02:07.560 --> 00:02:08.960
but maybe you want
to estimate it

00:02:08.960 --> 00:02:11.360
by getting the mean of a sample.

00:02:11.360 --> 00:02:14.290
And we learned that
in that first video.

00:02:14.290 --> 00:02:18.182
This is-- if that's the
whole population, that's

00:02:18.182 --> 00:02:20.640
millions of data points-- or
even data points in the future

00:02:20.640 --> 00:02:22.015
that you'll never
be able to get,

00:02:22.015 --> 00:02:23.430
because it's a random variable.

00:02:23.430 --> 00:02:24.513
So this is the population.

00:02:27.110 --> 00:02:32.630
You might just want to estimate
things by looking at a sample.

00:02:32.630 --> 00:02:35.540
And this is actually what
most of inferential statistics

00:02:35.540 --> 00:02:36.420
is all about.

00:02:36.420 --> 00:02:38.820
Figuring out descriptive
statistics about the sample,

00:02:38.820 --> 00:02:40.930
and making inferences
about the population.

00:02:40.930 --> 00:02:43.660
Let me try this
drug on 100 people.

00:02:43.660 --> 00:02:45.780
And if it seems to have
statistically significant

00:02:45.780 --> 00:02:47.490
results, this drug
will probably work

00:02:47.490 --> 00:02:48.940
on the population as a whole.

00:02:48.940 --> 00:02:49.940
So that's what it's all about.

00:02:49.940 --> 00:02:51.523
So it's really
important to understand

00:02:51.523 --> 00:02:53.710
this notion of a sample
versus a population,

00:02:53.710 --> 00:02:55.830
and being able to
find statistics

00:02:55.830 --> 00:02:58.280
on a sample that,
for the most part,

00:02:58.280 --> 00:03:01.580
can describe the population or
help us estimate-- they call

00:03:01.580 --> 00:03:04.220
it parameters for
the population.

00:03:04.220 --> 00:03:07.360
So what's the mean of a-- let
me rewrite these definitions.

00:03:07.360 --> 00:03:08.870
What's the mean of a population?

00:03:08.870 --> 00:03:11.670
I'll do it in that purple--
purple for population.

00:03:11.670 --> 00:03:14.120
The mean of a
population, you just

00:03:14.120 --> 00:03:16.330
take each of the data points.

00:03:16.330 --> 00:03:19.740
So you take each of the data
points in the population-- xi.

00:03:19.740 --> 00:03:21.860
You sum them up.

00:03:21.860 --> 00:03:23.360
You start with the
first data point,

00:03:23.360 --> 00:03:25.330
and you go all the way
to the N-th data point,

00:03:25.330 --> 00:03:27.467
and you divide by N.
You sum them all up

00:03:27.467 --> 00:03:29.050
and divide them by
N. That's the mean.

00:03:29.050 --> 00:03:30.490
So then you plug it
into this formula,

00:03:30.490 --> 00:03:32.310
and you just can see
how far each point is

00:03:32.310 --> 00:03:34.520
from that central
point, from that mean.

00:03:34.520 --> 00:03:36.390
And you get the variance.

00:03:36.390 --> 00:03:39.890
Now what happens if
we do it for a sample?

00:03:39.890 --> 00:03:42.890
Well, if we want to estimate
the mean of a population

00:03:42.890 --> 00:03:46.005
by somehow calculating a mean
for a sample, the best thing

00:03:46.005 --> 00:03:47.380
I can think of--
and really these

00:03:47.380 --> 00:03:49.250
are kind of engineered formulas.

00:03:49.250 --> 00:03:50.750
These are human
beings saying, well,

00:03:50.750 --> 00:03:51.840
what is the best
way to sample it?

00:03:51.840 --> 00:03:54.660
Well, all we can do is really
take an average of our sample.

00:03:54.660 --> 00:03:56.576
And that's the sample mean.

00:03:56.576 --> 00:03:57.950
And we learned in
the first video

00:03:57.950 --> 00:04:00.500
that notation-- the formula's
almost identical to this.

00:04:00.500 --> 00:04:01.999
It's just the
notation is different.

00:04:01.999 --> 00:04:05.180
Instead of writing mu, you
write x with a line over it.

00:04:05.180 --> 00:04:07.470
Sample mean is equal to--

00:04:07.470 --> 00:04:10.440
Once again, you take each of the
data points now in the sample,

00:04:10.440 --> 00:04:12.250
not in the whole population.

00:04:12.250 --> 00:04:16.951
You sum them up, from the first
one and then to the n-th one,

00:04:16.951 --> 00:04:17.450
right?

00:04:17.450 --> 00:04:20.740
They're saying that there are
n data points in this sample.

00:04:20.740 --> 00:04:23.420
And then you divide it by the
number of data points you have.

00:04:23.420 --> 00:04:24.520
Fair enough.

00:04:24.520 --> 00:04:25.820
It's really the same formula.

00:04:25.820 --> 00:04:27.140
The way I took the
mean of a population,

00:04:27.140 --> 00:04:28.240
I said, well, if I
just have a sample,

00:04:28.240 --> 00:04:29.865
let me just take the
mean the same way.

00:04:29.865 --> 00:04:32.160
And it might-- it's
probably a good estimate

00:04:32.160 --> 00:04:34.070
of the mean of the population.

00:04:34.070 --> 00:04:36.370
Now, it gets interesting
when we talk about variance.

00:04:36.370 --> 00:04:39.310
So your natural reaction
is, OK, I have this sample.

00:04:39.310 --> 00:04:42.960
If I want to estimate the
variance of the population,

00:04:42.960 --> 00:04:45.210
why don't I just apply this
same formula, essentially,

00:04:45.210 --> 00:04:46.360
to the sample?

00:04:46.360 --> 00:04:49.910
So I could say-- and this is
actually a sample variance.

00:04:49.910 --> 00:04:54.690
They use a formula-- s squared.

00:04:54.690 --> 00:04:58.254
So sigma is kind of the
Greek-letter equivalent of s.

00:04:58.254 --> 00:04:59.920
So now when we're
dealing with a sample,

00:04:59.920 --> 00:05:01.120
we just write the s there.

00:05:01.120 --> 00:05:02.370
So this is sample variance.

00:05:02.370 --> 00:05:03.328
Let me write that down.

00:05:03.328 --> 00:05:04.110
Sample variance.

00:05:12.590 --> 00:05:14.900
So we might just say,
well, maybe a good way

00:05:14.900 --> 00:05:17.500
to take the sample variance
is, do it the same way.

00:05:17.500 --> 00:05:23.720
Let's take the distance of each
of the points in the sample,

00:05:23.720 --> 00:05:26.440
find out how far it is
from our sample mean.

00:05:26.440 --> 00:05:28.396
Right here, we used
the population mean.

00:05:28.396 --> 00:05:29.770
But now we'll just
use the sample

00:05:29.770 --> 00:05:31.457
mean, because that's
all we can have.

00:05:31.457 --> 00:05:33.040
We don't know what
the population mean

00:05:33.040 --> 00:05:35.550
is without looking at
the whole population.

00:05:35.550 --> 00:05:36.550
Take the square of that.

00:05:36.550 --> 00:05:37.508
That makes it positive.

00:05:37.508 --> 00:05:40.361
It has other properties
which we'll go over later.

00:05:40.361 --> 00:05:42.860
And then take the average of
all of these squared distances.

00:05:42.860 --> 00:05:45.070
So you take it from--
you sum them all up.

00:05:45.070 --> 00:05:48.340
And there's n of them to
sum up, right-- lowercase n.

00:05:48.340 --> 00:05:51.209
And you divide by lowercase n.

00:05:51.209 --> 00:05:53.250
You say, well, you know,
this is a good estimate.

00:05:53.250 --> 00:05:55.030
Whatever this variance
is, that might

00:05:55.030 --> 00:05:57.071
be a good estimate for
the population as a whole.

00:05:57.071 --> 00:06:00.460
And actually this is what
some people often refer to,

00:06:00.460 --> 00:06:02.110
when they talk about
sample variance.

00:06:02.110 --> 00:06:05.280
And sometimes it'll actually
be referred to as this.

00:06:05.280 --> 00:06:07.537
They'll put a little
lowercase n there.

00:06:07.537 --> 00:06:09.870
And the reason why I do that
is because we divided by n.

00:06:09.870 --> 00:06:12.070
So you say, Sal, what's
the problem here?

00:06:12.070 --> 00:06:13.190
And then the problem--
and I'll give you

00:06:13.190 --> 00:06:14.360
the intuition, because
this is actually

00:06:14.360 --> 00:06:16.290
something that used
to boggle my mind.

00:06:16.290 --> 00:06:18.680
And I'm still,
frankly, struggling

00:06:18.680 --> 00:06:21.345
with the intuition behind it.

00:06:21.345 --> 00:06:23.490
Well, I have the
intuition, but more

00:06:23.490 --> 00:06:25.305
of kind of rigorously
proving it to myself,

00:06:25.305 --> 00:06:27.110
that this is
definitely the case.

00:06:27.110 --> 00:06:28.132
But think about this.

00:06:28.132 --> 00:06:30.590
If I have a bunch of numbers--
and I'll draw a number line,

00:06:30.590 --> 00:06:32.750
here.

00:06:32.750 --> 00:06:36.330
If I draw a number
line here-- let's

00:06:36.330 --> 00:06:39.446
say I have a bunch of
numbers in my population.

00:06:39.446 --> 00:06:41.320
So let's say-- I'm just
going to randomly put

00:06:41.320 --> 00:06:43.907
a bunch of numbers
in my population.

00:06:43.907 --> 00:06:45.990
And the ones to the right
are bigger than the ones

00:06:45.990 --> 00:06:46.490
to the left.

00:06:49.120 --> 00:06:51.510
And if I were to take a
sample of them, right?

00:06:51.510 --> 00:06:54.610
Maybe I take-- and the
sample, it's random.

00:06:54.610 --> 00:06:56.360
You actually want to
take a random sample.

00:06:56.360 --> 00:06:57.985
You don't want to be
skewed in any way.

00:06:57.985 --> 00:07:04.420
So maybe I take this one, this
one, this one, and that one,

00:07:04.420 --> 00:07:05.125
right?

00:07:05.125 --> 00:07:07.250
And then if I were to take
the mean of that number,

00:07:07.250 --> 00:07:08.340
that number, that
number, and that number,

00:07:08.340 --> 00:07:09.715
it'll be someplace
in the middle.

00:07:09.715 --> 00:07:11.220
It might be
someplace over there.

00:07:11.220 --> 00:07:13.761
And then, if I wanted to figure
out the sample variance using

00:07:13.761 --> 00:07:15.640
this formula, I'd
say, OK, this distance

00:07:15.640 --> 00:07:19.440
squared plus this distance
squared plus this distance

00:07:19.440 --> 00:07:22.630
squared plus that distance
squared, and average them

00:07:22.630 --> 00:07:23.325
all out.

00:07:23.325 --> 00:07:24.700
And then I would
get this number.

00:07:24.700 --> 00:07:27.750
And that probably would be
a pretty good approximation

00:07:27.750 --> 00:07:30.280
for the variance of
this entire population.

00:07:30.280 --> 00:07:31.880
The population of
the mean is probably

00:07:31.880 --> 00:07:34.649
going to be-- I don't know, it
might be pretty close to this.

00:07:34.649 --> 00:07:37.190
If we actually took all of the
data points and averaged them,

00:07:37.190 --> 00:07:39.160
maybe they're, like,
here someplace.

00:07:39.160 --> 00:07:40.045
And then, if you figure
out the variance,

00:07:40.045 --> 00:07:42.336
it probably would be pretty
close to the average of all

00:07:42.336 --> 00:07:46.771
of these lines, right, of the
sample variance distances.

00:07:46.771 --> 00:07:47.270
Fair enough.

00:07:47.270 --> 00:07:49.770
So you say, hey, Sal, this
looks pretty good now.

00:07:49.770 --> 00:07:52.030
But there's one little catch.

00:07:52.030 --> 00:07:54.530
What if-- I mean, there's always
a probability that, instead

00:07:54.530 --> 00:07:56.810
of picking these fairly
well-distributed numbers

00:07:56.810 --> 00:07:59.130
in my sample, what
if I happened to pick

00:07:59.130 --> 00:08:02.560
this number, this number, and
that number, and, let's say,

00:08:02.560 --> 00:08:05.580
that number, as my sample?

00:08:05.580 --> 00:08:08.430
Well, whatever your sample
is, your sample mean's

00:08:08.430 --> 00:08:10.350
always going to be in
the middle of it, right?

00:08:10.350 --> 00:08:13.110
So in this case, your sample
mean might be right here.

00:08:13.110 --> 00:08:14.360
So all of these numbers,
you might say, OK,

00:08:14.360 --> 00:08:16.180
this number's not too
far from that number.

00:08:16.180 --> 00:08:17.690
That number's not too far.

00:08:17.690 --> 00:08:19.200
And then that
number's not too far.

00:08:19.200 --> 00:08:21.540
So your sample variance,
when you do it this way,

00:08:21.540 --> 00:08:23.750
it might turn out
a little bit low.

00:08:23.750 --> 00:08:27.890
Because all of these numbers,
they're almost, by definition,

00:08:27.890 --> 00:08:30.500
going to be pretty close
to the mean of each other.

00:08:30.500 --> 00:08:34.340
But in this case, your
sample is kind of skewed,

00:08:34.340 --> 00:08:36.230
and the actual mean
of the population

00:08:36.230 --> 00:08:38.130
is out here someplace.

00:08:38.130 --> 00:08:40.350
So the actual variance
of the sample,

00:08:40.350 --> 00:08:41.872
if you had actually
known the mean--

00:08:41.872 --> 00:08:44.330
I know this is all a little
confusing-- if you had actually

00:08:44.330 --> 00:08:46.679
known the mean, you
would've said, oh, wow.

00:08:46.679 --> 00:08:48.220
You would have found
these distances,

00:08:48.220 --> 00:08:51.214
which would have
been a lot more.

00:08:51.214 --> 00:08:52.630
The whole point
of what I'm saying

00:08:52.630 --> 00:08:55.000
is, when you take
a sample, there's

00:08:55.000 --> 00:08:58.090
some chance that
your sample mean is

00:08:58.090 --> 00:09:00.497
pretty close to the
population mean.

00:09:00.497 --> 00:09:02.830
Maybe your sample mean is
here, and your population mean

00:09:02.830 --> 00:09:03.411
is here.

00:09:03.411 --> 00:09:04.910
And then this formula
would probably

00:09:04.910 --> 00:09:07.451
work out pretty well, at least
given your sample data points,

00:09:07.451 --> 00:09:09.300
of figuring out what
the variance is.

00:09:09.300 --> 00:09:13.520
But there's a reasonable
chance that your sample

00:09:13.520 --> 00:09:15.050
mean-- your sample
mean is always

00:09:15.050 --> 00:09:16.870
going to be within your
data sample, right?

00:09:16.870 --> 00:09:18.820
It's always going to be the
center of your data sample.

00:09:18.820 --> 00:09:21.330
But it's completely possible
that the population mean

00:09:21.330 --> 00:09:22.650
is outside of your data sample.

00:09:22.650 --> 00:09:23.683
It might have just
been-- you know,

00:09:23.683 --> 00:09:25.450
you just happened
to pick ones that

00:09:25.450 --> 00:09:28.290
don't contain the
actual population mean.

00:09:28.290 --> 00:09:31.680
And then this sample
variance, calculated this way,

00:09:31.680 --> 00:09:35.802
will actually underestimate
the actual population variance,

00:09:35.802 --> 00:09:38.260
because they're always going
to be closer to their own mean

00:09:38.260 --> 00:09:40.060
than they are to
the population mean.

00:09:40.060 --> 00:09:44.039
And if you're understanding,
frankly, even 10% of this,

00:09:44.039 --> 00:09:45.830
you are a very advanced
statistics student.

00:09:45.830 --> 00:09:48.490
But I'm saying all of
this to just give you,

00:09:48.490 --> 00:09:51.030
hopefully, some
intuition to realize

00:09:51.030 --> 00:09:53.560
that this will
often underestimate.

00:09:53.560 --> 00:09:56.580
This formula will
often underestimate

00:09:56.580 --> 00:09:59.107
the actual population variance.

00:09:59.107 --> 00:10:00.940
And there's a formula--
and this is actually

00:10:00.940 --> 00:10:02.430
proven more rigorously
than I'll do

00:10:02.430 --> 00:10:05.920
it-- that is considered to be
a better-- or, they'll call it,

00:10:05.920 --> 00:10:09.090
unbiased-- estimate of
the population variance,

00:10:09.090 --> 00:10:11.430
or the unbiased sample variance.

00:10:11.430 --> 00:10:14.190
And sometimes it's just
denoted by the s squared again.

00:10:14.190 --> 00:10:19.080
Sometimes it's denoted by
this-- s n minus 1 squared--

00:10:19.080 --> 00:10:20.770
and I'll show you why.

00:10:20.770 --> 00:10:22.390
It's almost the same thing.

00:10:22.390 --> 00:10:24.000
You take each of
the data points,

00:10:24.000 --> 00:10:28.080
figure out how far they
are from the sample mean,

00:10:28.080 --> 00:10:30.390
you square them, and then
you take the average of those

00:10:30.390 --> 00:10:33.570
squared, except for
one slight difference.

00:10:33.570 --> 00:10:35.800
i equals 1 to i equals n.

00:10:35.800 --> 00:10:38.420
Instead of dividing
by n, you divide

00:10:38.420 --> 00:10:41.970
by a slightly smaller number.

00:10:41.970 --> 00:10:44.390
You divide by n minus 1.

00:10:44.390 --> 00:10:47.620
So when you divide by n minus
1 instead of dividing by n,

00:10:47.620 --> 00:10:49.704
you're going to get a
slightly larger number here.

00:10:49.704 --> 00:10:51.286
And it turns out
that this is actually

00:10:51.286 --> 00:10:52.295
a much better estimate.

00:10:52.295 --> 00:10:54.378
And one day, I'm going to
write a computer program

00:10:54.378 --> 00:10:56.830
to at least prove it to
myself, experimentally,

00:10:56.830 --> 00:11:01.804
that this is a better estimate
of the population variance.

00:11:01.804 --> 00:11:03.470
And you would calculate
it the same way.

00:11:03.470 --> 00:11:05.193
You just divide by n minus 1.

00:11:05.193 --> 00:11:07.526
The other way to think about
it-- and actually-- no, no.

00:11:07.526 --> 00:11:08.530
I'm all out of time.

00:11:08.530 --> 00:11:10.330
I'll leave you there, now,
and then, in the next video,

00:11:10.330 --> 00:11:11.450
we'll do a couple
calculations, just

00:11:11.450 --> 00:11:13.400
so you don't get too
overwhelmed with these ideas,

00:11:13.400 --> 00:11:15.233
because we're getting
a little bit abstract.

00:11:15.233 --> 00:11:16.639
See you in the next video.


Statistics: Variance of a population

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.960
I realized I made a slight
error in the last video,

00:00:02.960 --> 00:00:06.920
when I talk about the
population and the sample mean.

00:00:06.920 --> 00:00:09.210
So I will rewrite the equations.

00:00:09.210 --> 00:00:11.037
I realize I made a
slight notational error,

00:00:11.037 --> 00:00:12.870
and it might have
confused you a little bit.

00:00:12.870 --> 00:00:14.900
So just to review a little
bit, it never hurts.

00:00:14.900 --> 00:00:17.210
The mean of a
population-- once again,

00:00:17.210 --> 00:00:21.330
that's mu-- the mean of
a population is equal to,

00:00:21.330 --> 00:00:24.620
you take the sum of
each of the data points.

00:00:24.620 --> 00:00:27.220
So you take the sum--
that's that big sigma is

00:00:27.220 --> 00:00:29.880
for-- of each of
the data points.

00:00:29.880 --> 00:00:30.760
So x sub i.

00:00:30.760 --> 00:00:32.380
I had written x sub n, before.

00:00:32.380 --> 00:00:33.937
And if you review
that last video,

00:00:33.937 --> 00:00:35.770
you can see why might
be a little confusing.

00:00:35.770 --> 00:00:38.680
And you start with
the first data point.

00:00:38.680 --> 00:00:40.921
So i is equal to 1.

00:00:40.921 --> 00:00:42.420
You start with the
first data point,

00:00:42.420 --> 00:00:45.510
and you take the sum all the
way to the n-th data point.

00:00:45.510 --> 00:00:46.010
Right?

00:00:46.010 --> 00:00:47.630
Where we have a big
capital N, where

00:00:47.630 --> 00:00:51.370
N is the total number of
elements in the population.

00:00:51.370 --> 00:00:53.740
And then you divide that by N.

00:00:53.740 --> 00:00:59.080
So that's another way of writing
x sub 1 plus x sub 2, plus--

00:00:59.080 --> 00:01:01.130
and you just keep
adding, bum, bum, bum,

00:01:01.130 --> 00:01:03.980
however many there
are-- x sub N.

00:01:03.980 --> 00:01:06.550
And then, you divide
that by N. And I think

00:01:06.550 --> 00:01:09.140
that's what you're familiar with
as just the arithmetic mean,

00:01:09.140 --> 00:01:09.841
or the average.

00:01:09.841 --> 00:01:11.340
You just add up all
of the elements,

00:01:11.340 --> 00:01:13.715
and you divide by the total
number of elements there are.

00:01:13.715 --> 00:01:15.620
That's just a fancy
way of writing that.

00:01:15.620 --> 00:01:18.476
And then, the sample mean is
essentially the same thing,

00:01:18.476 --> 00:01:20.350
although you use slightly
different notation.

00:01:20.350 --> 00:01:25.140
The sample mean is written
as x with a line over it.

00:01:25.140 --> 00:01:28.920
And that's equal to,
once again, the sum

00:01:28.920 --> 00:01:32.130
of the elements in
the sample, right,

00:01:32.130 --> 00:01:34.810
and then you have a slight
notational difference.

00:01:34.810 --> 00:01:36.910
You start at the first
element in the sample.

00:01:36.910 --> 00:01:39.750
And you go to the number
of elements in the sample.

00:01:39.750 --> 00:01:41.870
And that's why they
use that lowercase n.

00:01:41.870 --> 00:01:44.610
There are a big N elements
in the whole population.

00:01:44.610 --> 00:01:46.600
And if you took some
subset of that-- we're

00:01:46.600 --> 00:01:50.912
assuming that small n is
less than or equal to big N.

00:01:50.912 --> 00:01:52.870
And anyway, you divide
that by the total number

00:01:52.870 --> 00:01:55.980
of elements in the sample.

00:01:55.980 --> 00:02:00.500
So once again, this would be
x1 plus x2, plus dun, dun, dun,

00:02:00.500 --> 00:02:05.150
plus x lower case n,
divided by lowercase n.

00:02:05.150 --> 00:02:07.120
These are essentially
the same thing.

00:02:07.120 --> 00:02:09.404
If your sample was the
entire population, then

00:02:09.404 --> 00:02:11.070
these ends would be
equal to each other.

00:02:11.070 --> 00:02:13.028
And these numbers would
be equal to each other.

00:02:13.028 --> 00:02:14.730
But just the
notational difference,

00:02:14.730 --> 00:02:16.820
if you ever see this, you know
you're dealing with a sample.

00:02:16.820 --> 00:02:19.250
Here you know you're dealing
with the entire population.

00:02:19.250 --> 00:02:22.160
And similarly, big N,
entire population, small n,

00:02:22.160 --> 00:02:23.620
the sample.

00:02:23.620 --> 00:02:24.610
Fair enough.

00:02:24.610 --> 00:02:27.780
I think we're now ready to learn
a little bit about measures

00:02:27.780 --> 00:02:29.400
of dispersion.

00:02:29.400 --> 00:02:31.670
So the mean, and the
mode, and the median,

00:02:31.670 --> 00:02:33.910
that we covered in the first
video in this playlist,

00:02:33.910 --> 00:02:38.625
were all ways of measuring the
central tendency of a data set.

00:02:38.625 --> 00:02:40.000
Or kind of, picking
a number that

00:02:40.000 --> 00:02:43.046
is most representative
of all of the numbers.

00:02:43.046 --> 00:02:44.420
But we lose a lot
of information.

00:02:44.420 --> 00:02:46.340
We don't know whether all of
the numbers in the data set

00:02:46.340 --> 00:02:48.355
are close to that number,
close to the mean.

00:02:48.355 --> 00:02:50.480
Or maybe they're all really
far away from the mean.

00:02:50.480 --> 00:02:53.063
And that's why we want to come
up with measures of dispersion.

00:02:57.520 --> 00:02:59.650
And let me tell you, let
me show you what I mean.

00:02:59.650 --> 00:03:02.290
So let's say I have one set,
and let's say, I don't know,

00:03:02.290 --> 00:03:09.699
let's say it is a 2,
a 2, and a 3, and a 3.

00:03:09.699 --> 00:03:10.990
Let's say this is a population.

00:03:10.990 --> 00:03:14.844
Let's just deal with
population means and population

00:03:14.844 --> 00:03:15.760
disbursements for now.

00:03:15.760 --> 00:03:17.570
So what's the mean here?

00:03:17.570 --> 00:03:22.800
The mean here is going to
be 2 plus 2 plus 3 plus 3,

00:03:22.800 --> 00:03:26.900
all of that over 4.

00:03:26.900 --> 00:03:27.680
And what is that?

00:03:27.680 --> 00:03:29.890
That's equal to
2 and 1/2, right?

00:03:29.890 --> 00:03:33.180
4 plus 6, divided by 4, right.

00:03:33.180 --> 00:03:35.730
That's equal to 2.5.

00:03:35.730 --> 00:03:37.650
Fair enough.

00:03:37.650 --> 00:03:38.650
Fair enough.

00:03:38.650 --> 00:03:39.660
Now what if we had this?

00:03:39.660 --> 00:03:48.970
What if we had the numbers, I
don't know, 0, 0, and 5, and 5.

00:03:48.970 --> 00:03:50.470
So these are the
numbers in the set.

00:03:50.470 --> 00:03:52.230
I'll put commas,
just so you know

00:03:52.230 --> 00:03:54.200
these are separate numbers.

00:03:54.200 --> 00:03:55.850
What's the mean here?

00:03:55.850 --> 00:03:58.920
Well, the mean here-- let's
say this is the population

00:03:58.920 --> 00:04:00.700
that we're sampling,
This isn't a sample,

00:04:00.700 --> 00:04:02.020
this is the entire population.

00:04:02.020 --> 00:04:04.620
And you'll see why I'm making
that distinction later--

00:04:04.620 --> 00:04:08.485
so it'll be 0 plus
0 plus 5 plus 5.

00:04:08.485 --> 00:04:13.120
Well that's 10, divided
by 4 is equal to 2.5.

00:04:13.120 --> 00:04:17.100
So the arithmetic mean of
both of these populations

00:04:17.100 --> 00:04:18.060
are the same number.

00:04:18.060 --> 00:04:20.230
They're both 2.5.

00:04:20.230 --> 00:04:21.470
Right?

00:04:21.470 --> 00:04:24.480
But you see that these sets are
kind of, they are different.

00:04:24.480 --> 00:04:28.240
Here, all of the numbers are
pretty close to 2.5, right?

00:04:28.240 --> 00:04:32.620
Well, here, sure their mean,
their arithmetic mean, is 2.5,

00:04:32.620 --> 00:04:34.710
but they're further
away from 2.5.

00:04:34.710 --> 00:04:38.310
Or the distances of
each of these numbers,

00:04:38.310 --> 00:04:42.960
each of the numbers in the set,
their distance from the mean

00:04:42.960 --> 00:04:43.645
is further.

00:04:43.645 --> 00:04:46.020
So you can kind of view them
that they're more dispersed,

00:04:46.020 --> 00:04:47.460
they're further
away from the mean.

00:04:47.460 --> 00:04:48.960
Or another way you
could think of it

00:04:48.960 --> 00:04:52.790
is, the mean, although it does
measure its central tendency,

00:04:52.790 --> 00:04:55.620
it's not quite as indicative
of all the numbers.

00:04:55.620 --> 00:04:58.900
The numbers are much further
away from the mean, on average.

00:04:58.900 --> 00:05:00.390
So how do you measure that?

00:05:00.390 --> 00:05:02.930
Well, you measure
that with a variance.

00:05:02.930 --> 00:05:06.150
And this is something I
found, and even in my own,

00:05:06.150 --> 00:05:08.220
it seems complicated when
you first look at it.

00:05:08.220 --> 00:05:11.930
And most statistics textbooks
use fairly complex notations.

00:05:11.930 --> 00:05:15.270
But the idea is almost
as straightforward

00:05:15.270 --> 00:05:17.159
as the arithmetic mean.

00:05:17.159 --> 00:05:19.700
So what they'll do is, they'll
write, you know, the variance.

00:05:19.700 --> 00:05:21.550
And they'll write it
as this letter sigma,

00:05:21.550 --> 00:05:23.800
this Greek letter-- I wrote
the top part too long.

00:05:23.800 --> 00:05:25.389
Let me actually undo that.

00:05:25.389 --> 00:05:27.680
I don't want you to spend
the rest of your life writing

00:05:27.680 --> 00:05:31.316
with a big top part-- they
write it as sigma squared.

00:05:31.316 --> 00:05:32.690
And we'll talk,
in a few seconds,

00:05:32.690 --> 00:05:34.856
about why it's written as--
you know, why don't they

00:05:34.856 --> 00:05:36.210
just write v for variance?

00:05:36.210 --> 00:05:38.310
Why do they write this
weird letter squared?

00:05:38.310 --> 00:05:40.720
I'll talk about it in a second.

00:05:40.720 --> 00:05:43.640
But the variance of a
population is defined--

00:05:43.640 --> 00:05:47.020
and once again, these are
just human-derived constructs

00:05:47.020 --> 00:05:49.490
of trying to get our
minds around data, being

00:05:49.490 --> 00:05:51.860
able to describe a set
of data, without having

00:05:51.860 --> 00:05:53.004
to list all the numbers.

00:05:53.004 --> 00:05:54.420
And then, being
able to understand

00:05:54.420 --> 00:05:56.590
what that data might
represent, or what

00:05:56.590 --> 00:05:58.380
can represent that data.

00:05:58.380 --> 00:06:03.990
So what you do is,
you take the sum,

00:06:03.990 --> 00:06:08.460
and you start with all of
the points in the population.

00:06:08.460 --> 00:06:10.420
But instead of taking
the sum of the points,

00:06:10.420 --> 00:06:16.764
you take each point, x sub i,
and you subtract from that--

00:06:16.764 --> 00:06:19.180
and actually, it doesn't matter
if you subtract from that,

00:06:19.180 --> 00:06:21.580
or you subtract
that from the mean,

00:06:21.580 --> 00:06:25.710
the population mean,
and then you square it.

00:06:25.710 --> 00:06:26.460
So what is this?

00:06:26.460 --> 00:06:30.062
This is the distance between
each number and the mean.

00:06:30.062 --> 00:06:32.520
And then, when you square it,
it becomes a positive number.

00:06:32.520 --> 00:06:35.940
So you can kind of view it as
the squared absolute distance

00:06:35.940 --> 00:06:38.780
between each number and
the mean of that set.

00:06:38.780 --> 00:06:41.550
And then you take the
average of all of those.

00:06:41.550 --> 00:06:43.920
You divide that by n.

00:06:43.920 --> 00:06:46.260
Now that might seem like
a very complicated notion,

00:06:46.260 --> 00:06:49.630
but let's calculate it
for these two data sets.

00:06:49.630 --> 00:06:55.000
So here-- let me rewrite that
first data set-- it's 2, 2, 3,

00:06:55.000 --> 00:06:56.330
and 3.

00:06:56.330 --> 00:06:59.700
So what is-- let me, actually,
let me write it this way.

00:06:59.700 --> 00:07:03.940
I think this will help explain
it for you a little bit better.

00:07:03.940 --> 00:07:07.860
So if I wrote i.

00:07:07.860 --> 00:07:11.770
i1, i2, i3, i4.

00:07:11.770 --> 00:07:13.410
That's i.

00:07:13.410 --> 00:07:15.836
Then x sub i.

00:07:15.836 --> 00:07:17.730
And you know, it's
kind of arbitrary.

00:07:17.730 --> 00:07:18.640
It's just, you
know, this is saying

00:07:18.640 --> 00:07:20.641
the first term, the second
term, the third term.

00:07:20.641 --> 00:07:22.098
I could have had
this in any order.

00:07:22.098 --> 00:07:22.970
It doesn't matter.

00:07:22.970 --> 00:07:24.870
Maybe this was the first
term, and this was the second,

00:07:24.870 --> 00:07:25.380
and this was the third.

00:07:25.380 --> 00:07:25.800
It doesn't matter.

00:07:25.800 --> 00:07:27.550
Because we're just going to
add them all up, and then

00:07:27.550 --> 00:07:28.102
divide them.

00:07:28.102 --> 00:07:29.810
So it doesn't matter
what order we do it.

00:07:29.810 --> 00:07:39.130
But anyway, x sub 1 is equal
to 2, x sub 2 is equal to 2,

00:07:39.130 --> 00:07:41.590
x sub 3 is equal
to 3-- I'll stop

00:07:41.590 --> 00:07:44.320
writing this equal thing--
x sub 4 is equal to 4.

00:07:44.320 --> 00:07:45.230
Right?

00:07:45.230 --> 00:07:47.087
What's the mean?

00:07:47.087 --> 00:07:48.670
Well, we figured out
the mean up here.

00:07:48.670 --> 00:07:52.590
We just took these numbers and
added them and divided by 4.

00:07:52.590 --> 00:07:59.860
The mean is 2.5.

00:07:59.860 --> 00:08:05.151
So what is x sub
i minus the mean?

00:08:05.151 --> 00:08:05.650
Right?

00:08:05.650 --> 00:08:07.480
We're slowly building
up to this equation.

00:08:07.480 --> 00:08:11.640
What is x sub i minus the mean?

00:08:11.640 --> 00:08:14.754
Well, 2 minus 2.5,
that's minus 0.5.

00:08:14.754 --> 00:08:19.281
2 minus 2.5, that's
once again, minus 0.5.

00:08:19.281 --> 00:08:22.251
3 minus 2.5, that's 0.5.

00:08:22.251 --> 00:08:25.340
3 minus 2.5, that's 0.5.

00:08:25.340 --> 00:08:26.940
Fair enough.

00:08:26.940 --> 00:08:29.300
Now this equation, they
want us to square this.

00:08:29.300 --> 00:08:33.135
So x sub i minus
the mean squared.

00:08:35.287 --> 00:08:37.620
There's several other properties
we'll talk about later.

00:08:37.620 --> 00:08:40.429
But the most important thing
that the squaring does,

00:08:40.429 --> 00:08:42.669
and the absolute value
could have done it as well.

00:08:42.669 --> 00:08:44.770
But the squaring makes
all of these positive.

00:08:44.770 --> 00:08:49.620
So minus 0.5 squared
is positive 0.25.

00:08:49.620 --> 00:08:51.720
This is positive 0.25.

00:08:51.720 --> 00:08:54.770
Plus 0.5 squared is
also positive 0.25.

00:08:54.770 --> 00:08:57.610
And this is positive 0.25.

00:08:57.610 --> 00:09:03.250
And so what is-- so if we
wanted to know the sum from i

00:09:03.250 --> 00:09:08.990
is equal to 1 to 4 of x
sub i minus the mean, which

00:09:08.990 --> 00:09:11.030
is 2.5 squared.

00:09:11.030 --> 00:09:13.760
This is equal to, what, the
sum of all of these numbers.

00:09:13.760 --> 00:09:15.987
This is just saying
sum all of these.

00:09:15.987 --> 00:09:16.820
So sum all of these.

00:09:16.820 --> 00:09:19.770
0.25, so that's equal to 1.

00:09:19.770 --> 00:09:21.300
But this isn't the variance yet.

00:09:21.300 --> 00:09:23.779
The variance is this
thing-- let's look

00:09:23.779 --> 00:09:26.320
at the original formula-- the
variance is this thing, divided

00:09:26.320 --> 00:09:29.630
by the total number
of numbers you have.

00:09:29.630 --> 00:09:32.390
So you take this, and
then you-- so the variance

00:09:32.390 --> 00:09:36.800
is equal to this thing, divided
by the total number of numbers,

00:09:36.800 --> 00:09:37.490
which is 4.

00:09:37.490 --> 00:09:40.250
Which is equal to 0.25.

00:09:40.250 --> 00:09:42.180
And you see, I mean
here, the distance

00:09:42.180 --> 00:09:48.014
from every number to the
mean squared was 0.25.

00:09:48.014 --> 00:09:50.180
So the average of all of
these, which is essentially

00:09:50.180 --> 00:09:53.721
what the variance is, the
average was also 0.25.

00:09:53.721 --> 00:09:55.970
And I'll do another example
where these are different.

00:09:55.970 --> 00:09:57.130
The other example in this
video, actually, they're

00:09:57.130 --> 00:09:58.000
not different.

00:09:58.000 --> 00:10:01.830
But you see here, the average
squared distance from the mean,

00:10:01.830 --> 00:10:05.250
in that first data set, is 0.25.

00:10:05.250 --> 00:10:09.260
And here, what's the average
squared distance from the mean?

00:10:09.260 --> 00:10:10.460
So let's see.

00:10:10.460 --> 00:10:11.730
This is how far from the mean?

00:10:11.730 --> 00:10:18.400
It is-- So let's say x--
let me write-- x sub i.

00:10:18.400 --> 00:10:23.330
And then, x sub i minus the
mean for this population.

00:10:23.330 --> 00:10:26.770
So x sub i, there's a 0,
there's a 0, there's a 5,

00:10:26.770 --> 00:10:27.640
and a 5, right?

00:10:27.640 --> 00:10:35.230
There's the first term x sub 1,
this is x sub 2, and so forth.

00:10:35.230 --> 00:10:39.304
And then, each of these numbers
0 minus 2, that's minus 2.5.

00:10:39.304 --> 00:10:41.550
0 minus 2.5.

00:10:41.550 --> 00:10:42.050
Right?

00:10:42.050 --> 00:10:43.620
This could be 2.5, right?

00:10:43.620 --> 00:10:44.740
That's the mean.

00:10:44.740 --> 00:10:46.475
It's minus 2.5.

00:10:46.475 --> 00:10:51.500
5 minus 2.5 is 2.5,
5 minus 2.5 is 2.5.

00:10:51.500 --> 00:10:57.670
Now, if you took x sub i minus
the mean squared, 2.5 squared

00:10:57.670 --> 00:10:58.550
is what?

00:10:58.550 --> 00:11:00.320
6.25 and it becomes positive.

00:11:00.320 --> 00:11:02.160
So it's 6.25.

00:11:02.160 --> 00:11:05.290
That's the same thing, 6.25.

00:11:05.290 --> 00:11:10.550
That's already
positive, so 6.25, 6.25.

00:11:10.550 --> 00:11:12.460
And so, the variance
is the sum of all

00:11:12.460 --> 00:11:15.880
of these, divided by the total
number of numbers there are,

00:11:15.880 --> 00:11:16.751
right?

00:11:16.751 --> 00:11:19.250
So we take the sum of all-- so
this is the average of these,

00:11:19.250 --> 00:11:20.420
and that's pretty
easy to calculate.

00:11:20.420 --> 00:11:22.570
If you added all of these
up and divided by 4,

00:11:22.570 --> 00:11:24.820
you're just going to get 6.25.

00:11:24.820 --> 00:11:28.336
So the variance of this
population is 6.25.

00:11:28.336 --> 00:11:29.210
So there you have it.

00:11:29.210 --> 00:11:32.510
You have two data sets where
their means are the same.

00:11:32.510 --> 00:11:35.830
But the variance of this
data set is equal to,

00:11:35.830 --> 00:11:38.130
we figured out, was 0.25.

00:11:38.130 --> 00:11:44.772
While the variance of this
data set is equal to 6.25.

00:11:44.772 --> 00:11:47.230
And it's hard right now to have
an intuition of what this--

00:11:47.230 --> 00:11:49.250
you know, how does this
6 relate to the 0.25--

00:11:49.250 --> 00:11:51.720
but you know that this
is a larger number.

00:11:51.720 --> 00:11:55.070
This is a much larger
number than this is.

00:11:55.070 --> 00:11:57.990
Which tells you, just kind
of at a intuitive feel,

00:11:57.990 --> 00:12:01.040
that the numbers in this that
are, on average, much further

00:12:01.040 --> 00:12:04.510
away from the mean than the
numbers in this data set.

00:12:04.510 --> 00:12:05.800
Anyway, I'm out of time.

00:12:05.800 --> 00:12:06.640
I'll see you in the
next video, and we'll

00:12:06.640 --> 00:12:07.848
talk a little bit about this.

00:12:07.848 --> 00:12:10.624
And we'll talk-- and we'll move
into the standard deviation.

00:12:10.624 --> 00:12:12.040
And then, what
happens if you take

00:12:12.040 --> 00:12:14.600
these of a sample
instead of a population.

00:12:14.600 --> 00:12:16.800
Everything we're doing
here, we're taking the mean

00:12:16.800 --> 00:12:18.940
and the variance of every
number in the data set.

00:12:18.940 --> 00:12:20.315
Later we'll do it
for the sample.

00:12:20.315 --> 00:12:22.026
See you soon.


Statistics: The average

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:03.270
Welcome to the playlist
on statistics.

00:00:03.270 --> 00:00:06.380
Something I've been meaning
to do for some time.

00:00:06.380 --> 00:00:08.710
So anyway, I just want to get
right into the meat of it and

00:00:08.710 --> 00:00:12.220
I'll try to do as many examples
as possible and hopefully

00:00:12.220 --> 00:00:14.990
give you the feel for what
statistics is all about.

00:00:14.990 --> 00:00:16.850
And, really, just to kind of
start off in case you're not

00:00:16.850 --> 00:00:18.550
familiar with it -- although, I
think a lot of people have an

00:00:18.550 --> 00:00:20.660
intuitive feel for what
statistics is about.

00:00:23.200 --> 00:00:27.110
And essentially -- well in very
general terms it's kind of

00:00:27.110 --> 00:00:28.590
getting your head around data.

00:00:28.590 --> 00:00:30.540
And it can broadly
be classified.

00:00:30.540 --> 00:00:32.640
Well there are maybe
three categories.

00:00:32.640 --> 00:00:35.330
You have descriptive.

00:00:35.330 --> 00:00:39.150
So say you have a lot of data
and you wanted to tell someone

00:00:39.150 --> 00:00:41.480
about it without giving
them all of the data.

00:00:41.480 --> 00:00:45.360
Maybe you can kind of find
indicative numbers that

00:00:45.360 --> 00:00:47.560
somehow represent all of
that data without having to

00:00:47.560 --> 00:00:48.540
go over all of the data.

00:00:48.540 --> 00:00:50.370
That would be
descriptive statistics.

00:00:50.370 --> 00:00:51.510
There's also predictive.

00:00:51.510 --> 00:00:53.110
Well, I'll kind of
group them together.

00:00:53.110 --> 00:00:55.110
There's inferential statistics.

00:00:58.800 --> 00:01:00.920
And this is when you use
data to essentially make

00:01:00.920 --> 00:01:02.200
conclusions about things.

00:01:02.200 --> 00:01:06.500
So let's say you've sampled
some data from a population --

00:01:06.500 --> 00:01:08.890
and we'll talk a lot about
samples versus populations but

00:01:08.890 --> 00:01:11.390
I think you have just a basic
sense of what that is, right?

00:01:11.390 --> 00:01:13.800
If I survey three people
who are going to vote for

00:01:13.800 --> 00:01:16.500
president, I clearly haven't
surveyed the entire population.

00:01:16.500 --> 00:01:18.160
I've surveyed a sample.

00:01:18.160 --> 00:01:21.780
But what inferential statistics
are all about are if we can do

00:01:21.780 --> 00:01:24.890
some math on the samples, maybe
we can make inferences or

00:01:24.890 --> 00:01:27.740
conclusions about the
population as a whole.

00:01:27.740 --> 00:01:29.760
Well, anyway, that's just
a big picture of what

00:01:29.760 --> 00:01:30.800
statistics is all about.

00:01:30.800 --> 00:01:33.620
Let's just get into the
meat of it and we'll start

00:01:33.620 --> 00:01:34.746
with the descriptive.

00:01:37.950 --> 00:01:41.040
So the first thing that, I
don't know, that I would want

00:01:41.040 --> 00:01:44.330
to do or I think most people
would want to do when they are

00:01:44.330 --> 00:01:47.320
given a whole set of numbers in
they're told to describe it.

00:01:47.320 --> 00:01:51.030
Well, maybe I can come up with
some number that is most

00:01:51.030 --> 00:01:54.430
indicative of all of the
numbers in that set.

00:01:54.430 --> 00:01:57.092
Or some number that represents,
kind of, the central tendency

00:01:57.092 --> 00:01:59.900
-- this is a word you'll see
a lot in statistics books.

00:01:59.900 --> 00:02:03.040
The central tendency
of a set of numbers.

00:02:06.960 --> 00:02:09.280
And this is also
called the average.

00:02:09.280 --> 00:02:11.780
And I'll be a little bit more
exact here than I normally am

00:02:11.780 --> 00:02:16.060
with the word "average." When I
talk about it in this context,

00:02:16.060 --> 00:02:20.090
it just means that the average
is a number that somehow

00:02:20.090 --> 00:02:22.640
is giving us a sense of
the central tendency.

00:02:22.640 --> 00:02:25.430
Or maybe a number that is most
representative of a set.

00:02:25.430 --> 00:02:27.030
And I know that sounds all
very abstract but let's

00:02:27.030 --> 00:02:28.870
do a couple of examples.

00:02:28.870 --> 00:02:31.850
So there's a bunch of ways
that you can actually measure

00:02:31.850 --> 00:02:35.200
the central tendency or the
average of a set of numbers.

00:02:35.200 --> 00:02:37.950
And you've probably
seen these before.

00:02:37.950 --> 00:02:40.534
They are the mean.

00:02:40.534 --> 00:02:42.960
And actually, there's types
of means but we'll stick

00:02:42.960 --> 00:02:44.040
with the arithmetic mean.

00:02:48.000 --> 00:02:50.660
Later, when we talk about stock
returns and things, we'll do

00:02:50.660 --> 00:02:53.810
geometric means and maybe we'll
cover the harmonic

00:02:53.810 --> 00:02:55.040
mean one day.

00:02:55.040 --> 00:03:02.640
There's a mean, the
median, and the mode.

00:03:02.640 --> 00:03:07.050
And in statistics speak,
these all can kind of be

00:03:07.050 --> 00:03:10.620
representative of a data sets
or population central tendency

00:03:10.620 --> 00:03:12.650
or a sample central tendency.

00:03:12.650 --> 00:03:15.590
And they all are collectively
-- they can all be

00:03:15.590 --> 00:03:17.070
forms of an average.

00:03:17.070 --> 00:03:18.520
And I think when we see
examples, it'll make a

00:03:18.520 --> 00:03:19.470
little bit more sense.

00:03:19.470 --> 00:03:23.440
In every day speak, when people
talk about an average, I think

00:03:23.440 --> 00:03:26.100
you've already computed
averages in your life, they're

00:03:26.100 --> 00:03:28.710
usually talking about
the arithmetic mean.

00:03:28.710 --> 00:03:30.320
So normally when someone says,
"Let's take the average of

00:03:30.320 --> 00:03:32.530
these numbers." And they expect
you to do something, they want

00:03:32.530 --> 00:03:34.470
you to figure out the
arithmetic mean.

00:03:34.470 --> 00:03:36.490
They don't want you figure
out the median or the mode.

00:03:36.490 --> 00:03:38.780
But before we go any further,
let's figure out what

00:03:38.780 --> 00:03:41.110
these things are.

00:03:41.110 --> 00:03:43.230
Let me make up a
set of numbers.

00:03:43.230 --> 00:03:45.630
Let's say I have the number 1.

00:03:45.630 --> 00:03:50.220
Let's say I have
another 1, a 2, a 3.

00:03:50.220 --> 00:03:52.885
Let's say I have a 4.

00:03:55.410 --> 00:03:56.170
That's good enough.

00:03:56.170 --> 00:03:58.370
We just want a simple example.

00:03:58.370 --> 00:04:02.650
So the mean or the arithmetic
mean is probably what you're

00:04:02.650 --> 00:04:05.710
most familiar with when
people talk about average.

00:04:05.710 --> 00:04:07.600
And that's essentially -- you
add up all the numbers and you

00:04:07.600 --> 00:04:09.160
divide by the numbers
that there are.

00:04:09.160 --> 00:04:16.290
So in this case, it would be 1
plus 1 plus 2 plus 3 plus 4.

00:04:16.290 --> 00:04:19.420
And you're going to divide
by one, two, three,

00:04:19.420 --> 00:04:21.020
four, five numbers.

00:04:21.020 --> 00:04:21.540
It's what?

00:04:21.540 --> 00:04:23.470
1 plus 1 is 2.

00:04:23.470 --> 00:04:25.600
2 plus 2 is 4.

00:04:25.600 --> 00:04:27.640
4 plus 3 is 7.

00:04:27.640 --> 00:04:29.500
7 plus 4 is 11.

00:04:29.500 --> 00:04:32.550
So this is equal to 11/5.

00:04:32.550 --> 00:04:33.040
That's what?

00:04:33.040 --> 00:04:34.410
That's 2 1/5?

00:04:34.410 --> 00:04:38.320
So that's equal to 2.2.

00:04:38.320 --> 00:04:39.560
And so someone could
say, "Hey, you know.

00:04:39.560 --> 00:04:41.060
That is a pretty
good representative

00:04:41.060 --> 00:04:42.490
number of this set.

00:04:42.490 --> 00:04:44.680
That's the number that all of
these numbers you can kind of

00:04:44.680 --> 00:04:47.390
say are closest to." Or, 2.2
represents the central

00:04:47.390 --> 00:04:49.140
tendency of this set.

00:04:49.140 --> 00:04:51.400
And in common speak, that
would be the average.

00:04:51.400 --> 00:04:53.450
But if we're being a little
bit more particular, this

00:04:53.450 --> 00:04:55.410
is the arithmetic mean
of this set of numbers.

00:04:55.410 --> 00:04:56.740
And you see it kind
of represents them.

00:04:56.740 --> 00:04:59.210
If I didn't want to give you
the list of five numbers, I

00:04:59.210 --> 00:05:01.030
could say, "Well, you know, I
have a set of five numbers and

00:05:01.030 --> 00:05:03.680
their mean is 2.2." It kind of
tells you a little bit of at

00:05:03.680 --> 00:05:05.900
least, you know, where
the numbers are.

00:05:05.900 --> 00:05:08.510
We'll talk a little bit more
about how do you know how far

00:05:08.510 --> 00:05:12.500
the numbers are from that mean
in probably the next video.

00:05:12.500 --> 00:05:13.840
So that's one measure.

00:05:13.840 --> 00:05:17.150
Another measure, instead of
averaging it in this way, you

00:05:17.150 --> 00:05:19.510
can average it by putting the
numbers in order, which

00:05:19.510 --> 00:05:20.460
I actually already did.

00:05:20.460 --> 00:05:23.340
So let's just write them
down in order again.

00:05:23.340 --> 00:05:26.810
1, 1, 2, 3, 4.

00:05:26.810 --> 00:05:28.490
And you just take
the middle number.

00:05:28.490 --> 00:05:31.790
So let's see, there's one, two,
three, four, five numbers.

00:05:31.790 --> 00:05:34.010
So the middle number's going
to be right here, right?

00:05:34.010 --> 00:05:34.940
The middle number is 2.

00:05:34.940 --> 00:05:37.240
There's two numbers greater
than 2 and there's two

00:05:37.240 --> 00:05:38.610
numbers less than 2.

00:05:38.610 --> 00:05:39.720
And this is called the median.

00:05:39.720 --> 00:05:41.560
So it's actually very
little computation.

00:05:41.560 --> 00:05:43.440
You just have to essentially
sort the numbers.

00:05:43.440 --> 00:05:45.620
And then you find whatever
number where you have an

00:05:45.620 --> 00:05:48.260
equal number greater than
or less than that number.

00:05:48.260 --> 00:05:51.430
So the median of this set is 2.

00:05:51.430 --> 00:05:53.010
And you see, I mean,
that's actually fairly

00:05:53.010 --> 00:05:54.320
close to the mean.

00:05:54.320 --> 00:05:56.020
And there's no right answer.

00:05:56.020 --> 00:05:58.550
One of these isn't a better
answer for the average.

00:05:58.550 --> 00:06:01.890
They're just different ways
of measuring the average.

00:06:01.890 --> 00:06:05.020
So here it's the median.

00:06:05.020 --> 00:06:06.980
And I know what you might be
thinking. "Well, that was

00:06:06.980 --> 00:06:08.640
easy enough when we
had five numbers.

00:06:08.640 --> 00:06:12.160
What if we had six numbers?"
What if it was like this?

00:06:12.160 --> 00:06:14.260
What if this was our
set of numbers?

00:06:14.260 --> 00:06:19.880
1, 1, 2, 3, let's add
another 4 there.

00:06:19.880 --> 00:06:21.660
So now, there's no
middle number, right?

00:06:21.660 --> 00:06:24.870
I mean 2 is not the middle
number because there's two less

00:06:24.870 --> 00:06:26.600
than and three larger than it.

00:06:26.600 --> 00:06:28.820
And then 3's not the middle
number because there's three

00:06:28.820 --> 00:06:31.530
larger and -- sorry, there's
two larger and three

00:06:31.530 --> 00:06:32.550
smaller than it.

00:06:32.550 --> 00:06:33.990
So there's no middle number.

00:06:33.990 --> 00:06:36.390
So when you have a set with
even numbers and someone tells

00:06:36.390 --> 00:06:38.500
you to figure out the median,
what you do is you take the

00:06:38.500 --> 00:06:43.750
middle two numbers and then you
take the arithmetic mean

00:06:43.750 --> 00:06:45.050
of those two numbers.

00:06:45.050 --> 00:06:50.770
So in this case of this set,
the median would be 2.5.

00:06:50.770 --> 00:06:51.730
Fair enough.

00:06:51.730 --> 00:06:54.020
But let's put this aside
because I want to compare the

00:06:54.020 --> 00:06:56.680
median and the means and the
modes for the same

00:06:56.680 --> 00:06:57.620
set of numbers.

00:06:57.620 --> 00:07:00.160
But that's a good thing to
know because sometimes it

00:07:00.160 --> 00:07:01.340
can be a little confusing.

00:07:01.340 --> 00:07:03.760
And these are all definitions.

00:07:03.760 --> 00:07:05.930
These are all kind of
mathematical tools for getting

00:07:05.930 --> 00:07:08.470
our heads around numbers.

00:07:08.470 --> 00:07:11.720
It's not like one day someone
saw one of these formulas on

00:07:11.720 --> 00:07:13.660
the face of the sun and says,
"Oh, that's part of the

00:07:13.660 --> 00:07:16.980
universe that this is how the
average should be calculated."

00:07:16.980 --> 00:07:20.380
These are human constructs to
kind of just get our heads

00:07:20.380 --> 00:07:22.110
around large sets of data.

00:07:22.110 --> 00:07:24.670
This isn't a large set of data,
but instead of five numbers, if

00:07:24.670 --> 00:07:26.760
we had five million numbers,
you can imagine if you don't

00:07:26.760 --> 00:07:28.780
like thinking about every
number individually.

00:07:28.780 --> 00:07:31.640
Anyway, before I talk more
about that, let me tell

00:07:31.640 --> 00:07:33.410
you what the mode is.

00:07:33.410 --> 00:07:36.200
And the mode to some degree,
it's the one I think most

00:07:36.200 --> 00:07:39.650
people probably forget or never
learn and when they see it on

00:07:39.650 --> 00:07:41.930
an exam, it confuses them
because they're like, "Oh, that

00:07:41.930 --> 00:07:45.420
sounds very advanced." But in
some ways, it is the easiest of

00:07:45.420 --> 00:07:49.450
all of the measures of central
tendency or of average.

00:07:49.450 --> 00:07:53.810
The mode is essentially what
number is most common in a set.

00:07:53.810 --> 00:07:56.220
So in this example, there's
two 1's and then there's one

00:07:56.220 --> 00:07:57.510
of everything else, right?

00:07:57.510 --> 00:08:00.230
So the mode here is 1.

00:08:00.230 --> 00:08:02.840
So mode is the most
common number.

00:08:02.840 --> 00:08:04.890
And then you could kind of
say, "Whoa, hey Sal, what

00:08:04.890 --> 00:08:05.880
if this was our set?

00:08:05.880 --> 00:08:11.620
1, 1, 2, 3, 4, 4." Here I have
two 1's and I have two 4's.

00:08:11.620 --> 00:08:14.040
And this is where the mode gets
a little bit tricky because

00:08:14.040 --> 00:08:17.810
either of these would have been
a decent answer for the mode.

00:08:17.810 --> 00:08:20.270
You could have actually said
the mode of this is 1 or the

00:08:20.270 --> 00:08:23.135
mode of this is 4 and it gets
a little bit ambiguous.

00:08:23.135 --> 00:08:24.840
And you probably want
a little clarity from

00:08:24.840 --> 00:08:25.790
the person asking you.

00:08:25.790 --> 00:08:28.510
Most times on a test when they
ask you, there's not going

00:08:28.510 --> 00:08:29.190
to be this ambiguity.

00:08:29.190 --> 00:08:33.164
There will be a most
common number in the set.

00:08:33.164 --> 00:08:35.950
So now it's like oh, well you
know, why wasn't just one

00:08:35.950 --> 00:08:36.900
of these good enough?

00:08:36.900 --> 00:08:38.490
You know why we learned
averages, why don't

00:08:38.490 --> 00:08:40.270
we just use averages?

00:08:40.270 --> 00:08:43.220
Or why don't we use arithmetic
mean all the time?

00:08:43.220 --> 00:08:45.080
What's median and
mode good for?

00:08:45.080 --> 00:08:47.890
Well, I'll try to do one
example of that and see if

00:08:47.890 --> 00:08:50.710
it rings true with you.

00:08:50.710 --> 00:08:52.020
And then you can think
a little bit more.

00:08:52.020 --> 00:08:53.950
Let's say I had this
set of numbers.

00:08:53.950 --> 00:09:04.350
3, 3, 3, 3, 3, and,
I don't know, 100.

00:09:04.350 --> 00:09:08.960
So what's the
arithmetic mean here?

00:09:08.960 --> 00:09:12.070
I have one, two, three,
four, five 3's and 100.

00:09:12.070 --> 00:09:17.350
So it would be 115
divided by 6, right?

00:09:17.350 --> 00:09:20.090
I could have one, two, three,
four, five, six numbers.

00:09:20.090 --> 00:09:21.990
115 is just the sum
of all of these.

00:09:21.990 --> 00:09:27.270
So that's equal to -- how many
times does 6 go into 115?

00:09:27.270 --> 00:09:28.600
6 goes into it one time.

00:09:28.600 --> 00:09:30.520
1 times 6 is 6.

00:09:30.520 --> 00:09:32.320
55 goes into it 9 times.

00:09:32.320 --> 00:09:34.370
9 times 6 is 54.

00:09:34.370 --> 00:09:35.950
So it's equal to 19 1/6.

00:09:38.470 --> 00:09:39.140
Fair enough.

00:09:39.140 --> 00:09:40.610
I just added all the
numbers and divided by

00:09:40.610 --> 00:09:42.150
how many there are.

00:09:42.150 --> 00:09:44.840
But my question is, is this
really representative

00:09:44.840 --> 00:09:45.560
of this set?

00:09:45.560 --> 00:09:47.740
I mean, I have a ton of 3's
and then I have 100 all of a

00:09:47.740 --> 00:09:51.270
sudden, and we're saying that
the central tendency is 19 1/6.

00:09:51.270 --> 00:09:53.610
And, I mean, 19 1/6 doesn't
really seem indicative

00:09:53.610 --> 00:09:54.080
of the set.

00:09:54.080 --> 00:09:56.390
I mean maybe it does, depending
on your application, but it

00:09:56.390 --> 00:09:57.850
just seems a little
bit off, right?

00:09:57.850 --> 00:09:59.800
I mean, my intuition would be
that the central tendency is

00:09:59.800 --> 00:10:02.660
something closer to 3 because
there's a lot of 3's here.

00:10:02.660 --> 00:10:06.770
So what would the
median tell us?

00:10:06.770 --> 00:10:09.720
I already put these
numbers in order, right?

00:10:09.720 --> 00:10:11.375
If I give it to you out of
order, you'd want to put it

00:10:11.375 --> 00:10:13.480
in this order and you'd say
what's the middle number?

00:10:13.480 --> 00:10:16.375
Let's see, the middle two
numbers, since I have an

00:10:16.375 --> 00:10:18.410
even number, are 3 and 3.

00:10:18.410 --> 00:10:20.890
So if I take the average of
3 and 3 -- or I should be

00:10:20.890 --> 00:10:21.820
particular with my language.

00:10:21.820 --> 00:10:26.800
If I take the arithmetic
mean of 3 and 3, I get 3.

00:10:26.800 --> 00:10:30.390
And this is maybe a better
measurement of the central

00:10:30.390 --> 00:10:34.400
tendency or of the average of
this set of numbers, right?

00:10:34.400 --> 00:10:38.120
Essentially, what it does is by
taking the median, I wasn't so

00:10:38.120 --> 00:10:40.720
much affected by this really
large number that's very

00:10:40.720 --> 00:10:42.080
different than the others.

00:10:42.080 --> 00:10:43.765
In statistics they
call that an outlier.

00:10:43.765 --> 00:10:47.010
A number that, you know, if you
talked about average home

00:10:47.010 --> 00:10:51.580
prices, maybe every house in
the city is $100,000 and then

00:10:51.580 --> 00:10:54.140
there's one house that
costs $1 trillion.

00:10:54.140 --> 00:10:56.120
And then if someone told you
the average house price was, I

00:10:56.120 --> 00:10:58.440
don't know, $1 million, you
might have a very wrong

00:10:58.440 --> 00:10:59.760
perception of that city.

00:10:59.760 --> 00:11:03.640
But the median house price
would be $100,000 and you get

00:11:03.640 --> 00:11:06.440
a better sense of what the
houses in that city are like.

00:11:06.440 --> 00:11:08.720
So similarly, this median,
maybe, gives you a better

00:11:08.720 --> 00:11:11.820
sense of what the numbers
in this set are like.

00:11:11.820 --> 00:11:15.550
Because the arithmetic mean
was skewed by this, what

00:11:15.550 --> 00:11:18.030
they call an outlier.

00:11:18.030 --> 00:11:19.990
And being able to tell what
an outlier is, it's kind of

00:11:19.990 --> 00:11:22.130
one of those things that a
statistician will say, well,

00:11:22.130 --> 00:11:23.110
I know it when I see it.

00:11:23.110 --> 00:11:25.450
There isn't really a formal
definition for it but it tends

00:11:25.450 --> 00:11:28.290
to be a number that really kind
of sticks out and sometimes

00:11:28.290 --> 00:11:31.190
it's due to, you know, a
measurement error or whatever.

00:11:31.190 --> 00:11:33.020
And then finally, the mode.

00:11:33.020 --> 00:11:35.310
What is the most common
number in this set?

00:11:35.310 --> 00:11:38.590
Well there's five 3's
and there's 100.

00:11:38.590 --> 00:11:41.440
So the most common number
is, once again, it's a 3.

00:11:41.440 --> 00:11:44.905
So in this case, when you had
this outlier, the median and

00:11:44.905 --> 00:11:46.700
the mode tend to be, you know,
maybe they're a little bit

00:11:46.700 --> 00:11:50.650
better about giving you an
indication of what these

00:11:50.650 --> 00:11:51.650
numbers represent.

00:11:51.650 --> 00:11:53.220
Maybe this was just a
measurement error.

00:11:53.220 --> 00:11:54.370
But I don't know, we
don't actually know

00:11:54.370 --> 00:11:55.250
what these represent.

00:11:55.250 --> 00:11:57.530
If these are house prices, then
I would argue that these are

00:11:57.530 --> 00:12:00.700
probably more indicative
measures of what the

00:12:00.700 --> 00:12:03.050
houses in a area cost.

00:12:03.050 --> 00:12:05.520
But if this is something else,
if this is scores on a test,

00:12:05.520 --> 00:12:07.850
maybe, you know, maybe there is
one kid in the class -- one out

00:12:07.850 --> 00:12:09.750
of six kids who did really,
really well and everyone

00:12:09.750 --> 00:12:10.410
else didn't study.

00:12:10.410 --> 00:12:13.680
And this is more indicative
of, kind of, how students at

00:12:13.680 --> 00:12:14.680
that level do on average.

00:12:14.680 --> 00:12:17.830
Anyway, I'm done talking
about all of this.

00:12:17.830 --> 00:12:20.430
And I encourage you to play
with a lot of numbers and deal

00:12:20.430 --> 00:12:21.460
with the concepts yourself.

00:12:21.460 --> 00:12:24.960
In the next video, we'll
explore more descriptive

00:12:24.960 --> 00:12:25.480
statistics.

00:12:25.480 --> 00:12:27.510
Instead of talking about the
central tendency, we'll talk

00:12:27.510 --> 00:12:30.410
about how spread apart things
are away from the

00:12:30.410 --> 00:12:31.520
central tendency.

00:12:31.520 --> 00:12:33.370
See you in the next video.


Statistics: Sample vs. Population Mean

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:03.140
Before, at the end of the last
video, I actually said that

00:00:03.140 --> 00:00:06.420
we'd talk about measures of
dispersion or how things are

00:00:06.420 --> 00:00:08.400
distributed, but before I go
into that, I realize that I

00:00:08.400 --> 00:00:10.680
have more to talk about,
especially the mean.

00:00:10.680 --> 00:00:15.590
And before I do that, I want
to differentiate between a

00:00:15.590 --> 00:00:16.820
sample and a population.

00:00:23.150 --> 00:00:26.250
I touched on this a little
bit in the last video.

00:00:26.250 --> 00:00:28.630
Let's say I wanted to
know-- I don't know.

00:00:28.630 --> 00:00:31.590
Let's say I wanted to know
the average height of all

00:00:31.590 --> 00:00:33.530
men in America, right?

00:00:33.530 --> 00:00:37.600
So let me make the set
of all men in America.

00:00:37.600 --> 00:00:39.180
So that's all men in America.

00:00:39.180 --> 00:00:42.740
I know there's 300 million
people in the U.S., and half of

00:00:42.740 --> 00:00:45.240
them maybe roughly are men,
so this would be 150

00:00:45.240 --> 00:00:49.360
million men, right?

00:00:49.360 --> 00:00:53.120
And it would be nearly
impossible, even if I was

00:00:53.120 --> 00:00:56.620
intent on doing it, to actually
measure the average height

00:00:56.620 --> 00:00:58.100
of every man in America.

00:00:58.100 --> 00:01:00.770
Frankly, you know, every few
seconds, one of these men is

00:01:00.770 --> 00:01:02.490
being born and one of
these men is dying.

00:01:02.490 --> 00:01:04.590
So you know by the time I'm
done measuring everything,

00:01:04.590 --> 00:01:06.950
someone would have died, and
some new men would have

00:01:06.950 --> 00:01:09.760
been born, so it would
almost be impossible.

00:01:09.760 --> 00:01:15.510
And if not impossible, it would
be very tiresome to measure the

00:01:15.510 --> 00:01:18.930
average, or the mean, or the
median, or the mode of this

00:01:18.930 --> 00:01:20.860
entire population, right?

00:01:20.860 --> 00:01:24.240
So the best way I can get a
sense of this, because I'm

00:01:24.240 --> 00:01:27.570
interested in what the average
of the population is, maybe I

00:01:27.570 --> 00:01:30.040
can take the average
of a sample.

00:01:30.040 --> 00:01:32.970
So what I could do is I can go
up to, you know-- and I'd try

00:01:32.970 --> 00:01:34.100
to be pretty random about it.

00:01:34.100 --> 00:01:37.370
I don't want to like-- you
know, hopefully, my sample

00:01:37.370 --> 00:01:41.170
wouldn't be my college's
basketball team because that

00:01:41.170 --> 00:01:44.200
would be a skewed sample, but
I'd try to find random people

00:01:44.200 --> 00:01:46.930
and random situations where it
wouldn't be skewed

00:01:46.930 --> 00:01:48.020
based on height.

00:01:48.020 --> 00:01:50.860
And I'd maybe collect 10
heights, and I'd get, well,

00:01:50.860 --> 00:01:53.080
maybe-- you know, the more
people I get the more

00:01:53.080 --> 00:01:55.360
indicative it is, but if I got
10 heights, and those 10

00:01:55.360 --> 00:01:57.720
heights were-- I don't know.

00:01:57.720 --> 00:02:03.080
I'll do it in, you know, 5
feet, 6 feet, 5 and a half

00:02:03.080 --> 00:02:09.090
feet, 5.75 feet, and, well,
let's say I only do 6,

00:02:09.090 --> 00:02:11.140
or let's say in 6 and
a half feet, right?

00:02:11.140 --> 00:02:14.140
Those are the five people that
I'd sample, and we could talk

00:02:14.140 --> 00:02:17.060
more about what's a good way to
generate a random sample from a

00:02:17.060 --> 00:02:19.820
population so it's not skewed
one way or the other.

00:02:19.820 --> 00:02:22.150
But anyway, if I wanted to get
a sense of it and if I was kind

00:02:22.150 --> 00:02:24.230
of lazy, so I only took
five measurements, this

00:02:24.230 --> 00:02:25.130
is the way I would do it.

00:02:25.130 --> 00:02:28.030
This would be a sample.

00:02:28.030 --> 00:02:30.360
This would be a sample
of the population.

00:02:30.360 --> 00:02:33.960
So instead of taking the mean--
let's say how I wanted to

00:02:33.960 --> 00:02:36.410
calculate the average by
taking the arithmetic mean.

00:02:36.410 --> 00:02:38.540
Instead of taking the
arithmetic mean of this entire

00:02:38.540 --> 00:02:41.890
group of 150 million people, I
might just be happy taking the

00:02:41.890 --> 00:02:44.390
mean of this sample, and
that'll be called

00:02:44.390 --> 00:02:46.190
the sample mean.

00:02:46.190 --> 00:02:48.540
And I want to introduce you to
some notation, even though it's

00:02:48.540 --> 00:02:54.136
kind of-- so in statistics
speak, the mean, this mu, it's

00:02:54.136 --> 00:02:57.450
a Greek letter, essentially the
Greek letter that later turns

00:02:57.450 --> 00:03:02.640
into m, but mu is the
population mean, and this is

00:03:02.640 --> 00:03:07.150
just a convention
population mean.

00:03:09.660 --> 00:03:14.900
And x with a line over it, that
is equal to a sample mean.

00:03:17.930 --> 00:03:20.150
And these are just notations
that people might see, and you

00:03:20.150 --> 00:03:21.820
might have been confused
because sometimes you see

00:03:21.820 --> 00:03:24.500
something-- people talk about
means, and you see this mu, and

00:03:24.500 --> 00:03:28.240
sometimes you see this x with a
line over it, and it's nice

00:03:28.240 --> 00:03:29.020
to know the distinction.

00:03:29.020 --> 00:03:31.450
Here they're talking about
the mean of a sample of the

00:03:31.450 --> 00:03:35.100
population, and here they're
talking about the mean of

00:03:35.100 --> 00:03:37.500
the population as a whole.

00:03:37.500 --> 00:03:41.310
Now, the way you calculate
them is essentially the same.

00:03:41.310 --> 00:03:43.180
If you want to figure out the
population mean, you'd go to

00:03:43.180 --> 00:03:47.730
all 150 million people at one
moment and add up all their

00:03:47.730 --> 00:03:50.530
heights, and divide by 150
million to get the

00:03:50.530 --> 00:03:51.570
population mean.

00:03:51.570 --> 00:03:54.580
The sample mean, you just add
up the numbers in your sample

00:03:54.580 --> 00:03:57.570
and divide by the number
of data points you have.

00:03:57.570 --> 00:04:00.375
And the formulas I
want to show you.

00:04:00.375 --> 00:04:03.000
I think you know how to
calculate averages.

00:04:03.000 --> 00:04:05.840
It's a fairly straightforward
operation, and I want to show

00:04:05.840 --> 00:04:08.030
you how it's often written in
statistics books, so that

00:04:08.030 --> 00:04:09.960
you're not intimidated
when you see it.

00:04:09.960 --> 00:04:12.390
The population mean, they'll
write it as-- so just to do,

00:04:12.390 --> 00:04:13.900
you know, the convention.

00:04:13.900 --> 00:04:17.300
Each member of a-- well, let
me do the sample first.

00:04:17.300 --> 00:04:20.500
Each member of a sample, say
this is the first sample.

00:04:20.500 --> 00:04:22.430
They'll call that x sub 1.

00:04:22.430 --> 00:04:24.890
They'll call this x sub 2.

00:04:24.890 --> 00:04:29.580
They'll call this one x
sub 3, x sub 4, and this

00:04:29.580 --> 00:04:31.750
one x sub 5, right?

00:04:31.750 --> 00:04:33.370
And this is just a way
of referring to each

00:04:33.370 --> 00:04:34.310
of the samples.

00:04:34.310 --> 00:04:36.260
So in a sample mean, they'll
say, do you know what you do?

00:04:36.260 --> 00:04:38.350
You take the sum
of these numbers.

00:04:38.350 --> 00:04:41.070
And you know how to do that,
but the fancy way of writing

00:04:41.070 --> 00:04:43.400
it is to say, let's
write a capital Sigma.

00:04:43.400 --> 00:04:45.730
That means the sum.

00:04:45.730 --> 00:04:49.380
Sum of every x sub n, right?

00:04:49.380 --> 00:04:51.970
Take the sum of each of
these numbers, right?

00:04:51.970 --> 00:04:58.450
This is x sub 1, x sub 2, where
n goes from 1 to-- I mean, you

00:04:58.450 --> 00:05:01.160
could say to the size
of the population.

00:05:01.160 --> 00:05:04.700
You know, sometimes-- you know,
in this case it would be 5, or

00:05:04.700 --> 00:05:07.900
sometimes they'd write a big--
they'd write an n like that.

00:05:07.900 --> 00:05:11.030
And you'd divide it by the
number of members there are

00:05:11.030 --> 00:05:15.110
in that population,
so divided by n.

00:05:15.110 --> 00:05:16.920
You know, when you see this in
a book, you're like, wow, this

00:05:16.920 --> 00:05:18.480
is advanced mathematics.

00:05:18.480 --> 00:05:20.540
But essentially, they're saying
take the sum of all the data

00:05:20.540 --> 00:05:22.940
points, just sum up these
numbers, and divide by the

00:05:22.940 --> 00:05:23.930
number of numbers there are.

00:05:23.930 --> 00:05:27.400
So this would just be 5
plus 5 plus 5.5 plus 5.75

00:05:27.400 --> 00:05:30.050
plus 6.5 divided by 5.

00:05:30.050 --> 00:05:31.820
That's all this is telling you.

00:05:31.820 --> 00:05:33.680
For the population mean,
it's the same thing.

00:05:33.680 --> 00:05:36.160
They just use a slightly
different notation.

00:05:36.160 --> 00:05:42.060
They'll say that's equal to the
sum from n is equal to 1 to a

00:05:42.060 --> 00:05:46.240
big N-- and I'll explain why
they write a big N-- of each

00:05:46.240 --> 00:05:49.650
data point in the population,
not just the sample, all

00:05:49.650 --> 00:05:52.560
that divided by big N.

00:05:52.560 --> 00:05:54.820
And this is just a way of,
when they're at big N,

00:05:54.820 --> 00:05:57.140
they mean 150 million.

00:05:57.140 --> 00:05:59.960
They mean, you know, we want
you to get every data point

00:05:59.960 --> 00:06:01.380
in the entire population.

00:06:01.380 --> 00:06:03.850
So that's what they mean by--
and then divide by the number

00:06:03.850 --> 00:06:05.090
of the entire population.

00:06:05.090 --> 00:06:07.620
While the small n, they're kind
of-- it's just the convention,

00:06:07.620 --> 00:06:11.350
the notation, that they say,
hey, we just want you to get

00:06:11.350 --> 00:06:13.700
some smaller number, not
the entire population.

00:06:13.700 --> 00:06:16.090
But the way you calculate
them is, you know, they're

00:06:16.090 --> 00:06:18.540
essentially equivalent.

00:06:18.540 --> 00:06:21.970
Anyway, I wanted to leave you
with that just because this is

00:06:21.970 --> 00:06:24.710
something that if you don't get
it clarified early on-- it's a

00:06:24.710 --> 00:06:27.630
fairly simple concept-- later
on, it becomes very confusing

00:06:27.630 --> 00:06:29.260
when people want to
differentiate between the

00:06:29.260 --> 00:06:31.170
population and the sample mean.

00:06:31.170 --> 00:06:33.260
And you see these formulas
written slightly different.

00:06:33.260 --> 00:06:36.560
Sometimes you'll see a mu, and
sometimes you'll see an x with

00:06:36.560 --> 00:06:38.610
a line over it for
the sample mean.

00:06:38.610 --> 00:06:41.230
Anyway, I'll see in
the next video.