00:00Hi guys, so in this
00:01lesson we're going to look
00:02at the central limit theorem.
00:04Now this theorem is they
00:06say one of if not
00:08the most important theorem in
00:10statistics. Now I've even googled
00:14a little ago, so it
00:15was the most important theorem
00:16in statistics and this comes
00:17up essentially in a theorem
00:19straight away. Let's see it
00:22done a lot of these.
00:25The central number theorem is
00:26arguably the most important theorem
00:28statistics okay anyway it's important
00:30now why is it important
00:31what's what's going on well
00:34this is the theorem to
00:36begin with for any distribution
00:38for sufficiently large and the
00:41iB calls large 30 so
00:44that's the size of the
00:46sample so if you have
00:47a distribution and you take
00:49a large sample at least
00:51size 30 then the mean
00:56of the sample will be
01:00normally distributed with a mean
01:02mu and variance sigma squared
01:05over n. Now that's a
01:07big deal because we have
01:09a lot of skills and
01:12knowledge about the normal distribution
01:14when I say we I
01:15mean humanity and we can
01:20apply a lot of those
01:22to
01:24any other distribution that you
01:25can imagine. I want to
01:26show you I have this
01:28little app here. It's just
01:29an online app that you
01:30can go and play with
01:32yourself. It's the link here.
01:36Just type it in and
01:37go and have a play
01:38around. So essentially this is
01:42what's going on. The distribution
01:44you decide here. You can
01:46choose any distribution you want.
01:48You can literally just make
01:49these bigger smaller. Do it.
01:52Do it.
01:52everyone but this distribution is
01:54essentially, I'll stick with skewed.
01:58It essentially means zero is
02:00going to happen that many
02:01times. One is going to
02:02happen this many times. The
02:03mode is 0, 1, 2,
02:053. The mode is 3
02:06because of this. And it's
02:08less likely the things out
02:10here are going to occur
02:11like 32, etc. Now what
02:14we're going to do is
02:15we're going to take a
02:16sample from this. We're going
02:19to pick
02:20five data points from this
02:24population. Now obviously it's more
02:26likely that each sample or
02:30each data that we use
02:31for a sample, each piece
02:32of data will come from
02:33more towards the left, but
02:35you might get one from
02:35over here. You might even
02:37get this one, whatever. So
02:39what I'm gonna do is,
02:40I'm gonna do, let's say
02:41n is 10, we're gonna
02:42choose 10. Yes, I did
02:44say at least 30, but
02:46I'm gonna show you here
02:47how it works.
02:48Let's say 10. And then
02:53I'm going to click animated.
02:55So what this does is
02:56it takes 10 sample of
02:5810 from this. And then
03:02in this one, it puts
03:03the mean. So the mean
03:05of the sample is here.
03:06It's 1, 2, 3, 4,
03:085, 6, 7. The mean
03:10of this sample was 7.
03:12And I put a block
03:13there. Fine, we do it
03:15again. New sample.
03:16and the mean is there.
03:21Do it again. Another sample.
03:26Mean is there. Okay, now
03:28we keep doing that. I
03:29can do it five times
03:31in one go. That's what
03:33this button does here. Do
03:34it another five times. Another
03:36five times. Now what happens
03:38is you see what's happening
03:40here. This thing is starting
03:42to approach
03:44normal distribution and if I
03:48click 10 ,000 it actually
03:50does it 10 ,000 times
03:51and you see a nice
03:53normal distribution here if I
03:54click 100 ,000 it does
03:56that so you can see
03:58we have a normal distribution
04:00a period here you can
04:01actually fit a normal curve
04:02onto that and there you
04:04go so the mean hopefully
04:06guys that makes sense the
04:08me this is the mean
04:09these the this is the
04:12the mean of the sample.
04:16So let's do it again
04:17but this time I'm just
04:17gonna make up, let's just
04:19make up a totally random,
04:24crazy, totally crazy distribution. That
04:28means nothing. This is clearly,
04:35this is clearly not any
04:39kind of normal distribution
04:40not anything at all. I've
04:44just literally made it up
04:45to be some of yours.
04:45So I'm going to clear
04:46the bottom three here. Now
04:49let's do this. So let's
04:50take this time and is
04:52equal to, let's go with
04:55five. So you'll notice guys,
04:58the smaller the sample, the
05:01bigger the variance and the
05:02reason is, let's take five,
05:04one, two, three, four, five.
05:06So if I take, if
05:07I just take a sample
05:08of five,
05:08the mean, well, here it's
05:13kind of back in the
05:14middle, but I mean, if
05:16I took a sample of
05:17these five, it could be
05:17over here. So there's actually
05:19going to be more variance
05:21when n is smaller, and
05:23that's kind of, you can
05:24also see that from the
05:26formula as n is small,
05:28the variance is big. So
05:31let's do it again. We
05:36have another mean.
05:36There. Do it again. Now,
05:40the mean there is do
05:42it five times. Let's take
05:44off this curve. Let's do
05:45it another five times. Now,
05:47look what's happening. Let's do
05:51it 10 ,000 times. And
05:53you get a really nice
05:55normal distribution, 100 ,000 times.
05:58Now, look, from my crazy
06:00distribution here that I just
06:01made up, the distribution of
06:03the means,
06:04of the sample is a
06:06nice normal distribution. Okay, that's
06:10the central of the theorem
06:11guys. Hopefully that makes some
06:14kind of sense. Right. Let's
06:17go back to this lesson.
06:21So this is what we're
06:23going to use. It is
06:24not in the formula booklet
06:25guys. So we have to
06:27remember that. Let's look at
06:28these two examples. So customers
06:30in a coffee shop spend
06:31an average of 26 minutes
06:32in the shop.
06:33with a standard deviation of
06:34eight minutes. It doesn't even
06:35say that that's a normal
06:36distribution, it doesn't have to
06:38be. Calculate the probability that
06:4050 customers spend on average
06:42between 20 and 25 minutes,
06:43so 50 is gonna be
06:44our end, that's our kind
06:46of sample size there. So
06:49the mean is the saying,
06:52it says calculate the probability
06:53that a sample of 50
06:54customers spend an average, on
06:57average between the source of
06:59the average that we care
07:00about, and the average
07:01time is normally distributed with
07:05a mean of 26 and
07:08variance of the standard deviation
07:11is 8 to the variance
07:12is 8 squared to 64
07:14but divided by 50 because
07:18I'm dividing by n. Okay,
07:21now the probability that x
07:24is between 20 and 25,
07:26so x is between 20
07:29and 25 equals, we just
07:34use our calculator with normal
07:37distributions. So probability distributions, normal
07:41CDF, what did I say
07:4320 to 25? The mean
07:45is 26 and the standard
07:48deviation be careful, as always
07:50here guys, the standard deviation
07:52is the square root of
07:5464 over 50.
07:57and we have 0 .1883790
08:02.18383799. That is the probability
08:11that an average, the sample
08:15of 50 customers on average
08:17spend between 2025 and that's
08:19in the shop. Okay, next
08:22question. This is a binomial
08:25male distributions or even works
08:27for binomial distribution works for
08:28any distribution. That's the beauty
08:30of the central number theorem.
08:32So before each game, Steph
08:34Curry always takes 20 shots
08:35from the center circle. I
08:37don't know if that's true.
08:38I just kind of made
08:39it up, but let's pretend
08:40it is. And I know
08:41he does take some shots
08:42from the center circle. The
08:44number of times these scores
08:45can be modeled with the
08:46random variable this. Find the
08:49probability that over a season
08:50of 72 games, he averages
08:52five shots
08:53are less. Okay, actually guys,
08:57I wanted to change, I
08:58didn't want to do five
08:59shots or less there, I
09:00wanted to do 11 shots
09:03or less because five shots
09:05or less, the chances of
09:07that, of averaging, less to
09:09five shots is gonna be
09:10very, very small, like literally
09:12zero. So let's change that
09:15to 11. Okay, so first
09:19we need the mean of
09:21a binomial
09:21only distribution is n times
09:23p, which is 20 times
09:260 .6. This is in
09:27the formula book, like I
09:28said by the way, which
09:30is 12. Fine. The variance,
09:35let's see the square of
09:37the binomial distribution, is n
09:39p times 1 minus p.
09:41This is in the formula
09:42book that it's 12 times
09:461 minus 0 .6 is
09:470 .4, which
09:49is 4 .8. That's the
09:52mean, that's the variance. We
09:54can say then that the
09:57average, so the average he
09:58scores, now the average, what
10:00the average means here is
10:03that the, it's not on
10:05average he scores six shots,
10:07that's the mean of this.
10:09This is the mean of
10:10the sample. So that means
10:13if he did this, he
10:14did this like, well, 72
10:16times.
10:17Well, on average, how many
10:20would you get out of
10:2120? That's what this average
10:22means. Anyway, it's normally distributed
10:25with a mean of 12
10:30and a variance of 4
10:34.8 over 72. Okay. Find
10:41the probability. He averages 11
10:43shots at last. So the
10:45probability
10:45that x bar is less
10:48than or equal to 11
10:50is equal to, okay, menu
10:56probability distributions. Normal CDF between
11:03this, yes, and 11. The
11:06upper bound is 12 and
11:08the standard deviation is now
11:10the square root of this.
11:12So it's the square root
11:13of 4 .8 over 72,
11:164 .8 over 72, which
11:21is 0 .0054, 0 .0054.
11:31Now the reason that's so
11:32small, as I was the
11:33tiny, the reason that's so
11:36small is because the sample
11:39size is quite big now,
11:40this 72.
11:41The variance, which was 4
11:44.8 for just when he
11:46does it once, the variance
11:48of the average of the
11:50sample is now tiny. It's
11:524 .8 divided by 72.
11:57So if you take, if
11:59Steph Korea does this once,
12:01he's going to get, um,
12:04on average, he'll get 12
12:06shots. But he probably won't
12:08get 12. He might get
12:09You might get 11, you
12:10might get 10, you might
12:11get 13 when he does
12:13it once. But if he
12:14does it 72 times, on
12:17average, he's probably going to
12:19get close to 12. That's
12:20why this probability is so
12:21low. But anyway, I want
12:24to give you a binomial
12:26example just to show you
12:28that it works for this
12:29two. Now, so the important
12:30thing to remember is it
12:31works for any kind of
12:32distribution. You just need the
12:34mean and you need the
12:37very
12:37and then you know that
12:41it's the mean of the
12:44sample is a normal distribution.
12:47Okay, hope that makes sense
12:48guys. That is the central
12:50limit theorem and I'll see
12:52you guys in the next
12:53lesson.