IB · Mathematics Applications & Interpretation (AI)
AIHL 4.15.3 The central limit theorem
Learn AIHL 4.15.3 The central limit theorem in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 4.15—Central limit theorem.
Learn AIHL 4.15.3 The central limit theorem in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 4.15—Central limit theorem.
Learn AIHL 4.15.3 The central limit theorem in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 4.15—Central limit theorem.
The video discusses the Central Limit Theorem, which is considered one of the most important theorems in statistics. It states that for any distribution, if the sample size is sufficiently large (typically ), the mean of the sample will be approximately normally distributed with a mean and variance .
Key points include:
The video concludes by reinforcing the significance of the Central Limit Theorem in statistical analysis.
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
00:28statistics 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.