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# Inferring population mean from sample mean

Much of statistics is based upon using data from a random sample that is representative of the population at large. From that sample mean, we can infer things about the greater population mean. We'll explain. Created by Sal Khan.

Video transcript

Let's say you're
trying to design some type of a product for men. One that is somehow
based on their height. And the product is
for the United States. So ideally, you
would like to know the mean height of men
in the United States. Let me write this down. So how would you do that? And when I talk
about the mean, I'm talking about the
arithmetic mean. If I were to talk about
some other types of means-- and there are other
types of means, like the geometric mean--
I would specify it. But when people just
say mean, they're usually talking about
the arithmetic mean. So how would you
go about finding the mean height of men
in the United States? Well, the obvious one
is, is you go and ask every or measure every
man in the United States. Take their height,
add them all together, and then divide by
the number of men there are in the United States. But the question
you'd ask yourself is whether that is practical. Because you have on
the order-- let's see, there's about 300 million
people in the United States. Roughly half of them
will be men, or at least they'll be male, and so
you will have 150 million, roughly 150 million men
in the United States. So if you wanted
the true mean height of all of the men
United States, you would have to somehow
survey-- or not even survey. You would have to be
able to go and measure all 150 million men. And even if you did try to do
that, by the time you're done, many of them might
have passed away, new men will have been born,
and so your data will go stale immediately. So it is seemingly impossible,
or almost impossible, to get the exact height of
every man in the United States in a snapshot of time. And so, instead, what you
do is say, well, look, OK, I can't get every man, but
maybe I can take a sample. I could take a sample of the
men in the United States. And I'm going to make an effort
that it's a random sample. I don't want to just go
sample 100 people who happen to play basketball,
or played basketball for their college. I don't want to go
sample 100 people who are volleyball players. I want to randomly sample. Maybe the first
person who comes out of the mall in a random
town, or in several towns, or something like that. Something that should
not be based in any way, or skewed in any way, by height. So you take a sample
and from that sample you can calculate a mean
of at least the sample. And you'll hope that that is
indicative of-- especially if this was a reasonably
random sample-- you'll hope that was indicative of the
mean of the entire population. And what you're going to
see in much of statistics it is all about using
information, using things that we can calculate
about a sample, to infer things
about a population. Because we can't directly
measure the entire population. So for example, let's say--
And if you're actually trying to do this,
I would recommend doing at least 100
data points, or 1,000, and later on we'll
talk about how you can think about whether
you've measured enough or how confident you can be. But let's just say
you're a little bit lazy, and you just sample five men. And so you get
their five heights. Let's say one is 6.2 feet. Let's say one is 5.5 feet-- 5.5
feet would be 5 foot, 6 inches. Let's say one ends
up being 5.75 feet. Another one is 6.3 feet. Another is 5.9 feet. Now, if these are
the ones that you happen to sample,
what would you get for the mean of this sample? Well let's get our
calculator out. And we get 6.2 plus 5.5
plus 5.75 plus 6.3 plus 5.9. The sum is 29.65. And then we want to divide
by the number of data points we have. So we have five data points. So let's divide 29.65 divided
by 5, and we get 5.93 feet. So here, our sample
mean-- and I'm going to denote it with an
x with a bar over it, is-- and I already forgot
the number-- 5.93 feet. This is our sample mean, or,
if we want to make it clear, sample arithmetic mean. And when we're taking this
calculation based on a sample, and somehow we're
trying to estimate it for the entire population,
we call this right over here, we call it a statistic. Now, you might be saying,
well, what notation do we use if, somehow,
we are able to measure it for the population? Let's say we can't even
measure it for the population, but we at least want to denote
what the population mean is. Well if you want to do
that, the population mean is usually denoted by
the Greek letter mu. And so in a lot of
statistics, it's calculating a sample
mean in an attempt to estimate this thing
that you might not know, the population mean. And these calculations
on the entire population, sometimes you might
be able to do it. Oftentimes, you will
not be able to do it. These are called parameters. So what you're going to
find in much of statistics, it's all about calculating
statistics for a sample, finding these sample
statistics in order to estimate parameters
for an entire population. Now the last thing I want
to do is introduce you to some of the notation that
you might see in a statistics textbook that looks very
math-y and very difficult. But hopefully, after
the next few minutes, you'll appreciate that
it's really just doing exactly what we did here--
adding up the numbers and dividing by the
number of numbers you add. If you had to do
the population mean, it's the exact same thing. It's just many, many more
numbers in this context. You have to add up
150 million numbers and divide by 150 million. So how do mathematicians
talk about an operation like that-- adding
up a bunch of numbers and then dividing by
the number of numbers? Let's first think
about the sample mean, because that's where we
actually did the calculation. So a mathematician
might call each of these data points-- let's
say they'll call this first one right over here x sub 1. They'll call this one x sub 2. They'll call this one x sub 3. They'll call this
one-- when I say sub, I'm really saying subscript
1, subscript 2, subscript 3. They could call
this x subscript 4. They could call
this x subscript 5. And so if you had n of these
you would just keep going. x subscript 6, x
subscript seven, all the way to x subscript n. And so to take the
sum of all of these, they would denote it as let
me write it right over here. So they will say that the sample
mean is equal to the sum of all my x sub i's-- so the way
you can conceptualize it, these i's will change. In this case, the
i started at 1. The i's are going to
start at 1 until the size of our actual sample. So all the way until n. In this case n was equal to 5. So this is literally
saying this is equal to x sub 1 plus x
sub 2 plus x sub 3, all the way to the nth one. Once again, in this
case, we only had five. Now, are we done? Is this what the sample mean is? Well, no, we aren't done. We don't just add up
all of the data points. We then have to divide by
the number of data points there are. So this might look like
very fancy notation, but it's really just saying,
add up your data points and divide by the number
of data points you have. And this capital Greek letter
sigma literally means sum. Sum all of the x i's, from x
sub 1 all the way to x sub n, and then divide by the number
of data points you have. Now let's think about how we
would denote the same thing but, instead of for the
sample mean, doing it for the population mean. So the population mean,
they will denote it with mu, we already talked about that. And here, once again you're
going to take the sum, but this time it's going
to be the sum of all of the elements in
your population. So your x sub i's-- and you'll
still start at i equals 1. But it usually gets
denoted that, hey you're taking the whole
population, so they'll often put a capital N right over
here to somehow denote that this is a bigger number
than maybe this smaller n. But once again, we are not done. We have to divide by the
number of data points that we are actually summing. And so this, once
again, is the same thing as x sub 1 plus x sub 2 plus
x sub 3-- all the way to x sub capital N, all of that
divided by capital N. And once again,
in this situation, we found this practical. We found this impractical. We can debate whether
we took enough data points on our sample
mean right over here. But we're hoping that
it's at least somehow indicative of our
population mean.