Hi, I’m Adriene Hill, and Welcome back to Crash Course Statistics.
This is the episode you’ve been waiting for. The episode we designed this shelf for. The episode that
you have heard a lot about. (NORMAL DIST MONTAGE)
Well, today, we’ll get to see why we talk SO MUCH about the normal distribution.
INTRO
Things like height, IQ, standardized test scores, and a lot of mechanically generated
things like the weight of cereal boxes are normally distributed, but many other interesting
things from blood pressure, to debt, to fuel efficiency just aren’t.
One reason we talk so much about normal distributions is because distributions of means are normally
distributed, even if populations aren’t.
The normal distribution is symmetric, which means its mean, median and mode are all the
same value. And it’s most popular values are in the middle, with skinny tails to either side.
In general, when we ask scientific questions, we’re not comparing individual scores or
values like the weight of one blue jay, or the number of kills from one League of Legends game,
we’re comparing groups--or samples--of them. So we’re often concerned with the
distributions of the means, not the population.
In order to meaningfully compare whether two means are different, we need to know something
about their distribution: the sampling distribution of sample means. Also called the sampling
distribution for short.
And before we go any further, I want to say that the distribution of sample means is not
something we create, we don’t actually draw an infinite number of samples to plot and
observe their means. This distribution, like most distributions, is a description of a process.
Take income. Income is skewed….so we might think the distribution of all possible mean
incomes would also be skewed. But they’re actually normally distributed.
In the real population there are people that make a huge amount of money. Think Oprah,
Jeff Bezos, and Bill Gates. But when we take the mean of a group of three randomly selected
people, it becomes much less likely to see extreme mean incomes because in order to have
an income that’s as high as Oprah’s, you’d need to randomly select 3 people with pretty
high incomes, instead of just one.
Since scientific questions usually ask us to compare groups rather than individuals,
this makes our lives a lot easier, because instead of an infinite amount of different
distributions to keep track of, we can just keep track of one: the normal distribution.
The reason that sampling distributions are almost always normal is laid out in the Central
Limit Theorem. The Central Limit Theorem states that the distribution of sample means for
an independent, random variable, will get closer and closer to a normal distribution
as the size of the sample gets bigger and bigger, even if the original population distribution
isn’t normal itself.
As we get further into inferential statistics and making models to describe our data, this
will become more useful. Many inferential techniques in statistics rely on the assumption
that the distribution of sample means is normal, and the Central Limit Theorem allows us to
claim that they usually are.
Let’s look at a simulation of the Central Limit Theorem in action.
For our first example, imagine a discrete, uniform distribution. Like dice rolls. The
distribution of values for a single dice roll looks like this:
With a sample size of 1--the regular distribution of dice values--there’s one way to get a
1, one way to get a 2, one way to get a 3….and so on.
But we want to look at the mean of say...2 dice rolls, meaning our sample size is 2.
With two dice. Let’s first look at all the sums of the dice rolls we can get:
2,3,4,5,6,7,8,9,10,11,12
There’s only one way to get 2 and 12, either two ones, or two 6’s, but there’s 6 ways
to get 7, [1,6],[2,5], [3,4] or [6,1],[5,2], and [4,3]...which lends significance to the
number 7 - which is the number you’ll roll most often.
But back to means, we have the possible sums, but we want the mean, so we’ll divide each
total value by two, giving us this distribution:
Even though our population distribution is uniform, The distribution of sample means
is looking more normal, even with a sample size of 2. As our sample size gets bigger
and bigger, the middle values get more common, and the tail values are less and less common.
We can use the multiplication rule from probability to see why that happens.
If you roll a die one time, the probability of getting a 1--the lowest value--is ⅙.
When you increase the number of rolls to two, the probability of getting a mean of 1, is
now 1/36, or ⅙ times ⅙, since you have to get two 1’s to have a mean of 1.
Getting a mean value of 2 is a little bit easier since you can have a mean roll of 2
both by rolling two 2’s, but also by rolling a 3 and a 1, or a 1 and a 3. So the probability
is 3(1/36).
If we had the patience to roll a die 20 times, the probability of getting a mean roll value
of 1 would be (⅙)^20 since the only way to get a mean of 1 on 20 dice rolls is to
roll a one. Every. Single. Time. So you can see that even with a sample size of only 20,
the means of our dice rolls will look pretty close to normal.
The mean of the distribution of sample means is 3.5, the same as the mean of our original
uniform distribution of dice rolls, and this is always true about sampling distributions:
Their mean is always the same as the population they’re derived from. So with large samples,
the sample means will be a pretty good estimate of the true population mean.
There are two separate distributions we’re talking about.
There is the original population distribution that’s generating each individual die roll,
and there is a distribution of sample means that tells you the frequency of all the possible
sample means you could get by drawing a sample of a certain size--here 20--from that original
population distribution. Again, population distribution. And sampling distribution of sample means.
But while the mean of the distribution of sample
means is the same as the population’s, it’s standard deviation is not, which might be
intuitive since we saw how larger sample sizes render extreme values--like a mean roll value
of 1 or 6--very unlikely, while making values close to the mean more and more likely.
And it doesn’t just work for uniform population distributions. Normal population distributions
also give normal distributions of sample means, as do skewed distributions, and this weird looking guy:
In fact, with a large sample, any distribution with finite variance will have a distribution
of sample means that is approximately normal.
This is incredibly useful. We can use the nice, symmetric and mathematically pleasant
normal distribution to calculate things like percentiles, as well as how weird or rare
a difference between two sample means actually is.
The standard deviation of a distribution of sample means is still related to the original
standard deviation. But as we saw, the bigger the sample size, the closer your sample means
are to the true population mean, so we need to adjust the original population standard
deviation somehow to reflect this. The way we do it mathematically is to divide
by the square root of n--our sample size.
Since we divide by the square root of n, as n gets big, the standard deviation--or sigma--gets
smaller.. which we can see in these simulations of sampling distributions of size 20, 50,
and 100. The larger the sample size, the skinnier the distribution of sample means.
For example, say you grab 5 boxes of strawberries at your local grocery store--you’re making
the pies for a pie eating contest--and weigh them when you get home. The mean weight of
a box of strawberries from your grocery store is 15oz.
But that means that you don’t have quite enough strawberries. You thought that the
boxes were about 16oz, and you wonder if the grocery store got a new supplier that gives
you a little less.
You do a quick Google search and find a small farming company’s blog. They package boxes
of strawberries for a local grocery store, they list the mean weight of their boxes--16oz--and
the standard deviation--1.25 oz.
That’s all the information we need to calculate the distribution of sample means for a sample
of 5 boxes. Part of the mathematical pleasantness of the normal distribution is that if you
know the mean and standard deviation, you know the exact shape of the distribution.
So you grab your computer and pull up a stats program to plot the distribution of sample
means with a mean of 16oz and a standard deviation of 1.25 divided by the square root of 5--the
sample size.
We call The standard deviation of a sampling distribution the standard error so that we
don’t get it confused with the population standard deviation, it’s still a standard
deviation, just of a different distribution.
Our distribution of sample means for a sample of 5 boxes looks like this.
And now that we know what it looks like, we can see how different the mean strawberry
box weights of 15oz really is.
When we graph it over the distribution of sample means, we can see that it’s not too
close to the mean of 16oz, but it’s not too far either...We need a more concrete way
to decide whether the 15oz is really that far away from the mean of 16oz.
It might help if we had a measure of how different we expect one sample mean to be from the true
mean, luckily we do: the standard error which tells us the average distance between a sample
mean and the true mean of 16oz.
This is where personal judgement comes in. We could decide for example, that if a sample
mean was more than 2 standard errors away from the mean, we’d be suspicious. If that
was the case then maybe there was some systematic reduction in strawberries, because it’s
unlikely our sample mean was randomly that different from the true mean.
In this case our standard error would be 0.56. If we decided 2 standard errors was too far
away, we wouldn’t have much to be suspicious about. Maybe we should hold off leaving a
nasty comment on the strawberry farmers blog.
Looking at the distribution of sample means helped us compare two means, but we can also
use sampling distributions to compare other parameters like proportions, Regression Coefficients,
or standard deviations, which also follow the Central Limit Theorem.
The CLT allows us to use the same tools, like a distributions, with all different kinds
of questions. You may be interested in whether your favorite baseball team has better batting
averages, and your friend may care about whether Tylenol cures her headache faster than ibuprofen.
Thanks to the CLT you can both use the same tools to find your answers.
But when you look at things on a group level instead of the individual level, all these
diverse shapes and the populations that make them converge to one common distribution:
the normal distribution.
And the simplicity of the normal distribution allows us to make meaningful comparisons between
groups like whether hiring managers hire fewer single mothers, or whether male chefs make
more money. These comparisons help us know where things fit in the world.
Thanks for watching. I'll see you next time.