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• 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, ortimes 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

• 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.

Hi, I'm Adriene Hill, and Welcome back to Crash Course Statistics.

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# 正態分佈。統計速成班#19 (The Normal Distribution: Crash Course Statistics #19)

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林宜悉 發佈於 2021 年 01 月 14 日