The Central Limit Theorem says that the standard deviation of the sampling distribution is

We saw that the standard deviation of the sampling distribution is smaller when the sample size is larger. The Central Limit Theorem gives us an exact formula. The standard deviation of the sampling distribution of means equals the standard deviation of the population divided by the square root of the sample size. The standard deviation of the sampling distribution is called the “standard error of the mean.”

The formula for the standard error of the mean, sx̄, is

We can use this formula to calculate the standard error of any sampling distribution if we know the sample size and the population standard deviation.

Imagine you draw a sample of 100 cases from a population with a mean of 57 and a standard deviation of 20.

What are the mean mx̄ and the standard error sx̄ for the sampling distribution?

m = 57, s = 20

Sorry, this answer is incorrect!

Your answer describes the population, not the sampling distribution. Remember that the Central Limit Theorem states that for a given population and sample size:

The sampling distribution has the same mean as the underlying population, but a smaller standard deviation.

Notice how the sample size N is used in the formula, and then recalculate your answer.

m = 57, s = 200

Close, but not correct!

You seem to have the right idea, but you may have made a math error. Check the formula for the standard error of the mean. Be careful with decimal points.

m = 57, s = 4

Close, but not correct!

You seem to have the right idea, but you may have made a math error. Check the formula for the standard error of the mean. Did you remember to take the square root?

m = 57, s = 2

Yes, excellent!

The sampling distribution has less variance and is narrower with larger sample sizes. We divide the population standard deviation, sx, by the square root of the sample size, N, to get the standard error of the mean sx̄

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Learning Objectives

1. To become familiar with the concept of the probability distribution of the sample mean.
2. To learn the Central Limit Theorem and its formulas.

In practice, when we wish to estimate the mean $\mu$ of a population, we take a sample and use $\xbar$, the mean of the sample, as our estimate. Imagine however that we take many random samples, all of the same size $n$, and compute the sample mean $\xbar$ of each one. We will likely get a different value of $\xbar$ each time. The sample mean $\xbar$ is thus a random variable: it varies from sample to sample in a way that cannot be predicted with certainty. In this book, we will usually write $\Xbar$ when the sample mean is thought of as a random variable and $\xbar$ for the values that it takes, but it is common to use $\xbar$ for both, and we will occasionally do so.

The random variable $\Xbar$ has a probability distribution like any other random variable, with mean and standard deviation denoted $\muxbar$ and $\sigmaxbar$. The amazing thing is that in very general circumstances, $\Xbar$ is normally distributed, and the values of $\muxbar$ and $\sigmaxbar$ can be easily computed from the mean and standard deviation of the underlying random variable $X$—that is, from $\mu$ and $\sigma$. That is the content of the Central Limit Theorem, one of the fundamental theorems of statistics.

The Central Limit Theorem (CLT)

If the population is normally distributed or samples of size 30 or larger are taken, then the sample mean $\Xbar$ is approximately normally distributed with mean $\muxbar=\mu$ and standard deviation $\sigmaxbar=\frac{\sigma}{\sqrt{n}}$, where $n$ is the sample size. The larger the sample size, the better the approximation.

Notice that the CLT doesn’t assume anything about the distribution of $X$. It says that as long as we take large enough samples, $\Xbar$ will be approximately normally distributed with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$, no matter what. It doesn’t matter what the distribution of $X$ is. (It also says that if $X$ is itself normally distributed, then $\Xbar$ will be approximately normal even for small samples.)

The distribution of the random variable $\Xbar$ is called is called the sampling distribution of the sample mean. The Central Limit Theorem says that the sampling distribution looks more and more like a normal distribution as the sample size increases. Again, this will happen no matter how the values of $X$ are distributed (though if that distribution is really wild, then larger sample sizes may be needed).

It is easy for beginners to get confused when trying to apply the Central Limit Theorem. You must remember that there are two random variables involved, $X$ and $\Xbar$. A value of $X$ is a single measurement of a member of the population, while a value of $\Xbar$ is a sample mean. The mean and standard deviation of $X$ are $\mu$ and $\sigma$, while the mean and standard deviation of $\Xbar$ are $\muxbar$ and $\sigmaxbar$.

Example 1

Let $\Xbar$ be the mean of a random sample of size 50 drawn from a population with mean 112 and standard deviation 40.

1. Find the mean and standard deviation of $\Xbar$.
2. Find the probability that $\Xbar$ assumes a value between 110 and 114.
3. Find $P(\xbar\gt 113)$.
4. Find $P(X\gt 113)$.

Solution:

1. By the CLT, $\muxbar=\mu=112$ and $\sigmaxbar=\frac{40}{\sqrt{50}}\approx 5.65685$.
2. The CLT says that $\Xbar$ is approximately normally distributed, so we can use normalcdf: $P(110\lt\Xbar\lt 114)=$ normalcdf(110,114,112,5.65685) $\approx 0.2763$.
3. Here we have purposely written the question with a lower-case $x$ to remind you that problems are often written in this way. Again we use normalcdf: $P(\xbar\gt 113)=P(\Xbar\gt 113)=$normalcdf(113,1E99,112,5.65685) $\approx 0.4298$.
4. $P(X\gt 113)=$ normalcdf(113,1E99,112,50) $\approx 0.492$ (We don’t need the CLT here.)

Make sure that you understand why we could answer part (c) of Example 1 but not part (d). Part (d) asks about $X$, and we can’t answer because we don’t know the distribution of $X$. Part (c), on the other hand, asks about $\Xbar$, and we can answer, because the Central Limit Theorem guarantees that if we use large enough samples, then the distribution of $\Xbar$ will be approximately normal regardless of the distribution of $X$. This is tremendously powerful. It lets us say useful things about the means of real-world samples, even though we rarely know how the population is distributed. It is not overstating to say that, along with the Law of Large Numbers, the CLT is the basis for most of the techniques of inferential statistics in broad use today.

One last note. The standard deviation $\sigmaxbar$ of the sample mean is often called the standard error of the mean or just the standard error. This is unfortunate, as it confuses many people, but it is very common, and we will use it occasionally to help you get used to it.

Warm-up Exercises

1. Random samples of size 225 are drawn from a population with mean 100 and standard deviation 20.

   1. Find the mean and standard deviation of the sample mean $\Xbar$.
   2. Find $P(X\lt 97)$.
   3. Find $P(\Xbar\lt 97)$.

2. A population has mean 75 and standard deviation 12.

   1. Random samples of size 121 are taken. Find the mean and standard deviation of the sample mean.
   2. How would the answers to part (a) change if the size of the samples were 400 instead of 121?

Answers

1. 1. $\muxbar=100$, $\sigmaxbar=1.33$
   2. $P(X\lt 97)=$ normalcdf(-1E99,97,100,20) $\approx 0.4404$
   3. $P(\Xbar\lt 97)=$ normalcdf(-1E99,97,100,1.33) $\approx 0.0120$ (Don’t forget to use the standard deviation for $\Xbar$!)

2. 1. $\muxbar=75$, $\sigmaxbar=1.09$
   2. $\muxbar$ would stay the same but $\sigmaxbar$ would decrease to 0.6

What does the central limit theorem say about standard deviation?

What does the central limit theorem says about the standard deviation of the sampling distribution of the sample means?

What does the central limit theorem says about sampling distribution?

What is the standard deviation of the sampling distribution of the sample mean?