What happens to the sampling distribution of the mean when a population is normally distributed?

The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement

For the random samples we take from the population, we can compute the mean of the sample means:

and the standard deviation of the sample means:

Before illustrating the use of the Central Limit Theorem (CLT) we will first illustrate the result. In order for the result of the CLT to hold, the sample must be sufficiently large (n > 30). Again, there are two exceptions to this. If the population is normal, then the result holds for samples of any size (i..e, the sampling distribution of the sample means will be approximately normal even for samples of size less than 30).

Central Limit Theorem with a Normal Population

The figure below illustrates a normally distributed characteristic, X, in a population in which the population mean is 75 with a standard deviation of 8.

If we take simple random samples (with replacement)

The mean of the sample means is 75 and the standard deviation of the sample means is 2.5, with the standard deviation of the sample means computed as follows:

If we were to take samples of n=5 instead of n=10, we would get a similar distribution, but the variation among the sample means would be larger. In fact, when we did this we got a sample mean = 75 and a sample standard deviation = 3.6.

Central Limit Theorem with a Dichotomous Outcome

Now suppose we measure a characteristic, X, in a population and that this characteristic is dichotomous (e.g., success of a medical procedure: yes or no) with 30% of the population classified as a success (i.e., p=0.30) as shown below.

The Central Limit Theorem applies even to binomial populations like this provided that the minimum of np and n(1-p) is at least 5, where "n" refers to the sample size, and "p" is the probability of "success" on any given trial. In this case, we will take samples of n=20 with replacement, so min(np, n(1-p)) = min(20(0.3), 20(0.7)) = min(6, 14) = 6. Therefore, the criterion is met.

We saw previously that the population mean and standard deviation for a binomial distribution are:

Mean binomial probability:

Standard deviation:

The distribution of sample means based on samples of size n=20 is shown below.

The mean of the sample means is

and the standard deviation of the sample means is:

Now, instead of taking samples of n=20, suppose we take simple random samples (with replacement) of size n=10. Note that in this scenario we do not meet the sample size requirement for the Central Limit Theorem (i.e., min(np, n(1-p)) = min(10(0.3), 10(0.7)) = min(3, 7) = 3).The distribution of sample means based on samples of size n=10 is shown on the right, and you can see that it is not quite normally distributed. The sample size must be larger in order for the distribution to approach normality.

Central Limit Theorem with a Skewed Distribution

The Poisson distribution is another probability model that is useful for modeling discrete variables such as the number of events occurring during a given time interval. For example, suppose you typically receive about 4 spam emails per day, but the number varies from day to day. Today you happened to receive 5 spam emails. What is the probability of that happening, given that the typical rate is 4 per day? The Poisson probability is:

Mean = μ

Standard deviation =

The mean for the distribution is μ (the average or typical rate), "X" is the actual number of events that occur ("successes"), and "e" is the constant approximately equal to 2.71828. So, in the example above

Now let's consider another Poisson distribution. with μ=3 and σ=1.73. The distribution is shown in the figure below.

This population is not normally distributed, but the Central Limit Theorem will apply if n > 30. In fact, if we take samples of size n=30, we obtain samples distributed as shown in the first graph below with a mean of 3 and standard deviation = 0.32. In contrast, with small samples of n=10, we obtain samples distributed as shown in the lower graph. Note that n=10 does not meet the criterion for the Central Limit Theorem, and the small samples on the right give a distribution that is not quite normal. Also note that the sample standard deviation (also called the "standard error

CO-6: Apply basic concepts of probability, random variation, and commonly used statistical probability distributions.

LO 6.22: Apply the sampling distribution of the sample mean as summarized by the Central Limit Theorem (when appropriate). In particular, be able to identify unusual samples from a given population.

So far, we’ve discussed the behavior of the statistic p-hat, the sample proportion, relative to the parameter p, the population proportion (when the variable of interest is categorical).

We are now moving on to explore the behavior of the statistic x-bar, the sample mean, relative to the parameter μ (mu), the population mean (when the variable of interest is quantitative).

Let’s begin with an example.

Birth weights are recorded for all babies in a town. The mean birth weight is 3,500 grams, µ = mu = 3,500 g. If we collect many random samples of 9 babies at a time, how do you think sample means will behave?

Here again, we are working with a random variable, since random samples will have means that vary unpredictably in the short run but exhibit patterns in the long run.

Based on our intuition and what we have learned about the behavior of sample proportions, we might expect the following about the distribution of sample means:

Center: Some sample means will be on the low side — say 3,000 grams or so — while others will be on the high side — say 4,000 grams or so. In repeated sampling, we might expect that the random samples will average out to the underlying population mean of 3,500 g. In other words, the mean of the sample means will be µ (mu), just as the mean of sample proportions was p.

Spread: For large samples, we might expect that sample means will not stray too far from the population mean of 3,500. Sample means lower than 3,000 or higher than 4,000 might be surprising. For smaller samples, we would be less surprised by sample means that varied quite a bit from 3,500. In others words, we might expect greater variability in sample means for smaller samples. So sample size will again play a role in the spread of the distribution of sample measures, as we observed for sample proportions.

Shape: Sample means closest to 3,500 will be the most common, with sample means far from 3,500 in either direction progressively less likely. In other words, the shape of the distribution of sample means should bulge in the middle and taper at the ends with a shape that is somewhat normal. This, again, is what we saw when we looked at the sample proportions.

Comment:

• The distribution of the values of the sample mean (x-bar) in repeated samples is called the sampling distribution of x-bar.

Let’s look at a simulation:

Video: Simulation #3 (x-bar) (4:31)

Did I Get This?: Simulation #3 (x-bar)

The results we found in our simulations are not surprising. Advanced probability theory confirms that by asserting the following:

The Sampling Distribution of the Sample Mean

If repeated random samples of a given size n are taken from a population of values for a quantitative variable, where the population mean is μ (mu) and the population standard deviation is σ (sigma) then the mean of all sample means (x-bars) is population mean μ (mu).

As for the spread of all sample means, theory dictates the behavior much more precisely than saying that there is less spread for larger samples. In fact, the standard deviation of all sample means is directly related to the sample size, n as indicated below.

Since the square root of sample size n appears in the denominator, the standard deviation does decrease as sample size increases.

Learn by Doing: Sampling Distribution (x-bar)

Let’s compare and contrast what we now know about the sampling distributions for sample means and sample proportions.

Now we will investigate the shape of the sampling distribution of sample means. When we were discussing the sampling distribution of sample proportions, we said that this distribution is approximately normal if np ≥ 10 and n(1 – p) ≥ 10. In other words, we had a guideline based on sample size for determining the conditions under which we could use normal probability calculations for sample proportions.

When will the distribution of sample means be approximately normal? Does this depend on the size of the sample?

It seems reasonable that a population with a normal distribution will have sample means that are normally distributed even for very small samples. We saw this illustrated in the previous simulation with samples of size 10.

What happens if the distribution of the variable in the population is heavily skewed? Do sample means have a skewed distribution also? If we take really large samples, will the sample means become more normally distributed?

In the next simulation, we will investigate these questions.

Video: Simulation #4 (x-bar) (5:02)

Did I Get This?: Simulation #4 (x-bar)

To summarize, the distribution of sample means will be approximately normal as long as the sample size is large enough. This discovery is probably the single most important result presented in introductory statistics courses. It is stated formally as the Central Limit Theorem.

We will depend on the Central Limit Theorem again and again in order to do normal probability calculations when we use sample means to draw conclusions about a population mean. We now know that we can do this even if the population distribution is not normal.

How large a sample size do we need in order to assume that sample means will be normally distributed? Well, it really depends on the population distribution, as we saw in the simulation. The general rule of thumb is that samples of size 30 or greater will have a fairly normal distribution regardless of the shape of the distribution of the variable in the population.

Applet: Sampling Distribution for a Sample Mean

Comment:

• For categorical variables, our claim that sample proportions are approximately normal for large enough n is actually a special case of the Central Limit Theorem. In this case, we think of the data as 0’s and 1’s and the “average” of these 0’s and 1’s is equal to the proportion we have discussed.

Before we work some examples, let’s compare and contrast what we now know about the sampling distributions for sample means and sample proportions.

Learn by Doing: Using the Sampling Distribution of x-bar

Household size in the United States has a mean of 2.6 people and standard deviation of 1.4 people. It should be clear that this distribution is skewed right as the smallest possible value is a household of 1 person but the largest households can be very large indeed.

(a) What is the probability that a randomly chosen household has more than 3 people?

A normal approximation should not be used here, because the distribution of household sizes would be considerably skewed to the right. We do not have enough information to solve this problem.

(b) What is the probability that the mean size of a random sample of 10 households is more than 3?

By anyone’s standards, 10 is a small sample size. The Central Limit Theorem does not guarantee sample mean coming from a skewed population to be approximately normal unless the sample size is large.

(c) What is the probability that the mean size of a random sample of 100 households is more than 3?

Now we may invoke the Central Limit Theorem: even though the distribution of household size X is skewed, the distribution of sample mean household size (x-bar) is approximately normal for a large sample size such as 100. Its mean is the same as the population mean, 2.6, and its standard deviation is the population standard deviation divided by the square root of the sample size:

To find

we standardize 3 to into a z-score by subtracting the mean and dividing the result by the standard deviation (of the sample mean). Then we can find the probability using the standard normal calculator or table.

Households of more than 3 people are, of course, quite common, but it would be extremely unusual for the mean size of a sample of 100 households to be more than 3.

The purpose of the next activity is to give guided practice in finding the sampling distribution of the sample mean (x-bar), and use it to learn about the likelihood of getting certain values of x-bar.

Learn by Doing: Using the Sampling Distribution of x-bar #2

Did I Get This?: Using the Sampling Distribution of x-bar