Central Limit Theorem Explained

Why Sample Means Can Be Normal Even When the Population Is Skewed

In “Sampling Distribution When the Population Is Normal,” we saw that a Normal population makes the sampling distribution of \(\bar{x}\) exactly Normal for any sample size, provided observations are independent. But many measurements are not Normally distributed. Wait times, for example, often have a long right tail: most waits may be moderate, while a few are unusually long.

The Central Limit Theorem (CLT) explains why the distribution of sample means can still be approximately Normal when samples are large enough. It concerns the sampling distribution of \(\bar{x}\), not the distribution of individual observations. A skewed population does not become Normal; rather, the means of repeated random samples from it tend to have a more nearly Normal distribution as the sample size increases.

The requirement that the observations come from the same population matters. It means they share a common distribution; in particular, they have the same population mean and standard deviation. Independence and matching means and standard deviations alone are not enough to state the usual CLT. The theorem also requires a population with finite mean and standard deviation.

For sampling without replacement from a finite population, observations are not literally independent. As covered in “The 10% Condition for Sample Means,” when the sample is less than 10% of the population, it is reasonable to treat the observations as approximately independent for the standard-deviation calculation. For an AP Statistics application, also describe how the sample was obtained and consider whether the population’s shape makes a Normal approximation reasonable.

Here, the second parameter in \(N(\mu,\sigma/\sqrt{n})\) is the standard deviation. The center and spread come from earlier results: as in “Mean of the Sampling Distribution of x-bar,” the mean of \(\bar{x}\) is \(\mu\); as in “Standard Deviation of the Sample Mean,” its standard deviation is \(\sigma/\sqrt{n}\) when independence is appropriate. The CLT adds the important claim about the sampling distribution’s shape: for a sufficiently large sample, it is approximately Normal.

Using the CLT to Find Probabilities

Once the CLT makes a Normal approximation reasonable, calculate a probability about \(\bar{x}\) using a Normal model centered at \(\mu\) with standard deviation \(\sigma/\sqrt{n}\). Keep the sample size attached to the model: changing \(n\) changes the spread of sample means, even though their center remains \(\mu\).

The CLT does not say that every sample mean will be close to \(\mu\), or that a sample of a particular size must produce a Normal-looking distribution. It describes the shape of the distribution across repeated samples. The probability calculations account for the fact that sample means vary from sample to sample.

What Sample Size Does—and Does Not—Tell You

As \(n\) increases, \(\sigma/\sqrt{n}\) decreases, so sample means tend to be less variable. This spread formula was established in “How Sample Size Affects Variability of x-bar.” The CLT addresses a separate question: whether the distribution’s shape is approximately Normal. A larger sample generally helps, but the quality of the approximation also depends on the population’s shape.

A mildly skewed population may need a less substantial sample for its sample means to look approximately Normal than a population with extreme skewness or unusually influential values. The next tutorial considers how to judge whether a particular sample size is large enough. For now, do not treat any one number as a universal rule or use the CLT to claim exact Normality.

Also distinguish a probability about one observation from a probability about an average. For an individual wait, the population’s right-skewed distribution is relevant. For the mean of a sufficiently large random sample, the CLT supports using an approximately Normal model with a smaller standard deviation.

Reading the Model Carefully

In the examples, the Normal model describes \(\bar{x}\), not a customer’s individual wait. For instance, a right-skewed distribution of individual waits does not become symmetric just because 64 or 100 waits were collected. Instead, if we repeatedly took random samples of that size and plotted their means, the resulting sampling distribution would tend to be more nearly Normal.

The standard deviation \(\sigma/\sqrt{n}\) is the typical distance of sample means from \(\mu\) under the independence model. It is not a guarantee that a particular sample mean will be within that distance, and it is not the standard deviation of individual waits. Keep the parameter and statistic distinct: \(\mu\) and \(\sigma\) describe the population; \(\bar{x}\) and its sampling distribution describe sample means.

Common Mistakes and AP Exam Tip

• Applying the CLT to unrelated observations: The usual AP Statistics statement concerns a random sample from the same population. Do not claim that independence and equal means and standard deviations by themselves guarantee the CLT.
• Saying the population becomes Normal: The CLT concerns the sampling distribution of \(\bar{x}\), not the distribution of individual measurements.
• Calling the result exact: For a non-Normal population, the CLT gives an approximate Normal model. Exact Normality for every sample size comes from a Normal population, as in the previous tutorial.
• Using \(\sigma\) instead of \(\sigma/\sqrt{n}\): The former is the standard deviation of individual observations; the latter is the standard deviation of sample means under the independence model.
• Assuming one sample-size cutoff works for all populations: The amount of skewness and other population features affect how quickly the sampling distribution becomes approximately Normal. A sample-size claim should be justified in context.
• Omitting the sampling assumptions: State that the sample is random and comes from the same population. For sampling without replacement, check and describe the 10% condition when appropriate.

For a full-credit explanation, identify the random-sampling and independence assumptions, name the CLT when the population is not Normal, give the approximate mean and standard deviation of \(\bar{x}\), show the probability calculation, and interpret the result in context. Use “approximately” when reporting a probability based on the CLT.