Wiki Page, Math 13 - Perez Sandbox : Module 22 - Distribution of Sample Means (3a of 4)

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2026-01-17T21:15:36-08:00

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

Comparison of sampling distributions for sample means and sample proportions

Variable | Parameter | Statistic | Sampling Distribution
Center | Spread | Shape
Categorical (example: gender) | p = population proportion | = sample proportion | p | | Normal when np ≥ 10 and n(1 – p) ≥ 10

Quantitative (example: age)

= population mean,
= population standard deviation

= sample mean

When will the distribution of sample means be approximately normal?

Here are the common features of the sampling distributions for proportions and for means:

The mean of the sampling distribution is the population parameter.

The standard deviation of the sampling distribution is affected by sample size, with the statistics from larger samples having less variability.

Under certain conditions the shape of the sampling distribution is normal.

We now investigate the conditions that guarantee a normal sampling distribution for sample means.

Shape of the Sampling Distribution of Means

When we discussed the sampling distribution of sample proportions, we learned that this distribution is approximately normal under certain conditions; if and . In other words, we had a guideline based on sample size for determining the conditions under which we could use a normal curve to do probability calculations for sample proportions.

Now we investigate these questions:

When will the distribution of sample means be approximately normal?

Does it depend on the size of the sample?

What happens if the distribution of the variable in the population is heavily skewed?

The following simulation video helps us investigate these questions.

WalkThrough Simulation

Discussion of the simulation

Are you surprised that a variable with a skewed distribution in the population can have a sampling distribution that is approximately normal?

This discovery is probably the single most important result presented in introductory statistics courses. It is so important that it has a name: the Central Limit Theorem. The Central Limit Theorem says that for large samples, the sampling distribution of sample means is approximately normal.

This theorem is important because inference procedures, such as hypothesis tests and confidence intervals, are based on a normal model for the sampling distribution. The central limit theorem assures us that we can use a normal probability model for sample means without knowing anything about the shape of the distribution of the variable in the population. All we have to do is collect large samples.

How large a sample size do we need to assume that sample means will be normally distributed?

It really depends on the population distribution, as we saw in the simulation. The more skewed the distribution in the population, the larger the samples we need in order to use a normal model for the sampling distribution.

The general guideline is that samples of size greater than 30 will have a fairly normal distribution regardless of the shape of the distribution of the variable in the population. But if a population is strongly skewed, it is safer to use larger samples.

Summary

We are now ready to complete the table comparing the sampling distributions for proportions and for means by describing the conditions for a normal shape.

Comparison of sampling distributions for sample means and sample proportions

Variable | Parameter | Statistic | Sampling Distribution
Center | Spread | Shape
Categorical (example: left-handed or not) | p= population proportion | = sample proportion | p | | Normal if np ≥ 10 and n(1 - p) ≥ 10

Quantitative (example: age)

= population mean,
= population standard deviation

= sample mean

Normal if n > 30 (always normal if population is normal)

In summary, for large enough sample size, the sampling distribution of means is approximately normal (even if distribution of the variable in the population is not normal).

If a variable has a skewed distribution for individuals in the population, a larger sample size is needed to ensure that the sampling distribution has a normal shape. The general rule is that if the sample size n is more than 30, then the sampling distribution of means will be approximately normal.

If the variable is normally distributed in the population, then any sample size will produce a normal-shaped sampling distribution.

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