Wiki Page, Math 13 - Perez Sandbox : Module 17 - Distribution of Sample Proportions (2c of 6)

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2026-01-17T21:15:21-08:00

Let's summarize what we have observed about the variability in sample proportions using language that a statistician uses.

What does a statistician mean by "error"?

We don’t expect a sample proportion to be exactly equal to the population proportion. Statisticians refer to the difference between a sample proportion and the population proportion as "error".

When a statistician describes "error in a sample," she is thinking of the sample proportion as an estimate of the population proportion. The "error" is how far off the estimate it is.

For example, if the population proportion is 0.60 and a sample proportion equals 0.58, it is off by 0.02, so this is the error in the estimate. When a statistician says "there is error in the estimate," she is not referring to a mistake; she is simply describing how much the sample statistic deviates from the population parameter.

In fact, statisticians refer to the standard deviation of the sample proportions as the standard error.

EXAMPLE

Now let's return to our example. We based our investigation on the assumption that 60% of part-time college students are female.

But how much error do we expect when selecting random samples from this population?

We now see that the answer to this question depends on the size of the sample.

If we select a random sample of 25 students, the distribution of sample proportions has a standard deviation of about 0.1. A statistician would say the standard error is 0.1.

If we select a random sample of 100 students, the distribution of sample proportions has less variability. It has a standard deviation of about 0.05. A statistician would say the standard error is 0.05.

Statisticians use the standard error to describe a margin of error. Typically, the margin of error is two standard deviations. So the margin of error will also depend on the sample size.

The margin of error helps us identify unusual outcomes.

EXAMPLE

If we select a random sample of 25 students, the distribution of sample proportions has a standard error of about 0.1 and a margin of error of 2 × 0.10 = 0.20. Sample proportions with an error greater than 0.20 are considered unusual. For example, a sample proportion less than 0.40 (0.60 – 0.20 = 0.40) or greater than 0.80 (0.60 + 0.20 = 0.80) will be unlikely to occur when the sample proportion has 25 students in it.

If we select a random sample of 100 students, the distribution of sample proportions has a standard error of about 0.05 and a margin of error of 2 × 0.05 = 0.10. Sample proportions with an error greater than 0.10 are considered unusual. For example, a sample proportion less than 0.50 (0.60 – 0.10 = 0.50) or greater than 0.70 (0.60 + 0.10 = 0.70) will be unlikely to occur when the random sample has 100 students in it.

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