Wiki Page, Math 13 - Perez Sandbox : Module 17 - Distribution of Sample Proportions (4 of 6)

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

The simulations on the previous page reinforce what we have observed about patterns in random sampling.

Proportions from random samples approximate the population proportion, p, so sample proportions average out to the population proportion.

Larger random samples better approximate the population proportion, so large samples have sample proportions closer to p. In other words, a sampling distribution for large samples has less variability.

The distribution of sample proportions appears normal (at least for the examples we have investigated).

We can describe the sampling distribution with a mathematical model that has these same features.

The Sampling Distribution of Sample Proportions

For a categorical variable, imagine a population with a proportion p of successes. (For example, for the variable gender, imagine a population of part-time college students with p = 0.60 female. Note that a success is the category of interest. It is what we are counting. Here a success is a female.) We create a mathematical model that describes the sample proportions from all possible random samples of size n from this population. The model has the following center, spread, and shape.

Center: Mean of the sample proportions is p, the population proportion.

Spread: The standard deviation of the sample proportions is . The standard deviation of the sampling distribution is also called the standard error.

Shape: A normal model is a good fit if the expected number of successes and failures is at least 10. We can translate these conditions into formulas: and .

Comment

The distribution of sample proportions for ALL samples of the same size is called the sampling distribution of sample proportions.

In a simulation, we collect thousands of random samples to examine the distribution of sample proportions. But when we model this distribution, our model describes the sampling distribution that comes from ALL possible random samples of the same size.

The model is a curve that comes from a sophisticated mathematical equation that is beyond the scope of this course.

EXAMPLE

Let's apply this model to our previous example about the population of part-time college students to see how it compares to our simulation. Recall that we assumed the population of part-time college students is 60% female. We selected samples of 25 part-time college students and calculated the proportion of females in each sample.

If you like, you can refresh your memory of this simulation.

Comparison of a simulation and a mathematical model of a sampling distribution

N = 25

Simulation
Thousands of random samples
each with 25 individuals

Mathematical Model
ALL possible samples,
each with 25 individuals

Mean of the
sample proportions

0.6

0.6

Standard deviation of
the sample proportions

0.096

0.098

Shape of distribution
of sample proportions

Approximately normal

Normal because conditions
are met:

Compare the mean and standard deviation we observed in the simulation to the mathematical model. Notice that the conditions are met, so a normal model is a good fit. We see that the model is a good description of the center, spread, and shape we observed in the simulation.

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