Wiki Page, Math 13 - Perez Sandbox : Module 17 - Parameters vs. Statistics (2a of 2)

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2026-01-17T21:15:22-08:00

EXAMPLE Means from random samples vary

Now let’s consider a quantitative variable with this same population of 200 students at a small college. Let’s also suppose that the mean amount of financial aid received by students at the college is $1,500.

In this simplified situation, we have

Population: 200 students at the college.

Variable: Financial aid amount ($) is a quantitative variable, so we use a mean as a summary.

Population mean: $1,500.

How accurate are random samples at predicting this population mean of $1,500?

To answer this question, we randomly select 8 students and determine the mean amount of financial aid received by the students. We repeat this process several times. Here are the results for 3 random samples:

Random sample #1

Student # | 38 | 49 | 61 | 68 | 120 | 139 | 165 | 200
Aid ($) | $2,100 | $2,300 | $3,100 | $0 | $0 | $3,100 | $2,300 | $3,800

= $2,087.50

Random sample #2

Student # | 1 | 45 | 48 | 82 | 86 | 100 | 148 | 83
Aid ($) | $2,000 | $2,400 | $0 | $1,500 | $0 | $0 | $3,400 | $1,300

= $1,325.00

Random sample #3

Student # | 25 | 56 | 63 | 75 | 130 | 131 | 193 | 195
Aid ($) | $0 | $0 | $2,400 | $0 | $0 | $1,600 | $0 | $1,500

= $687.50

Notice that observations we made earlier about sample proportions are true for sample means.

Each random sample came from a population for which the mean amount of financial aid received by individual students is $1,500. But the sample means vary: Each random sample has a different mean.

Some sample means are larger than the population mean of $1,500. Some sample means are smaller than the population mean.

Some samples give better estimates of the population mean than others. For example, $1,325.00 is a much better estimate than $687.50.

A lot of variability occurs in the sample means. It is not surprising, therefore, that a sample of 8 students may give an inaccurate estimate of the mean amount of financial aid received by the population. Again, it makes sense that small samples of only 8 students may not represent the population accurately. We investigate the factors that affect the variability of means from random samples in the module Distribution of Sample Means.

The variability we see in the means from random samples is due to chance.

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