Why does the class size of 40 not affect the calculation of the sample variance?

Conditions

• The problem asks for the sample variance, not the population variance.
• The sample consists of 6 randomly selected students from a class of 40.

Reasoning, step by step

1. Identify that the 40 students represent the population.
2. Recognize that the 6 pulse rates represent the sample.
3. Recall the formula for corrected sample variance: $S^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$.
4. Observe that the formula only uses the sample size $n=6$ and the sample data, not the population size 40.

Example

The script states: "Neither the whole class size 40 nor the sample size 6 replaces that denominator." The denominator is strictly $n-1 = 5$.

Common misconceptions

• Believing that the population size should be used in the denominator of the sample variance formula.
• Confusing the sample variance with the standard error of the mean, which does involve population size in finite population sampling.

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Connected concepts

Explore next

• Why is the denominator of the corrected sample variance n-1 instead of n?
• How is the sample arithmetic mean calculated for the six pulse-rate data points?
• What is the difference between sample and population variance?