Education Technology

Why Divide by n-1?

Published on 10/06/2011

Activity Overview

Students will investigate calculating a sample variance using both n and n-1 as the divisor for samples drawn with and without replacement.

Objectives

  • Students will recognize that the mean of all the sample variances for samples of a given size drawn with replacement calculated using n-1 as a divisor will give the population variance calculated using n as a divisor.
  • Students will recognize that the mean of all the sample variances for samples of a given size drawn without replacement calculated using n-1 as a divisor will give a value equal to the population variance if calculated using n-1 as a divisor.
  • Students will recognize that as the sample size increases the difference in calculating the variances for a sample using n and n-1 as a divisor will decrease.
  • Students will recognize that when a sample variance is calculated using n-1 as a divisor, the sample variance produces an unbiased estimator of the population variance.

Vocabulary

• bias
• mean
• parameter
• population
• sample size
• sampling distribution
• standard deviation
• variance

About the Lesson

This lesson involves investigating calculating a sample variance using both n and n-1 as the divisor for samples drawn with and without replacement.
As a results, students will:

  • Investigate the sampling distribution of the variances of all possible samples of size three, drawn with and without replacement from a given population, using as divisors both n and n-1.
  • Compare the means of these distributions to the variance for the population.
  • Investigate the impact of sample size on the variance calculated using each of the divisors.