Why use n-1 instead of n as the divisor for sample variance?

Conditions

• The observations are independent and identically distributed (iid).
• The population has finite variance.
• The sample size n is greater than 1.

Reasoning, step by step

1. Recognize that the sample mean is used to calculate the squared deviations.
2. Understand that using the sample mean introduces a downward bias in the squared-deviation total.
3. Apply the divisor n-1 to correct this bias.
4. Conclude that this makes the sample variance unbiased for the population variance.

Example

For iid observations with finite variance and $n>1$, dividing by $n-1$ makes the sample variance unbiased for population variance. This corrects the average downward bias from estimating the mean in the same sample.

Common misconceptions

• Believing that dividing by n always provides an unbiased estimate of the population variance.
• Thinking that the bias correction is only necessary for very small samples.

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Connected concepts

Explore next

• What does it mean for an estimator to be unbiased?
• How does the sample size affect the difference between dividing by n and n-1?

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