What is Sample Variance?
Sample variance (s²) measures how much values in a sample differ from the sample mean. It quantifies the spread or dispersion of data points in a sample and is used to estimate the population variance when the full population is not available. A higher value indicates that the data points are more spread out; a lower value means they’re more tightly clustered around the mean.
Sample variance is commonly used in descriptive statistics, hypothesis testing, and as a key component in other formulas, such as the standard deviation, t-tests, and ANOVA. Unlike population variance, the sample value uses a slightly different formula to avoid underestimating variability. Variance values are not as interpretable as the standard deviation.
Sample Variance Formula
To calculate sample variance (s²), use the following formula:
Where:
- n is the number of observations in the sample
- xᵢ is each individual value in the sample
- x̄ is the sample mean
- (xᵢ − x̄)² is the squared deviation of each value from the mean
This formula works by first finding the difference between each value and the sample mean, squaring each difference to eliminate negatives, summing those squared deviations, and then dividing by n − 1.
The use of n − 1 instead of n is called Bessel’s correction. It helps reduce bias by compensating for the tendency of sample data to underestimate the correct value, especially in small samples.
Example Calculations Using the Sample Formula
Let’s use the formula to find the sample variance. Suppose we have a sample of 5 exam scores:
82, 85, 87, 90, and 96.
The mean is 88.
The squared deviations from the mean are: (82–88)² = 36, (85–88)² = 9, (87–88)² = 1, (90–88)² = 4, and (96–88)² = 64.
Sum of squared deviations: 36 + 9 + 1 + 4 + 64 = 114
Divide by n – 1 = 5 – 1 = 4
Sample variance = 114 ÷ 4 = 28.5
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