01 Key Concepts
What Variance Measures
The average of the squared differences between each data point and the mean.
Why Squaring
Squaring the differences makes them all positive (so they don't cancel out) and emphasizes larger deviations more heavily.
Population Variance
Divides the sum of squared deviations by n, the total number of values in the population.
Sample Variance
Divides the sum of squared deviations by (n-1) instead of n — this correction (Bessel's correction) gives a better estimate of the population variance from a sample.
Units of Variance
Variance is expressed in squared units of the original data (e.g., dollars-squared), which is part of why standard deviation is often preferred for interpretation.
02 Key Formulas
- Population variance = [sum of (x - mean)^2] / n
- Sample variance = [sum of (x - mean)^2] / (n-1)
03 Solved Examples
Example 1 Find the population variance of 2, 4, 6, 8.
- Mean = (2+4+6+8)/4 = 5.
- Deviations from mean: -3, -1, 1, 3.
- Squared deviations: 9, 1, 1, 9. Sum = 20.
- Variance = 20/4.
Answer: Population variance = 5
Example 2 Find the sample variance of the same dataset: 2, 4, 6, 8.
- Sum of squared deviations = 20 (from the previous example).
- Sample variance divides by (n-1) = 3.
- Variance = 20/3.
Answer: Sample variance ≈ 6.67
Example 3 Explain why sample variance uses (n-1) instead of n.
- Using the sample mean (instead of the true population mean) slightly underestimates the true spread.
- Dividing by (n-1) corrects for this bias, giving a more accurate estimate of the population variance.
Answer: Dividing by (n-1) corrects for the bias introduced by estimating the mean from the sample itself
04 Practice Questions
1Find the population variance of 1, 2, 3, 4, 5.
2
2Find the sample variance of 1, 2, 3, 4, 5.
2.5
3Find the population variance of 10, 10, 10, 10.
0
4Why does a variance of 0 make sense for identical values?
There's no spread at all — every value equals the mean
5What units is variance expressed in relative to the original data?
Squared units
📄 Variance — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.