Sample variance and standard deviation (n − 1)
For a sample, divide the sum of squared deviations by n − 1, not n. The standard deviation is its square root.
Worked example
These values come from a sample. Work out the sample variance s² and the standard deviation s.
13; 14; 5; 15; 18; 6; 17
Worked solution
- 1.x̄ = 88 / 7 = 12.5714
- 2.Σ(xᵢ − x̄)² = 157.7143
- 3.s² = Σ(xᵢ − x̄)² / (n − 1) = 157.7143 / 6 = 26.29
- 4.s = √s² = 5.13
Typical mistake: Dividing by n instead of n − 1 for a sample.
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task
These values come from a sample. Work out the sample variance s² and the standard deviation s.
Data (n = 5)
s² (2 decimal places)
s (2 decimal places)
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- Sample: s² = Σ(xᵢ − x̄)² / (n − 1)
- Population: σ² = Σ(xᵢ − μ)² / N
- Standard deviation: s = √s² or σ = √σ²
Typical mistake: Dividing by n instead of n − 1 for a sample.
Frequently asked questions
What is the typical mistake in "Sample variance and standard deviation (n − 1)"?
Dividing by n instead of n − 1 for a sample.
Why divide by n − 1 for a sample?
The sample mean is estimated from the same data, so dividing by n would make the spread systematically too small; n − 1 corrects this (unbiased estimate).