Variance and standard deviation
Measures of spread show how far values lie from the mean. The standard deviation is the square root of the variance and has the units of the data.
Formulas
- Sample: s² = Σ(xᵢ − x̄)² / (n − 1)
- Population: σ² = Σ(xᵢ − μ)² / N
- Standard deviation: s = √s² or σ = √σ²
Your turn
Three problems on this topic. 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 = 6)
s² (2 decimal places)
s (2 decimal places)
Comma or point; fractions like 1/6 work too.
Worked example
These values come from a sample. Work out the sample variance s² and the standard deviation s.
3; 6; 11; 7; 6; 11; 4
Worked solution
- 1.x̄ = 48 / 7 = 6.8571
- 2.Σ(xᵢ − x̄)² = 58.8571
- 3.s² = Σ(xᵢ − x̄)² / (n − 1) = 58.8571 / 6 = 9.81
- 4.s = √s² = 3.13
Typical mistake: Dividing by n instead of n − 1 for a sample.
A course has exactly these exam scores (the whole population). Work out the variance σ² and the standard deviation σ.
13; 11; 12; 10; 2
Worked solution
- 1.μ = 48 / 5 = 9.6000
- 2.σ² = Σ(xᵢ − μ)² / N = 77.2000 / 5 = 15.44
- 3.σ = √σ² = 3.93
Typical mistake: Dividing by N − 1 for a population; that is only for samples.
Frequently asked questions
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).