Population variance and standard deviation
When you have every value of the population, divide by N. Mixing this up with the sample variance is a common exam mistake.
Worked example
A course has exactly these exam scores (the whole population). Work out the variance σ² and the standard deviation σ.
14; 8; 8; 7; 10; 11; 11
Worked solution
- 1.μ = 69 / 7 = 9.8571
- 2.σ² = Σ(xᵢ − μ)² / N = 34.8571 / 7 = 4.98
- 3.σ = √σ² = 2.23
Typical mistake: Dividing by N − 1 for a population; that is only for samples.
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task
A course has exactly these exam scores (the whole population). Work out the variance σ² and the standard deviation σ.
Data (n = 6)
σ² (2 decimal places)
σ (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 − 1 for a population; that is only for samples.
Frequently asked questions
What is the typical mistake in "Population variance and standard deviation"?
Dividing by N − 1 for a population; that is only for samples.
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).