Variance and standard deviation · Grade 12
In Grade 12 you get fluent with: coding data.
Standard deviation measures how far the data typically are from the mean. A small standard deviation means the values are bunched together. A large one means they are spread out. The variance is the standard deviation squared.
Key facts
- Variance: σ² = (Σ(x − x̄)²)/n = (Σx²)/n − x̄².
- Standard deviation σ = √variance, in the same units as the data.
- From a frequency table: σ² = (Σfx²)/(Σf) − x̄².
- Adding a constant to every value moves the mean but does not change σ.
- Multiplying every value by k multiplies the mean and σ by k (and the variance by k²).
- Some courses also use the sample standard deviation, dividing by n − 1. Check which one a question asks for.
Worked examples
Example 1
Daily temperatures have mean 15 °C and standard deviation 4 °C. Find the mean and standard deviation in °F, using F = 1.8C + 32.
Answer: Mean 59 °F, σ = 7.2 °F