Standard deviation formulas
Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
- Find the mean (add the values and divide by how many there are).
- Subtract the mean from each value and square the result.
- Add the squared differences.
- Divide by N (population) or n − 1 (sample) to get the variance.
- Take the square root.
Example: For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 40 ÷ 8 = 5. The squared differences are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32. Population variance = 32 ÷ 8 = 4, so σ = 2. Sample variance = 32 ÷ 7 ≈ 4.571, so s ≈ 2.138.
Mean, median and mode
- Mean — the arithmetic average. Sensitive to outliers.
- Median — the middle value when sorted (average of the two middle values for an even count). Robust to outliers, which is why it’s used for incomes and house prices.
- Mode — the most frequent value. A data set can have several modes or none.
- Range — maximum minus minimum.
Frequently asked questions
Should I use sample or population standard deviation?
Use population standard deviation (σ) when your data includes every member of the group you care about — for example, the test scores of all 25 students in a class. Use sample standard deviation (s) when your data is a sample used to estimate a larger population, such as a survey of 500 voters. Most real-world statistics use the sample version.
Why does sample standard deviation divide by n − 1?
A sample’s values tend to be closer to the sample mean than to the true population mean, which makes the spread look smaller than it really is. Dividing by n − 1 (Bessel’s correction) compensates for this bias.
What does standard deviation tell me?
It measures how spread out values are around the mean. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two.
What is variance?
Variance is the square of the standard deviation — the average squared distance from the mean. It is used in many formulas, but standard deviation is easier to interpret because it has the same units as the data.