STATISTICS GUIDE
Standard deviation formula: sample vs population
Standard deviation measures how spread out values are around the mean. The key choice is whether your data represents a sample or the full population.Explore statistics & data analysis →FORMULA OR CORE IDEA
Population: σ = √(Σ(x − μ)² ÷ n); Sample: s = √(Σ(x − x̄)² ÷ (n − 1))WORKED EXAMPLE
For 58, 64, 72, 72, 78, 84, 91, the mean is about 74.14. In sample mode, variance is about 128.14 and sample standard deviation is about 11.32.STEP BY STEP
Enter the data in the right format
Use raw values for a normal list, frequency rows when exact values repeat, grouped intervals when you only have class ranges, or paired X/Y values for regression.
Calculate the mean
Add the values and divide by the number of values. In grouped data, the calculator estimates the mean from class midpoints and frequencies.
Find each deviation from the mean
Subtract the mean from each value. These deviations show how far each value sits above or below the center of the data.
Square and add the deviations
Squaring removes negative signs and gives more weight to larger distances from the mean. The total squared deviation is the base for variance.
Choose sample or population
Use n for population standard deviation when the entered values are the entire group. Use n − 1 for sample standard deviation when the values represent a sample.
Take the square root
The square root of variance gives standard deviation, which is easier to interpret because it uses the same unit as the original data.