Standard Deviation Calculator
TOOLEnter a list of numbers to get the population and sample standard deviation, variance and mean.
Separate values with commas, spaces or new lines. Decimals and negatives are fine.
What standard deviation measures
The mean tells you where a data set is centred, but not how spread out it is. Standard deviation fills that gap: it measures the typical distance of each value from the mean. A small standard deviation means the values huddle close to the average; a large one means they are scattered widely.
The formula
Both versions start the same way. Find the mean, subtract it from each value to get a deviation, square each deviation, and add the squares. This total is the sum of squared deviations. Then divide:
- Population variance σ² = Σ(xᵢ − μ)² ÷ n
- Sample variance s² = Σ(xᵢ − x̄)² ÷ (n − 1)
The standard deviation is the square root of the variance, which returns the figure to the same units as your data. The only difference between the two is the divisor: dividing by n − 1 for a sample is called Bessel's correction, and it compensates for the fact that a sample tends to underestimate the spread of the population it came from.
For the list 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Dividing by n = 8 gives a population variance of 4 and a population standard deviation of 2. Dividing the same 32 by n − 1 = 7 gives a sample variance of ≈ 4.571 and a sample standard deviation of ≈ 2.138.
Visualising spread
Standard deviation is easiest to grasp visually. A histogram shows how tightly values cluster around the centre — a tall, narrow shape means a small standard deviation, while a wide, flat shape means a large one. When you are comparing two variables and want to see how loosely or tightly they track together, a scatter plot reveals that spread directly. Pairing the number with a picture makes the concept far more intuitive.