- 1Enter your data points in the text area, separated by commas, spaces, or line breaks.
- 2The calculator displays two results side by side: the sample standard deviation (s) and the population standard deviation (σ).
- 3The mean and dataset size (N) are shown below the results for reference.
Standard deviation measures how spread out values are from the mean (average) of a dataset. A low standard deviation means the values are clustered close to the mean, while a high standard deviation means the values are more widely dispersed. This calculator shows both the sample standard deviation and the population standard deviation.
Used when the data represents the entire population. Divides by N.
σ = √[ Σ(xᵢ − μ)² / N ]Used when the data is a sample drawn from a larger population. Divides by (N − 1) to correct for estimation bias (Bessel's correction).
s = √[ Σ(xᵢ − x̄)² / (N − 1) ]Mean = 26 / 5 = 5.2 → Squared differences: 1.44, 7.84, 0.64, 0.04, 4.84 → Sum = 14.8 → σ = √(14.8/5) ≈ 1.7205 → s = √(14.8/4) ≈ 1.9235Mean = 5; Σ(x−5)² = 32; Population variance = 32 / 8 = 4; σ = √4Sample variance = 32 / (8 − 1) ≈ 4.571; s = √4.571- Use Population (σ) when your data includes every member of the group you are analyzing (e.g., all students in a specific class, all scores on a specific exam).
- Use Sample (s) when your data is a subset drawn from a larger group and you want to estimate the variability of the full group (e.g., a survey of 100 people representing a city).
- The sample standard deviation (s) is always slightly larger than the population standard deviation (σ) for the same data because dividing by (N − 1) instead of N corrects for the tendency of a sample to underestimate variability.
- With only one data point, the sample standard deviation is undefined because the denominator (N − 1) equals zero.
Why does the calculator show two different standard deviations?
The population standard deviation (σ) divides by N, while the sample standard deviation (s) divides by (N − 1). Both are provided so you can choose the correct one for your situation.
What does it mean when the sample standard deviation says "Undefined (N < 2)"?
The sample standard deviation requires at least 2 data points because its formula divides by (N − 1). With only 1 data point, the denominator is zero and the result is undefined.
What is the difference between population and sample standard deviation?
Both start from the squared distances to the mean. Population divides the total by N; sample divides by N − 1, which makes the sample value slightly larger. The calculator shows both.
How does it relate to variance?
Standard deviation is the square root of the variance, so it is expressed in the same units as the original data.