Standard deviation calculator

Standard deviation describes how spread out values are around the mean. A small standard deviation means the values cluster tightly; a large one means they are scattered.

The calculator gives both versions:

• Population standard deviation (σ): use it when your data includes every member of the group you care about.
• Sample standard deviation (s): use it when your data is a sample from a larger group. It divides by n − 1 instead of n, which corrects the tendency of samples to underestimate spread.

It also shows variance, mean, median, mode, minimum, maximum, range, quartiles, the coefficient of variation, the standard error of the mean and a 95% confidence interval. Everything is computed in your browser.

σ = √( Σ(x − μ)² / n ) and s = √( Σ(x − x̄)² / (n − 1) )

Data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8)

1. Mean = 40 / 8 = 5
2. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32
3. Population variance = 32 / 8 = 4, so σ = 2
4. Sample variance = 32 / 7 ≈ 4.571, so s ≈ 2.138

The median is 4.5, the mode is 4 and the range is 9 − 2 = 7.

For data that follows a roughly normal (bell-shaped) distribution, the empirical rule applies:

So if adult heights in a group average 69 inches with a standard deviation of 3 inches, about 95% fall between 63 and 75 inches. The coefficient of variation (SD divided by the mean) lets you compare spread between data sets with different units or scales.

The standard error of the mean is s / √n. It shrinks as the sample grows, because averages of bigger samples are more precise. In the example, SE = 2.138 / √8 ≈ 0.756.

The 95% confidence interval for the mean uses Student's t-distribution with n − 1 degrees of freedom: mean ± t × SE. With 7 degrees of freedom t ≈ 2.365, giving roughly 3.21 to 6.79. This assumes the data comes from a roughly normal population; with small, skewed samples, treat it with care.

The lower quartile (Q1) and upper quartile (Q3) mark the 25th and 75th percentiles. They are computed with linear interpolation, the same method as the QUARTILE.INC function in common spreadsheets, so results match Excel and Google Sheets.

Separate values with commas, spaces or line breaks; you can paste a column from a spreadsheet. Use a period as the decimal point. Non-numeric entries are skipped and listed so you can check them. For averages with weights, use the average calculator.

• Using the wrong formula. Most real-world data is a sample, so s (n − 1) is usually right. Spreadsheet functions STDEV.S and STDEV.P match the two versions.
• Confusing standard deviation with standard error. SD describes the spread of individual values; SE describes how precise the mean is.
• Reading too much into small samples. With only a handful of values, the standard deviation can change a lot when one value changes.