Standard Deviation Calculator
Type or paste your numbers, separated by commas or spaces, pick Population or Sample and press Calculate. The calculator shows the standard deviation, the mean, the variance and the count of values, rounded to 4 decimal places. It needs at least two numbers. An entry that is not a number gets a message that names it.
How to use the Standard Deviation Calculator
- Enter your numbers in the Standard Deviation Calculator above.
- Press the Calculate button to see the result.
- To try other numbers, change any value and press the button again.
The standard deviation formulas
σ (population) = √(Σ(x − μ)² / N) · · · s (sample) = √(Σ(x − x̄)² / (n − 1))
Standard deviation measures how far the values spread around the mean. The population formula divides by N. The sample formula divides by n − 1, a step called Bessel's correction. It removes the bias from the sample variance as an estimate of the population variance, and only part of the bias from the standard deviation. The calculator works out the mean and the variance from your numbers as exact fractions and rounds once, at the end.
Worked example
Data: 4, 8, 6, 5, 3. The mean is 5.2. The squared deviations are 1.44, 7.84, 0.64, 0.04 and 4.84, which add up to 14.8. Population: variance 14.8 / 5 = 2.96 and SD √2.96 = 1.7205. Sample: variance 14.8 / 4 = 3.7 and SD √3.7 = 1.9235.
Frequently asked questions
Should I use population or sample SD?
Sample SD (n − 1) when your data is a subset used to estimate a larger population. Population SD (N) when you have the entire population's data.
What does standard deviation measure?
How spread out values are around the mean. About 68% of values fall within ±1 SD of the mean in a normal distribution; 95% within ±2 SD; 99.7% within ±3 SD.
What about non-normal data?
SD is still a valid spread measure but the 68/95/99.7 rule doesn't apply. For skewed or heavy-tailed data, consider interquartile range (IQR) as an alternative.