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Standard Deviation Calculator

Calculate mean, variance, and standard deviation for population and sample datasets.

Input Information

Result

Enter numbers to calculate

Standard Deviation Calculator is a statistics tool that calculates the standard deviation, a measure of data dispersion.

Standard deviation indicates how spread out the data is from the mean. A small standard deviation means data is clustered around the mean, while a large one means data is widely spread.

This calculator computes both population standard deviation (σ) and sample standard deviation (s); note the different denominators:

  • Population: When all data is included (denominator: N)
  • Sample: When a subset is used as a sample (denominator: N-1, Bessel correction)

Term Glossary

Standard Deviation
The most widely used statistic for data dispersion; the square root of variance.
Variance
The average of squared differences between data and mean.
Population vs Sample
Population is the entire group of interest; sample is a subset drawn from it. Sample uses N-1 to correct bias.

  1. Enter the data values separated by commas or spaces. (e.g. 2, 4, 4, 4, 5, 5, 7, 9)
  2. Click the calculate button.
  3. Read the mean and the population/sample standard deviation results.
  4. Interpret the spread by how large the standard deviation is relative to the mean.

2, 4, 4, 4, 5, 5, 7, 9

The mean is 5 and the standard deviation is about 2.138. The data is relatively close to the mean.

10, 12, 23, 23, 16, 23, 21, 16

The mean is 18 and the standard deviation is about 5.26.

Constant data 5, 5, 5

If all values are identical, every difference from the mean is 0, so the standard deviation is 0.

Mean

μ=ΣxN\mu = \dfrac{\Sigma x}{N}

Sum all data and divide by the count.

Population Standard Deviation

σ=Σ(x−μ)2N\sigma = \sqrt{\dfrac{\Sigma(x - \mu)^2}{N}}

Uses all data; denominator is the count (N).

Sample Standard Deviation

s=Σ(x−xˉ)2n−1s = \sqrt{\dfrac{\Sigma(x - \bar{x})^2}{n - 1}}

Uses sample data; denominator is n-1 (Bessel correction). Used when estimating population variance.

Interpreting Standard Deviation

  • 68-95-99.7 Rule:
  • Mean ± 1σ: ~68% of data
  • Mean ± 2σ: ~95% of data
  • Mean ± 3σ: ~99.7% of data

Real-Life Applications

  • Finance: Measuring volatility of stock returns
  • Quality control: Checking consistency of product dimensions
  • Test scores: Analyzing grade distribution within a class
  • Weather: Measuring temperature variability

Population vs Sample: When to use which?

If you have all the data of interest, use population (σ); if you observed only a subset, use sample (s). For example, "all students in our class" is population, "10 randomly selected students" is sample.

QWhat is the difference between population and sample standard deviation?

Population standard deviation (σ) uses all data of interest with denominator N. Sample standard deviation (s) uses only a subset and uses denominator N-1 (Bessel correction) to better estimate the population.

QWhat is the relationship between variance and standard deviation?

Standard deviation is the square root of variance. Variance is the average of (value - mean)², and the standard deviation returns it to the same unit as the original data.

QWhat does standard deviation measure?

Standard deviation measures how spread out the data is from the mean. A smaller value means data is tightly clustered around the mean; a larger value means it is widely spread.

QWhen should I use standard deviation?

Use it to compare the spread or variability of data. It is used in finance for stock volatility, in quality control for product consistency, and in test score analysis for grade distribution.

QWhat is the standard deviation when all values are identical?

If all values are identical, every difference from the mean is 0, so the variance is 0 and the standard deviation is also 0.

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