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Standard Deviation Calculator

Calculate standard deviation, variance, mean, and data spread instantly. Use our free Standard Deviation Calculator for statistics, research, education, and data analysis.

Standard deviation measures the average statistical dispersion of data points around the arithmetic mean. It provides both Population Standard Deviation $(\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}})$ for complete census datasets and Sample Standard Deviation $(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}})$ utilizing Bessel's correction $(n-1)$ to eliminate bias when inferring population parameters from subset samples.

Standard Deviation Calculator

Calculate standard deviation, variance, mean, range, and other statistical measures from a dataset.

Statistical Inference StandardReviewed: January 2026

Gaussian Normal Distribution & Empirical Rule (68–95–99.7)

68.27% (±1σ) | 95.45% (±2σ) | 99.73% (±3σ)

For standard normal distributions $\mathcal{N}(\mu, \sigma^2)$, approximately 68.3% of observations lie within 1 standard deviation of the mean, 95.5% within 2 standard deviations, and 99.7% within 3 standard deviations (the 3-sigma process capability threshold).

Reference: NIST/SEMATECH e-Handbook of Statistical Methods (Section 1.3.5: Quantitative Measures)

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Standard Deviation Formula

σ = √( Σ(x − μ)² / N )

Standard deviation measures how spread out data values are from the mean. A low standard deviation indicates data points are close to the mean, while a high standard deviation indicates greater variability.

Example Calculation

Input: 10, 20, 30, 40, 50

Output: Mean = 30, Standard Deviation ≈ 14.14

Common Uses

  • • Statistics
  • • Research
  • • Data Analysis
  • • Education
  • • Business Analytics
  • • Data Science
  • • Survey Analysis
  • • Mathematics
  • • Scientific Studies
  • • Quality Control

How the Standard Deviation Calculator Works

1

Enter Data Values

Enter numbers separated by commas.

2

Calculate Mean

The calculator finds the average of all values.

3

Calculate Variance

Variance is calculated using deviations from the mean.

4

View Standard Deviation

See standard deviation, variance, mean, and count.

Questions about this calculator

Standard deviation measures how much data values differ from the average value.

A low standard deviation indicates that data points are clustered close to the mean.

A high standard deviation indicates that data values are spread out over a wider range.

Variance is the average of squared differences from the mean and is used to calculate standard deviation.

Standard deviation is used in statistics, finance, research, data science, engineering, and quality control.

Yes. Many statistical analyses use sample standard deviation when working with a subset of a population.

It saves time, reduces calculation errors, and helps analyze data variability quickly.

Population standard deviation divides by N (total population size), while sample standard deviation divides by n - 1 (Bessel's correction) to correct for bias when estimating population variability from a sample.

What standard deviation tells you

Standard deviation measures how far values typically spread from their mean. A smaller result indicates values clustered near the average; a larger result indicates greater dispersion. It uses the same unit as the original data.

Population and sample formulas

Population variance divides the sum of squared deviations by N. Sample variance divides by N - 1 to estimate variability in a larger population. Choose the version that matches whether your entries are the complete population or a sample.

Interpretation cautions

Standard deviation is sensitive to outliers and does not describe distribution shape by itself. The 68–95–99.7 guideline applies approximately to normally distributed data, not automatically to every dataset.

Population and sample standard deviation are different

Population standard deviation divides the sum of squared deviations by the full count N. Sample standard deviation uses N − 1 to estimate variability in a wider population from a sample. Choosing the wrong version can materially change a result when the dataset is small, so decide whether the entered values are the complete population or observations sampled from something larger.

Standard deviation shares the same unit as the original data and is sensitive to extreme values. Review the mean, median, range and the underlying distribution rather than judging spread from one statistic alone. Empty datasets and a one-value sample do not provide a valid sample standard deviation.

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