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Standard Deviation

Standard deviation measures how spread out data values are from the mean. It is the most widely used measure of statistical variability, appearing in quality control, finance (portfolio volatility), scientific research, and machine learning. Our calculator handles both population and sample standard deviation, shows the full variance table, and visualizes the distribution.

Numbers (comma or space-separated)

Sample Std Dev (s)

Population Std Dev (σ)

Mean

Variance

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Math

About the Standard Deviation Calculator

Standard deviation measures how spread out data values are from the mean. It is the most widely used measure of statistical variability, appearing in quality control, finance (portfolio volatility), scientific research, and machine learning. Our calculator handles both population and sample standard deviation, shows the full variance table, and visualizes the distribution.

How to use it

  1. Enter your data values separated by commas or line breaks.
  2. Choose population (σ) or sample (s) standard deviation.
  3. See mean, median, variance, standard deviation, and range.
  4. View the frequency distribution and normal curve overlay.

Formula & methodology

Mean: μ = Σx / n. Population variance: σ² = Σ(x−μ)² / n. Sample variance: s² = Σ(x−μ)² / (n−1). Standard deviation = √variance. The n−1 denominator (Bessel's correction) makes the sample variance an unbiased estimator of the population variance.

Common use cases

  • Finance: measuring stock volatility (higher σ = higher risk)
  • Quality control: Six Sigma (process within ±3σ = 99.73% conformance)
  • Academic grading: normalizing test scores to a curve
  • Scientific experiments: reporting measurement uncertainty
  • A/B testing: determining statistical significance of results

Frequently asked questions

Use population σ when your data IS the entire population (e.g., all 12 monthly revenues for a full year). Use sample s when your data is a subset (e.g., 50 customer survey responses representing all customers). Sample SD uses n−1 (Bessel's correction) to compensate for underestimating variability in small samples.
For a normal distribution: ~68% of data falls within ±1σ of the mean, ~95% within ±2σ, ~99.7% within ±3σ (the 68-95-99.7 rule). So if test scores have mean 70 and σ = 10, about 68% of students scored between 60 and 80.

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