Standard Deviation Calculator

Calculate mean, variance and standard deviation (population and sample) for any data set instantly.

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Tip: Use population σ when you have the entire population; use sample s when working with a subset.

Formulas

x̄ = Σxᵢ / n
σ² = Σ(xᵢ−x̄)² / n
s² = Σ(xᵢ−x̄)² / (n−1)
σ = √σ²
68-95-99.7 rule:
±1σ → ~68%
±2σ → ~95%
±3σ → ~99.7%

How to use?

  1. 1
    Enter your dataType numbers separated by commas or spaces (e.g. 4, 7, 13, 2, 1).
  2. 2
    Click CalculateMean, variance, population and sample standard deviation appear instantly.
  3. 3
    Interpret the resultLow σ means values cluster near the mean; high σ means data is spread out.

FAQ

What is the difference between population and sample standard deviation?
Population σ divides the sum of squared deviations by n (the total count). Sample s divides by n−1 (Bessel's correction) to give an unbiased estimate when you only have a subset. For large n the difference is small; for small samples it matters.
What does standard deviation tell you?
Standard deviation measures how spread out values are around the mean. In a normal distribution, about 68% of values fall within ±1σ, 95% within ±2σ, and 99.7% within ±3σ (the 68-95-99.7 rule).
What is variance?
Variance is the average of squared deviations from the mean. Standard deviation is simply the square root of variance — it returns the result in the same units as the original data, making it more interpretable.

Standard Deviation and Variance Explained

Standard deviation (σ or s) is the most widely used measure of data dispersion — it tells you how far, on average, individual values lie from the mean. A small standard deviation means values cluster tightly around the mean; a large one means they are spread out.

Step-by-Step Calculation

  1. Calculate the mean: x̄ = Σxᵢ / n
  2. Compute each squared deviation: (xᵢ − x̄)²
  3. Population variance: σ² = Σ(xᵢ − x̄)² / n
  4. Sample variance: s² = Σ(xᵢ − x̄)² / (n−1)
  5. Standard deviation = √variance

The 68-95-99.7 Rule

For normally distributed data: ~68% of values fall within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ of the mean. This rule is used extensively in quality control, finance and experimental sciences.

Population vs. Sample

Use population σ when you have data for every member of the group (e.g. heights of all students in a class). Use sample s when your data is a subset drawn from a larger population (e.g. a survey of 50 people to estimate national averages). Dividing by n−1 corrects the tendency to underestimate variability in small samples.

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