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📊 Standard Deviation Calculator

Calculate mean, median, variance, and standard deviation for any data set. Supports both population and sample standard deviation. Enter numbers separated by commas or new lines.

Standard Deviation Formulas

Population Standard Deviation
σ = √( Σ(xᵢ − μ)² / N )
Sample Standard Deviation
s = √( Σ(xᵢ − x̄)² / (N − 1) )
Mean
μ = Σxᵢ / N

Use population SD (σ) when you have data for the entire population. Use sample SD (s) when your data is a sample from a larger population — the (N−1) denominator corrects for bias.

How to Calculate Standard Deviation

  1. 1
    Choose Population or Sample
    Select population SD (σ) if you have all the data, or sample SD (s) if your data is a subset.
  2. 2
    Enter Your Numbers
    Type or paste your numbers separated by commas, spaces, or new lines. Decimals are supported.
  3. 3
    Read Your Results
    The calculator instantly shows count, mean, median, min, max, range, variance, and standard deviation.
  4. 4
    Review Sorted Data
    The sorted data list helps you spot outliers and understand the distribution visually.

Real-World Example

Data set: 4, 8, 15, 16, 23, 42

Count = 6
Mean = (4+8+15+16+23+42) ÷ 6 = 18
Median = (15+16) ÷ 2 = 15.5
Population SD = 13.3
Range = 42 − 4 = 38

Frequently Asked Questions

Standard deviation measures how spread out data values are from the mean. A low SD means data points are clustered close to the average; a high SD means they are widely spread. It is fundamental in statistics, finance, and science.

Use population SD (σ) when you have data for every member of the group (e.g. all test scores in one class). Use sample SD (s) when your data represents a subset of a larger group (e.g. a survey sample). Sample SD uses N−1 to reduce bias.

Variance is the square of the standard deviation. It represents the average squared distance from the mean. While variance is useful in calculations, standard deviation is easier to interpret because it is in the same units as the original data.

For a normal (bell-curve) distribution: about 68% of data falls within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. This rule helps identify outliers — values beyond 3 SD are statistically unusual.

Outliers have a large effect on SD because each deviation is squared before averaging, amplifying the impact of extreme values. If you have suspected outliers, consider reporting both the full SD and the SD with outliers removed.

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