MasterTool

Standard Deviation Calculator

Measure how spread out a data set is, with sample and population standard deviation side by side.

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What this tool does

Measure how spread out a data set is, with sample and population standard deviation side by side. Use the sample SD (n−1) when your data are a sample from a larger population; use the population SD (N) when the data are the entire population of interest.

How to use the Standard Deviation Calculator

  1. Enter or select numbers (comma or whitespace separated).
  2. Read the calculated result; change any measurement to compare alternatives.

Formula

sample s = sqrt( Σ(x−x̄)² / (n−1) ); population σ = sqrt( Σ(x−μ)² / N )
data
Numbers (comma or whitespace separated)

Use the sample SD (n−1) when your data are a sample from a larger population; use the population SD (N) when the data are the entire population of interest.

Worked example

For standard deviation calculator, the following measurements illustrate the exact method: Numbers (comma or whitespace separated): 2,4,4,4,5,5,7,9.

Inputs

  • Numbers (comma or whitespace separated)2,4,4,4,5,5,7,9

Result

  • Sample standard deviation (s)2.14
  • Population standard deviation (σ)2
  • Mean5
  • Sample variance (s²)4.57
  • Population variance (σ²)4
  • Count8

Results explained

Sample standard deviation (s)
Sample standard deviation (s) from the formula above. Use the sample SD (n−1) when your data are a sample from a larger population; use the population SD (N) when the data are the entire population of interest.
Population standard deviation (σ)
Population standard deviation (σ) from the formula above. Use the sample SD (n−1) when your data are a sample from a larger population; use the population SD (N) when the data are the entire population of interest.
Mean
Mean from the formula above. Use the sample SD (n−1) when your data are a sample from a larger population; use the population SD (N) when the data are the entire population of interest.
Sample variance (s²)
Sample variance (s²) from the formula above. Use the sample SD (n−1) when your data are a sample from a larger population; use the population SD (N) when the data are the entire population of interest.
Population variance (σ²)
Population variance (σ²) from the formula above. Use the sample SD (n−1) when your data are a sample from a larger population; use the population SD (N) when the data are the entire population of interest.
Count
Count from the formula above. Use the sample SD (n−1) when your data are a sample from a larger population; use the population SD (N) when the data are the entire population of interest.

Frequently asked questions

The population SD divides squared deviations by N and describes a complete population. The sample SD divides by n−1 (Bessel's correction), which corrects the bias when a sample mean stands in for the unknown population mean.

A sample's deviations are measured from its own mean, which sits closer to the data than the true population mean, so raw squared deviations run small. Dividing by n−1 instead of n inflates the estimate just enough to make it unbiased.

Every value in the data set is identical, so there is no spread at all. Any data set with at least two different values has a positive standard deviation.

Yes. If your data are test scores, the SD is in score points; variance, by contrast, is in squared units, which is why the SD is usually easier to interpret.

No. It only means more spread. Whether that matters depends on context — a large SD in exam scores may signal uneven learning, while a large SD in daily temperatures across a year is expected.