Standard Deviation Calculator: Your Handy Data Tool

Enter numbers separated by commas, spaces, or line breaks:

How the Calculator Works

This calculator computes statistical measures for your dataset:

  • Mean: The average of all numbers
  • Variance: The average of squared differences from the mean
  • Standard Deviation: The square root of variance (measures data spread)

Population vs Sample:

  • Population: Use when your data represents the entire group (divides by N)
  • Sample: Use when your data is a subset of a larger group (divides by N-1)

Example: For the numbers 2, 4, 6, 8, 10:

  • Mean = 6
  • Population Standard Deviation = 2.83
  • Sample Standard Deviation = 3.16

Population vs. Sample: Why the Divisor Changes

Both variance calculations sum the squared differences between each value and the mean, but they divide that sum differently. Population variance divides by n (the count) because you're treating your data as the entire group of interest. Sample variance divides by n-1 (Bessel's correction) because a sample tends to underestimate the true variance of the larger population it was drawn from - dividing by a smaller number compensates by making the estimate slightly larger.

Worked Example

Dataset: 5, 7, 9, 9, 10 (five test scores). Mean: (5+7+9+9+10) / 5 = 40/5 = 8. Squared differences from the mean: (5-8) squared=9, (7-8) squared=1, (9-8) squared=1, (9-8) squared=1, (10-8) squared=4. Sum: 9+1+1+1+4 = 16.

Population variance: 16 / 5 = 3.2, so population standard deviation = square root of 3.2, approximately 1.7889. Sample variance: 16 / 4 = 4.0, so sample standard deviation = square root of 4 = 2.0000 exactly. Notice how the sample figure is always a bit larger than the population figure for the same data - that gap shrinks as your dataset grows.

Which One to Pick

Use Population when your numbers are the complete set you care about (every student in one specific class). Use Sample when your numbers are a subset meant to represent a bigger group (a survey of 50 customers standing in for your whole customer base).

Minimum Input

At least two numbers are required - a single value has no spread to measure, so variance and standard deviation would both be undefined.