Standard Deviation Calculator

Quickly calculate standard deviation, variance, mean, and more for both population and sample datasets with our calculator.

Enter a set of numbers separated by commas or spaces (e.g., 20,160,40,80)
Standard Deviation Type
Use when your data represents the entire population
Use when your data is a sample from a larger population
Display Options
Display the step-by-step calculation process
Widget
Related Tools
Confidence Interval Calculator

What Is Standard Deviation

Standard deviation (? or s) measures how spread out data points are from the mean. A low value indicates data clusters closely around the mean, while a high value suggests greater variability. It is widely used in statistics to quantify uncertainty and assess consistency in datasets.

How to Calculate Standard Deviation

Follow these steps to calculate standard deviation manually:

  1. Find the mean: \(\text{Mean} (\mu) = \frac{\sum x_i}{n}\)
  2. Calculate deviations: Subtract the mean from each data point
  3. Square deviations: \((x_i - \mu)^2\)
  4. Compute variance: \(\text{Variance} (\sigma^2) = \frac{\sum (x_i - \mu)^2}{n}\) (population) or \(s^2 = \frac{\sum (x_i - \mu)^2}{n-1}\) (sample)
  5. Take the square root: \(\sigma = \sqrt{\sigma^2}\) or \(s = \sqrt{s^2}\)

Example: For data [5, 7, 3, 7, 10] with mean 6.4:
Squared deviations: \((5-6.4)^2 + (7-6.4)^2 + ... = 28.8\)
Population ?: \(\sqrt{28.8 / 5} = 2.4\)

Fields Where Standard Deviation Is Applied

Standard deviation is essential across numerous disciplines:

  • Finance: Measures investment risk and market volatility.
  • Quality Control: Assesses product consistency in manufacturing.
  • Sports: Analyzes player performance consistency.
  • Education: Evaluates scoring variability in standardized tests.

Our Fast & Free Standard Deviation Calculator

While manual calculations are educational, our online tool simplifies the process. Enter your dataset, and it instantly computes population or sample standard deviation. No sign-up required – perfect for students, researchers, and professionals needing quick results.