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Standard Deviation Calculator: Sample and Population

This free standard deviation calculator measures how spread out your numbers are. Choose sample or population, paste your data and get the standard deviation, variance, mean and a table of the working.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

Standard deviation tells you how much a set of numbers varies around its average. Two classes can both have an average mark of 60, yet one has marks between 55 and 65 and the other between 20 and 100. The standard deviation shows that difference at a glance.

This tool computes the standard deviation and variance for either a whole population or a sample, and lists the squared deviations so you can follow the sum.

How to Use This Tool

  1. Paste your numbers, separated by commas, spaces or lines.
  2. Pick Sample or Population.
  3. Press Calculate.
  4. Read the standard deviation, then the variance and the step table.

How the Calculation Works

First find the mean. Then subtract the mean from each value and square the result. Add the squares. For a population, divide by n to get the variance; for a sample, divide by n − 1. The standard deviation is the square root of the variance.

Dividing by n − 1 for samples is called Bessel’s correction. It makes the estimate of the spread less biased when you only see part of the data.

Worked Example

Take 4, 8, 6, 5, 3, 7. The mean is 5.5. The squared deviations are 2.25, 6.25, 0.25, 0.25, 6.25 and 2.25, which add up to 17.5. As a population, the variance is 17.5 ÷ 6 = 2.9167 and the standard deviation is 1.7078. As a sample, it is 17.5 ÷ 5 = 3.5 and the standard deviation is 1.8708.

Common Uses

  • Judging how consistent test marks or measurements are.
  • Comparing risk between investments.
  • Quality control in manufacturing.
  • Preparing z-scores and confidence intervals.

Tips and Common Mistakes

  • You need at least two numbers for a sample standard deviation.
  • Outliers raise the standard deviation a lot, so check for typing mistakes.
  • About 68% of values in a bell-shaped data set lie within one standard deviation of the mean.

Next, you may need the average calculator, z-score calculator, percent error calculator, weighted average calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

It is the typical distance of the values from their mean. A small value means the numbers are close together and a large value means they are spread out.
Use population if your list is every value you care about. Use sample if the list is only part of a larger group, which is the usual case in surveys and experiments.
Variance is the standard deviation squared. It is the average squared distance from the mean, so its unit is the square of the data unit.
No. It is a square root, so it is zero or positive. It is zero only when every number is the same.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.