This free online standard deviation calculator measures how spread out a set of numbers is from its mean, reporting both population and sample variance and standard deviation. Enter your data set to see a complete statistical breakdown instantly.
How to Use the Standard Deviation Calculator
Type or paste your list of numbers into the "Enter Numbers" box, separated by commas, spaces, or new lines.
Click "Calculate".
Review the count, mean, population variance/standard deviation, and sample variance/standard deviation in the results table.
How the Calculator Works
The tool reads every number in the box, works out the mean, and then measures how far each value sits from that mean. The steps are:
Mean = sum of all values ÷ n, where n is the count of values.
For each value, find its deviation (value − mean) and square it, so negative and positive gaps do not cancel out.
Add the squared deviations. This total is called the sum of squares.
Population variance = sum of squares ÷ n. Sample variance = sum of squares ÷ (n − 1).
Standard deviation is the square root of the variance, which brings the answer back to the same unit as your data (marks, rupees, kilograms and so on).
All results are rounded to 4 decimal places. Any entry that is not a number, such as a word or a stray symbol, is skipped, so check that the Count matches the number of values you meant to enter.
Worked Example
Take the six values 4, 8, 15, 16, 23 and 42. Their sum is 108, so the mean is 108 ÷ 6 = 18.
Value
Value − Mean
Squared
4
−14
196
8
−10
100
15
−3
9
16
−2
4
23
5
25
42
24
576
Total
0
910
Population variance = 910 ÷ 6 = 151.6667 and population standard deviation = 12.3153. Sample variance = 910 ÷ 5 = 182 and sample standard deviation = 13.4907. These are the figures the calculator shows for this input. Notice that the single value 42 contributes 576 of the 910, which is why one extreme value can inflate the standard deviation so much.
Reading the Result
A small standard deviation means the values sit close to the mean. Two classes can both average 65 marks, yet the class with a standard deviation of 5 is far more uniform than one with 20.
For data that is roughly bell-shaped, about 68% of values fall within one standard deviation of the mean and about 95% within two. This rule does not hold for strongly skewed data such as incomes.
To compare spread between data sets with very different averages, divide the standard deviation by the mean to get the coefficient of variation. Investors use the same idea when they quote the volatility of a fund's returns.
Standard deviation is never negative. It is zero only when every value is identical.
Frequently Asked Questions
Population standard deviation divides the sum of squared differences from the mean by n (the count of numbers), assuming your data is the entire population. Sample standard deviation divides by n - 1 instead (Bessel's correction), which is used when your numbers are only a sample drawn from a larger population.
The mean is the sum of all numbers divided by the count. Variance is the average of the squared differences between each number and the mean, and standard deviation is simply the square root of variance.
Sample variance and sample standard deviation require at least two numbers, so with only one value the table shows "N/A — need at least 2 numbers" for those two rows, while population statistics are still shown.
If the numbers cover everyone you care about, such as the marks of all 40 students in one class, use the population figure. If they are a sample meant to represent a bigger group, such as 200 survey responses from a city, use the sample figure. Most school statistics textbooks and spreadsheet functions like STDEV.S use the sample formula by default.
Variance is in squared units, so for heights in centimetres it is measured in square centimetres, which has no everyday meaning. Taking the square root returns the standard deviation to centimetres, so you can say values typically differ from the average by about that many centimetres.