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- Measures how spread out a set of numbers is around the mean.
- Gives sample (s) or population (σ) standard deviation and variance.
- Also shows mean, count, sum, range, standard error and coefficient of variation.
- Includes a step-by-step table of each deviation and its square.
How to use the standard deviation calculator
Enter your data
Paste or type your numbers, separated by commas, spaces or new lines.
Choose sample or population
Pick sample if your numbers are a subset of a larger group, or population if you have every value.
Read the result
The standard deviation, variance and mean appear instantly.
Check the working
Open the step-by-step table to see each value minus the mean, squared, then added up.
How standard deviation is calculated
Population standard deviation
σ = √( Σ(x − μ)² ÷ N )
μ is the mean of all N values in the population.
Sample standard deviation
s = √( Σ(x − x̄)² ÷ (n − 1) )
Dividing by n − 1 (Bessel’s correction) stops a sample from underestimating the spread of the full population.
In words: find the mean, subtract it from every value, square each result, add the squares, divide by n (population) or n − 1 (sample) to get the variance, then take the square root.
Example 1: sample of 8 values
Data: 10, 12, 23, 23, 16, 23, 21, 16.
- Mean 144 ÷ 8 = 18
- Sum of squared deviations 192
- Sample variance 192 ÷ 7 = 27.428571
- Sample standard deviation 5.237229
If the same 8 numbers were a full population, the variance would be 192 ÷ 8 = 24 and the standard deviation 4.898979.
Example 2: a classic population
Data: 2, 4, 4, 4, 5, 5, 7, 9.
- Mean 5
- Population variance 4
- Population standard deviation 2
This textbook data set is handy for checking your own manual working.
The 68-95-99.7 rule
For data that follows a normal (bell-shaped) distribution:
- Within 1 SD of the mean about 68% of values
- Within 2 SD about 95% of values
- Within 3 SD about 99.7% of values
Use this as a rough guide only. Skewed data does not follow it closely.
Where a standard deviation calculator helps
Statistics homework
Check sample and population answers and copy the step table into your work.
Test scores and grades
See whether a class scored consistently or had a wide spread of results.
Investing and risk
Measure how much returns move around their average, a common measure of volatility.
Quality control
Track how much product measurements vary from the target size or weight.
For the mean, median and mode of the same data, use the average calculator. The rounding calculator helps you report results to sensible precision, and the percent error calculator compares a measured mean with a known value.
Standard deviation calculator FAQ
Should I use sample or population standard deviation?
Use population when your data includes every member of the group you care about. Use sample when your data is a subset used to estimate a larger group, which is the most common case.
Why is the sample standard deviation larger?
It divides by n − 1 instead of n. That correction makes up for the fact that a sample tends to sit closer to its own mean than to the true population mean.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean. Standard deviation is its square root, so it is in the same units as your data and easier to interpret.
What does a standard deviation of 0 mean?
Every value is the same, so there is no spread at all.
What is the coefficient of variation?
It is the standard deviation divided by the mean, shown as a percent. It lets you compare spread between data sets with different scales.
How many numbers can I enter?
Up to 5,000 values. A sample needs at least 2 numbers.