Standard Deviation Calculator
Enter your data values separated by commas and this standard deviation calculator instantly computes the mean, median, sample standard deviation, population standard deviation, and variance. All results shown together in one place.
How to Use This Standard Deviation Calculator
Type or paste your numbers into the input box, separated by commas or spaces. Hit Calculate and the tool shows six statistics at once: count, mean, median, sample standard deviation, population standard deviation, and variance. The histogram below the numbers gives a quick visual sense of how spread out your data is.
Use sample standard deviation (s) when your numbers are a sample taken from a larger population. Use population standard deviation (sigma) when you have complete data for the entire group you are measuring. When in doubt, sample standard deviation is the safer choice for most school and research work.
Standard Deviation Formulas
Step 1, Find the Mean: Add all values together and divide by the count.
Mean (x̄) = (x1 + x2 + ... + xn) / n
Step 2, Find Each Squared Deviation: Subtract the mean from each value and square the result.
(xi - x̄)²
Step 3, Population Standard Deviation (σ): Take the square root of the average squared deviation.
σ = sqrt[ Sum(xi - x̄)² / n ]
Step 4, Sample Standard Deviation (s): Same formula but divide by n-1 instead of n. This is called Bessel's correction and reduces bias when working from a sample.
s = sqrt[ Sum(xi - x̄)² / (n-1) ]
Variance is simply the standard deviation squared. So if σ = 4, variance = 16.
Standard Deviation Examples
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Test scores: 70, 75, 80, 85, 90
Mean = 80. Deviations: -10, -5, 0, 5, 10. Squared: 100, 25, 0, 25, 100. Sum = 250. Population variance = 50. Population std dev = 7.07. Sample std dev = 7.91. -
Small data set: 2, 4, 4, 4, 5, 5, 7, 9
Mean = 5. Population std dev = 2. This is the classic textbook example because the answer comes out exactly as a whole number. - Interpreting the result: A standard deviation of 2 on a data set with a mean of 50 means most values fall between 48 and 52, a tight cluster. A standard deviation of 20 on the same mean means values spread widely from 30 to 70.
Frequently Asked Questions
What does standard deviation tell you?
Standard deviation measures how spread out the values in a data set are from the mean. A small standard deviation means the numbers are clustered tightly around the average. A large one means there is a lot of variation. In practical terms, a class where all students scored between 78 and 82 has a low standard deviation; one where scores range from 40 to 98 has a high one.
When should I use sample vs population standard deviation?
Population standard deviation (σ) is correct when you have data for every member of a group. Sample standard deviation (s) is for when you have measured only some members and want to draw conclusions about the whole group. For homework and most research assignments, sample standard deviation is typically what is expected unless the problem states otherwise.
What is variance?
Variance is the average of the squared differences from the mean. It is always the square of the standard deviation. Variance is useful in statistical formulas because it avoids square roots in intermediate steps, making certain calculations cleaner. For interpretation, standard deviation is easier to use because it is in the same units as the original data.
How many numbers can I enter?
The tool handles large data sets comfortably. Paste a hundred or a thousand values and it processes them instantly. There is no hard limit built in, though very large data sets may take a fraction of a second longer on older devices.
Guides for the Standard Deviation Calc
Longer reads on the maths and the practical side of this tool.