How to use the standard deviation calculator
- 1 Type or paste your numbers, separated by commas, spaces or new lines.
- 2 Choose whether the data is a sample or the whole population.
- 3 Read the standard deviation, variance and mean.
- 4 Check the steps and the 68-95-99.7 ranges.
Worked example
The numbers 10, 12, 23, 23, 16, 23, 21, 16 as a sample. The mean is 144 ÷ 8 = 18. The squared deviations add up to 192, so the sample variance is 192 ÷ 7 = 27.43 and the sample standard deviation is 5.2372. Treated as a whole population, the variance is 192 ÷ 8 = 24 and the standard deviation is 4.899.
Using standard deviation well
- Pick sample or population before you calculate.
- Check for outliers: one extreme value can inflate the result.
- Compare spreads with the coefficient of variation when means differ.
- Report it with the mean, for example 18 ± 5.2.
Find the center of your data with the Mean, Median and Mode Calculator, compare one value with the Z-Score Calculator, and estimate the true mean with the Confidence Interval Calculator.
Features
Sample and population
Both results, with the right divisor explained.
Every step
Mean, squared deviations, variance and the square root.
Extra statistics
Coefficient of variation, standard error and the empirical-rule ranges.
When to use it
Statistics homework
Check each step of a standard deviation problem. Find the mean, median and mode with the Mean, Median and Mode Calculator.
Test scores
See how spread out scores are, then compare one score with the Z-Score Calculator.
Research and surveys
Report the spread of your data, then build a confidence interval with the Confidence Interval Calculator.
Quality control
Track how much a measurement varies from batch to batch.
Related Math tasks
Frequently asked questions
What is standard deviation?
A measure of how far values typically are from the mean. A small standard deviation means the values are close together; a large one means they are spread out.
How do you calculate standard deviation?
Find the mean, subtract it from each value and square the result, add the squares, divide by n − 1 for a sample (or n for a population), then take the square root.
What is the difference between sample and population standard deviation?
A population includes every member of the group; a sample is part of it. The sample formula divides by n − 1 instead of n (Bessel's correction), which gives a slightly larger value that better estimates the population.
Which one should I use?
Use the sample standard deviation when your data is a sample from a bigger group, which is most real data. Use the population formula only when you have every value, such as all students in one class.
What is variance?
The average squared distance from the mean: the standard deviation squared. Variance is in squared units, so the standard deviation is easier to interpret.
What does the 68-95-99.7 rule mean?
In a normal distribution, about 68% of values are within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
What is a good standard deviation?
There is no single good value: it depends on the data and its units. The coefficient of variation (standard deviation ÷ mean) helps compare spread between data sets of different sizes.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is zero or positive. It is zero only when all values are the same.
What is the standard error?
The standard deviation divided by the square root of n. It measures how much the sample mean would vary from sample to sample.
How do I find standard deviation in Excel or Google Sheets?
Use STDEV.S (or STDEV) for a sample and STDEV.P for a population. The results match this calculator.
Does one outlier change the standard deviation a lot?
Yes. Because deviations are squared, a single extreme value can raise the standard deviation sharply. The interquartile range is less affected.
Built and reviewed by Hamza Farooq, founder and lead developer
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