Free Standard Deviation Calculator
Calculate sample or population standard deviation, variance, mean, median, range and standard error from numeric data. Use advanced options and copy or download your results. Browser-based and no signup required.
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How to use the standard deviation calculator
Calculate sample or population standard deviation, variance, mean, median, range and standard error from numeric data. No signup required.
- Paste up to 10,000 numeric values, separated by whitespace, commas, semicolons or newlines.
- Choose sample or population interpretation. Use a decimal point and omit thousands separators.
- Review standard deviation, variance, mean, median, range and the sample standard-error estimate when available.
- Change display precision if needed and copy or download the data and result report.
Where standard deviation helps
An average alone can hide a lot. Two classes can both average 75 on a test, yet one class may have scores tightly grouped around 75 while the other ranges from 40 to 100. Standard deviation puts a number on that spread.
Students use it for statistics homework and lab reports. Quality teams use it to see whether part sizes or fill weights stay consistent. Small business owners use it to judge how much daily sales swing, and researchers use it with the standard error to describe how precise a sample mean is.
A worked example with five values
Enter 12, 15, 11, 18, 14 and keep Sample selected. The count is 5, the sum is 70 and the mean is 70 divided by 5, which is 14. The deviations from the mean are minus 2, 1, minus 3, 4 and 0. Squared, they are 4, 1, 9, 16 and 0, which add up to 30.
For a sample, the variance is 30 divided by 5 minus 1, so 30 divided by 4, which is 7.5. The standard deviation is the square root of 7.5, about 2.739. The standard error of the mean is 2.739 divided by the square root of 5, about 1.225.
Switch to Population and the variance becomes 30 divided by 5, which is 6, with a standard deviation of about 2.449. The median is 14, the minimum 11, the maximum 18 and the range 7 in both modes.
How the calculator computes the results
The formula is variance equals the sum of squared deviations from the mean divided by n minus 1 for a sample or n for a population. Standard deviation is the square root of the variance. Internally the tool uses Welford's method, which updates the mean and the sum of squares one value at a time. This avoids the rounding errors that a naive formula can produce with large or very close numbers.
The median comes from the sorted list: the middle value for an odd count, or the average of the two middle values for an even count. The standard error is shown only in sample mode, because it estimates how far a sample mean is likely to sit from the population mean.
You can show results with 0 to 10 decimal places. Very large or very small results switch to scientific notation so they stay readable.
Common mistakes to avoid
Do not type thousands separators. Commas separate values, so 1,250 is read as two numbers, 1 and 250. Write 1250 instead, and use a period for decimals.
Do not choose Population just because you have a lot of data. The question is whether your values are the whole group or a sample drawn from it. A survey of 500 customers is still a sample of all customers.
Also check for stray text such as units or labels. The calculator stops and names any token that is not a valid number rather than skipping it.
Useful tips and limitations
- Sample variance divides by n minus 1; population variance divides by n.
- Scientific notation is supported. Invalid tokens are rejected rather than silently dropped.
- Paste a column straight from a spreadsheet, since newlines are accepted as separators.
- Compare standard deviation with the mean, because a spread of 5 is large for values near 10 but small for values near 1,000.
Common questions
When should I choose sample standard deviation rather than population?
Use sample when the data are a sample used to estimate a wider population. Use population when the values are the full population you want to describe.
Why is sample standard deviation undefined for a single value?
Sample variance divides by n minus 1. With one value the denominator is zero. Population deviation for one value is zero.
How should decimals and thousands be entered in the statistics tool?
Use a period for decimals and no thousands grouping. Commas separate values, so 1,000 is interpreted as two entries rather than one thousand.
What is the standard error of the mean?
It is the sample standard deviation divided by the square root of the count. It estimates how much a sample mean would vary from sample to sample.
Can I enter negative numbers or scientific notation?
Yes. Values such as -3.5 and 2.4e-3 are accepted, as long as their absolute value does not exceed 1e100.
How many values can I enter?
You can enter between 1 and 10,000 numbers, separated by spaces, commas, semicolons or new lines.
Your input is processed in your browser without a ToolWorld Online account. Copy or download anything you want to keep before leaving the page.