About Standard Deviation Calculator
This standard deviation calculator is free, needs no sign-up and runs in your browser, so your data is never uploaded. Paste a list of numbers, choose sample or population, and read the standard deviation, variance, mean, count, sum, standard error and margin of error. A table underneath shows every deviation from the mean and its square, so you can check the arithmetic line by line.
Standard deviation measures how spread out numbers are around their mean. A small value means the numbers cluster near the mean, a large value means they are scattered. The sample version divides the sum of squared deviations by n - 1 and the population version divides by n. Use sample when your numbers are a subset of a larger group, and population when they are every value you care about.
You can also enter a frequency table, one value and its count per line, or class intervals such as 10-20 with a count, which are counted at their midpoint. The coefficient of variation, the standard deviation as a percentage of the mean, is shown alongside for comparing spread between data sets with different units.
How to use Standard Deviation Calculator
- Paste or type your numbers, separated by commas, spaces or new lines. Use a dot for decimals.
- Pick Sample (n - 1) if the numbers are a sample, or Population (n) if they are the whole group.
- For grouped data, switch Data entry to Frequency table and enter one "value, frequency" pair per line. A class interval such as 10-20, 4 is counted at 15.
- Read the standard deviation at the top, then the variance, mean, sum, count, standard error and margin of error.
- Open the working and the deviations table to see each (x - mean) and its square, and the totals that feed the formula.
Features
- Sample and population standard deviation side by side, with the one you chose as the headline.
- Variance, mean, count, sum, range, standard error and margin of error at 90%, 95% or 99% confidence.
- Step-by-step working in written-out formulas, with a table of deviations and squared deviations.
- Frequency table and class interval input, with a weighted calculation instead of an expanded list.
- Coefficient of variation as a percentage of the mean.
- Copy the results as text. Works with long lists and on a phone.
The standard deviation formulas
Find the mean, subtract it from every value, square each difference, add the squares to get the sum of squares (SS), divide by n or n - 1 for the variance, and take the square root.
For a frequency table each value x is weighted by its frequency f, so SS = sum of f times (x - mean) squared and n = sum of f.
mean = sum(x) / n
SS = sum((x - mean)^2)
population: variance = SS / n, sigma = sqrt(SS / n)
sample: variance = SS / (n - 1), s = sqrt(SS / (n - 1))
standard error = s / sqrt(n)
coefficient of variation = sd / |mean| x 100%Worked examples
Data: 2, 4, 4, 4, 5, 5, 7, 9 gives Population sd 2, sample sd 2.1381
Mean 5, sum of squares 32. Population variance 32 / 8 = 4, so sigma = 2. Sample variance 32 / 7 = 4.5714.
Test scores: 72, 85, 90, 68, 77, 95, 81, 88, 79, 84 gives Sample sd 8.2523, mean 81.9
Variance 68.1, standard error 2.61, coefficient of variation 10.08%.
Frequency table: 1 (4 times), 2 (7), 3 (10), 4 (6), 5 (3) gives n = 30, mean 2.9, sample sd 1.1847
Each value is weighted by its frequency, so there is no need to write out the 30 numbers.
Good to know
Sample or population: which one do I use?
If your numbers are everything you are describing, such as the exam marks of a whole class when the class is the subject, use population. If they are a sample used to estimate a larger group, such as 30 customers out of thousands, use sample. Dividing by n - 1 instead of n makes the sample variance an unbiased estimate of the population variance, because a sample sits closer to its own mean than to the true mean.
For large data sets the two answers are nearly equal. For small ones they can differ a lot, which is the most common reason two calculators disagree.
What the margin of error means here
The margin of error is the half-width of a confidence interval for the mean: a critical value times the standard error s / sqrt(n). For a sample it uses the t distribution with n - 1 degrees of freedom, and for a population setting it uses the normal distribution. Treat it as a guide for the mean of your data, not as a statement about individual values.
Privacy: how it works
- All the arithmetic happens in your browser tab. The numbers you paste are not sent to Ulkaa or to any other server.
Tips
- One extreme value can inflate the standard deviation a lot. Check the deviations table for a very large squared deviation.
- Standard deviation has the same unit as your data. Variance is in squared units, which is harder to interpret.
- To compare spread between data sets with different means or units, use the coefficient of variation.
- A sample needs at least two values. With one value the sample standard deviation is undefined.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I calculate standard deviation by hand?
Find the mean, subtract it from each value, square the results, add them, divide by n for a population or n - 1 for a sample, and take the square root. The deviations table shows each step for your data.
What is the difference between sample and population standard deviation?
The population version divides by n and the sample version by n - 1. Use the sample version when your data is a subset of a larger group.
Can the standard deviation be negative?
No. It is a square root, so it is zero or positive. It is zero only when every value is the same.
What is a good standard deviation?
There is no universal good value. It depends on the unit and the mean. Compare it with the mean using the coefficient of variation, or with another data set measured the same way.
How do I find standard deviation from a frequency table?
Switch to Frequency table and enter each value with its count. The calculator weights each deviation by its frequency, and counts a class interval at its midpoint.
Why do two calculators give different answers?
Almost always one is using sample (n - 1) and the other population (n). Check which one each uses.
Popular Standard Deviation Calculator pages
- Sample Standard Deviation CalculatorPaste your numbers to get the sample standard deviation, which divides by n - 1.
- Population Standard Deviation CalculatorPaste the whole population to get the standard deviation sigma, which divides by n.
- Standard Deviation From a Frequency TableEnter each value with its frequency, or class intervals with counts, and get the standard deviation.
- Coefficient of Variation CalculatorPaste your numbers to get the coefficient of variation, the standard deviation as a percentage of the mean.