What is population standard deviation?
The population standard deviation, written \(\sigma\) (sigma), measures how spread out a set of numbers is when those numbers represent the entire population rather than a sample. A small \(\sigma\) means the values cluster tightly around the mean; a large \(\sigma\) means they are widely scattered. Use the population formula when your data covers every member of the group you care about — for a subset (a sample), use the sample standard deviation, which divides by \(N-1\) instead of \(N\).
How to use this calculator
Type your numbers into the box separated by commas or spaces, for example 4, 8, 15, 16, 23, 42. The calculator returns the standard deviation \(\sigma\), plus the count \(N\), sum, mean \(\mu\), and variance \(\sigma^2\) so you can check each step.
The formula explained
First compute the mean of the \(N\) values:
$$\mu = \frac{1}{N}\sum_{i=1}^{N} x_i$$Then average the squared distance of each value from the mean and take the square root:
$$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2}$$Here \(x_i\) = each data value, \(N\) = number of values, and \(\mu\) = the population mean.
Worked example
For the data \(2, 4, 4, 4, 5, 5, 7, 9\) there are \(N = 8\) values that sum to 40, so:
$$\mu = \frac{40}{8} = 5$$The squared deviations sum to \(9+1+1+1+0+0+4+16 = 32\), giving:
$$\sigma = \sqrt{\frac{32}{8}} = \sqrt{4} = 2$$
FAQ
When do I use population vs sample? Use population (divide by \(N\)) when your data is the complete set; use sample (divide by \(N-1\)) when it is a sample drawn from a larger group.
What is variance? Variance \(\sigma^2\) is the standard deviation squared — the average squared deviation from the mean before taking the root.
Can I enter decimals or negatives? Yes. Separate values with commas or spaces; negative numbers and decimals are fully supported.
