Vertoq
Standard Deviation Calculator
Enter your values separated by commas or spaces to see the mean, standard deviation, and variance.
What standard deviation measures
Standard deviation measures how spread out the values in a dataset are around the mean. A low standard deviation means the values sit close to the mean, a high standard deviation means they spread out much more widely.
Population or sample, which one to use
If your dataset is the entire population you care about, every value that exists, use the population standard deviation. If your dataset is just a sample drawn from a larger population, which you want to draw conclusions about, use the sample standard deviation. The difference lies in the denominator: population divides by n (the count), sample divides by n−1, giving a slightly larger value to compensate for the uncertainty of only having a sample.
The formulas
σ = √(Σ(x − x̄)² / n)
s = √(Σ(x − x̄)² / (n−1))
x̄ is the mean, x is each individual value, and n is the count of values. Both formulas sum the squared difference between each value and the mean, then take the square root of the average.
Worked example, step by step
For the dataset 2, 4, 4, 4, 5, 5, 7, 9 (8 values), the mean is 5. The differences from the mean are −3, −1, −1, −1, 0, 0, 2, 4, and their squares are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. The population standard deviation is √(32/8) = √4 = 2. The sample standard deviation is √(32/7) ≈ 2.14.
The connection to variance
Variance is standard deviation squared, meaning it's exactly that sum before the square root is taken. In the example above, the population variance is 4 (the square of 2). Variance is used often in statistical calculations, but it's harder to interpret directly since its unit is squared, which is why standard deviation is usually more intuitive to present.
Frequently asked questions
Why is the sample standard deviation always bigger than the population one?
Because the sample divides the sum by n−1 instead of n, a smaller number to divide by gives a bigger result. That compensates for a sample tending to underestimate the true spread in the full population.
What counts as a valid input format?
Enter the numbers separated by commas or spaces, like "2, 4, 6, 8" or "2 4 6 8". Both formats work, and you can even mix them.
What happens if I enter just one number?
The calculator requires at least two values. With just one value there's no spread to measure, and the standard deviation (especially the sample one, which divides by n−1) becomes undefined.