Precision Utility
Standard Deviation
Calculator
σ Population
0
s Sample
0
Enter a list of numbers separated by commas or spaces and get instant results. This calculator computes population and sample standard deviation, variance, mean and count from any data set.
Your Data
Separate values with commas, spaces, or new lines
Population Standard Deviation
0
Standard Deviation
0
Variance
0
Mean
0
Count
0
How the standard deviation calculator works
Type or paste your numbers into the input box. You can separate values with commas, spaces or new lines — the calculator handles all three formats automatically.
Choose between Population and Sample mode using the toggle. Population mode divides by N (the total number of data points), while Sample mode divides by N−1 to apply Bessel's correction for bias.
Results update instantly as you type. You will see the standard deviation, variance, mean and count displayed in the result card. The hero stats at the top always show both population and sample standard deviation side by side for quick comparison.
The formula used is: find the mean, subtract the mean from each value, square each difference, average those squared differences (using the chosen divisor), then take the square root.
What you need to know about standard deviation
Standard deviation is one of the most important concepts in statistics. It tells you how much individual data points typically differ from the average (mean) of the set.
Key facts:
- A low standard deviation means data points are clustered tightly around the mean
- A high standard deviation means data is spread out over a wider range
- Standard deviation is always non-negative — it equals zero only when every value is identical
- About 68% of data in a normal distribution falls within one standard deviation of the mean
- About 95% falls within two standard deviations, and 99.7% within three (the 68-95-99.7 rule)
Variance is the square of the standard deviation. While variance is useful in mathematical derivations, standard deviation is preferred for interpretation because it shares the same unit as the original data.
Use population standard deviation (σ) when your data covers the entire group. Use sample standard deviation (s) when you have a subset — the N−1 divisor corrects the tendency for samples to underestimate the true spread.
Frequently asked questions
What is standard deviation?
Standard deviation is a measure of how spread out the values in a data set are from the mean. A low standard deviation means the values are clustered close to the average, while a high standard deviation means they are more spread out.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides by N (the total number of values) and is used when your data includes every member of the group. Sample standard deviation (s) divides by N−1 to correct for bias when your data is only a subset of a larger population.
How do you calculate standard deviation step by step?
First, find the mean (average) of your data. Then subtract the mean from each value and square the result. Next, find the average of those squared differences (divide by N for population, N−1 for sample). Finally, take the square root of that average to get the standard deviation.
What is variance?
Variance is the average of the squared differences from the mean. It is the standard deviation squared. Variance uses the same divisor as standard deviation: N for population variance and N−1 for sample variance.
When should I use sample vs population standard deviation?
Use population standard deviation when your data set includes every single member of the group you are studying. Use sample standard deviation when your data is a subset or sample taken from a larger population, which is the more common scenario in statistics.
What does a standard deviation of zero mean?
A standard deviation of zero means every value in your data set is identical. There is no spread or variation at all — all data points equal the mean.