Standard Deviation Calculator
Enter a list of numbers below to instantly calculate the mean, variance, and both population and sample standard deviation — with the full step-by-step work shown.
What Is Standard Deviation?
Standard deviation measures how spread out a set of numbers is from its average (mean). A small standard deviation means the values are clustered tightly around the mean; a large standard deviation means the values are spread out over a wider range. It's the single most common way to describe variability in statistics, used everywhere from test scores and quality control to finance and scientific research.
How to Calculate Standard Deviation, Step by Step
- Find the mean (average) of all the values.
- Subtract the mean from each value to find each value's difference from the mean.
- Square each of those differences (this removes negative signs).
- Add up all the squared differences.
- Divide that sum by n for population standard deviation, or by n − 1 for sample standard deviation. This gives you the variance.
- Take the square root of the variance. That result is the standard deviation.
Worked example
Data set: 4, 8, 6, 5, 3 (n = 5)
Mean = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2
Squared differences from the mean: (4−5.2)² = 1.44, (8−5.2)² = 7.84, (6−5.2)² = 0.64, (5−5.2)² = 0.04, (3−5.2)² = 4.84
Sum of squared differences = 14.8
Population variance = 14.8 / 5 = 2.96 → population standard deviation = √2.96 ≈ 1.72
Sample variance = 14.8 / 4 = 3.7 → sample standard deviation = √3.7 ≈ 1.92
Population vs. Sample Standard Deviation
Population standard deviation (σ) is used when your data represents the entire group you care about — dividing the sum of squared differences by n, the total count.
Sample standard deviation (s) is used when your data is only a sample drawn from a larger population you're trying to estimate — dividing by n − 1 instead of n. This small adjustment (called Bessel's correction) corrects for the fact that a sample tends to slightly underestimate the true variability of the full population. In practice, most real-world data analysis uses sample standard deviation, since it's rare to have every member of a true population.
Frequently Asked Questions
What is a good standard deviation?
There's no universal "good" value — it depends entirely on what you're measuring and the scale of your data. A standard deviation is only meaningful relative to the mean and the context: a standard deviation of 5 is small for data averaging 1,000 but large for data averaging 10. Compare it to the mean (as a coefficient of variation) or to similar datasets, rather than judging the raw number alone.
What does a standard deviation of 0 mean?
A standard deviation of 0 means every value in the dataset is identical — there is no spread or variation at all. The mean and every data point are the same number.
What's the difference between variance and standard deviation?
Variance is the average of the squared differences from the mean. Standard deviation is the square root of variance. Standard deviation is more commonly used for interpretation because it's expressed in the same units as the original data, while variance is in squared units.
Should I use population or sample standard deviation?
Use population standard deviation (dividing by n) when your dataset is the entire group you care about — for example, every student's score in one specific class. Use sample standard deviation (dividing by n − 1) when your dataset is a sample drawn from a larger population you're trying to estimate — for example, 50 surveyed voters standing in for an entire city. Sample standard deviation is used far more often in real-world statistics, since most data collected is a sample.
Can standard deviation be negative?
No. Standard deviation is calculated from squared differences and a square root, both of which always produce a non-negative result. The smallest possible standard deviation is 0 (when all values are identical).
How do you find standard deviation by hand, without a calculator?
Find the mean of your data. Subtract the mean from each value and square the result. Add up all the squared differences. Divide that sum by n (population) or n − 1 (sample). Take the square root of that number — that's your standard deviation. The step-by-step table in the calculator above shows this exact process for your own numbers.
Related Calculators
- Percentile Calculator — Statistics
- Mean, Median, Mode Calculator — Statistics
- Variance Calculator — Statistics
- Quartile Calculator — Statistics