How It Works
Standard deviation measures how spread out values are from the mean — a larger SD means more variability.
σ = √(Σ(xᵢ − x̄)² / N) | s = √(Σ(xᵢ − x̄)² / (N−1))
- Population SD (σ) divides by N — use when you have data for the entire group being studied.
- Sample SD (s) divides by N−1 (Bessel’s correction) — use when your data is a sample representing a larger population.
- Most practical applications (surveys, experiments, quality control) use sample SD because full population data is rarely available.
- Variance is the square of standard deviation; SD is in the same units as the data while variance is in squared units.
Worked Example
Dataset: 10, 12, 8, 15, 11 (n = 5).
Mean
(10+12+8+15+11) / 5 = 11.2
Squared deviations
(10−11.2)² + (12−11.2)² + (8−11.2)² + (15−11.2)² + (11−11.2)²
Sum of sq. dev
1.44 + 0.64 + 10.24 + 14.44 + 0.04 = 26.80
Population variance
26.80 / 5 = 5.36 → σ = 2.315
Sample variance
26.80 / 4 = 6.70 → s = 2.588
The dataset has a sample SD of 2.588, meaning values typically deviate from the mean (11.2) by about 2.6 units.
Understanding Standard Deviation
Measuring the spread around the mean
This tool measures how spread out a set of numbers is. You enter your values and a count, and it returns the sample and population standard deviation, the matching variances, the mean, and the range. In short, it tells you not just where the data centers but how widely it scatters around that center.
Spread matters as much as the average in surveys, lab results, test scores, quality control, and finance. Two datasets can share the same mean yet behave completely differently — the standard deviation is the number that captures that difference.
How standard deviation is calculated
The method has four steps: find the mean, subtract it from each value to get the deviations, square those deviations and average them (that average is the variance), then take the square root to return to the original units. The square root is what makes the standard deviation directly comparable to your data.
Squaring is the key step. It removes negative signs so deviations do not cancel out, and it gives extra weight to values that sit far from the mean. That is also why a single outlier can move the standard deviation a lot.
Sample versus population
Use the population standard deviation (σ) when your data is the entire group you care about — every item, with nothing left out. Use the sample standard deviation (s) when your data is a sample meant to represent a larger group, which is the usual case in research.
The only difference is the divisor: population divides by n, while sample divides by n−1. That smaller divisor makes the sample value slightly larger, correcting for the tendency of a sample to understate the true spread of the whole population.
What an SD value tells you
Standard deviation is in the same units as your data, so an SD of 2.6 on test scores means values typically sit about 2.6 points from the mean. A small SD means the data clusters tightly; a large one means it is widely scattered.
For roughly bell-shaped data, the empirical rule helps: about 68% of values fall within one SD of the mean, about 95% within two, and about 99.7% within three. That turns a single number into a quick map of where most of your data lives.
A worked example
For the dataset 10, 12, 8, 15, 11, the mean is 11.2. The squared deviations are 1.44, 0.64, 10.24, 14.44, and 0.04, which sum to 26.8. Dividing by n (5) gives a population variance of 5.36 and σ ≈ 2.315.
For the sample version, divide the same 26.8 by n−1 (4) to get a variance of 6.70 and s ≈ 2.588. So the typical value lies about 2.6 units from the mean of 11.2 — and notice the sample SD is a bit larger, exactly as the n−1 divisor predicts.
Outliers, skew, and the eight-value cap
Standard deviation assumes meaningful numeric data and at least two values; a single value has no spread. It is also sensitive to outliers and can be misleading for heavily skewed distributions, where the median and interquartile range often describe the data better.
This calculator handles up to eight values, which is plenty for learning and quick checks but not for large datasets — use a spreadsheet or statistics package for those. Always set the count to match exactly how many values you entered.
Sources & References
Figures on this page are checked against primary, authoritative sources. Links open in a new tab.