A standard deviation calculator measures how spread out a set of numbers is around its mean. A small standard deviation means the values cluster tightly near the average; a large one means they scatter widely. It is the most-used measure of variability in statistics, appearing everywhere from test grading and quality control to investing and scientific research.
Paste your numbers, choose sample or population mode, and the calculator returns the standard deviation, variance, mean, count, and the sum of squared deviations — every intermediate quantity you need to show your work. The sample/population choice matters: the two formulas divide by different amounts and give different answers.
The Formulas: σ vs s
Both formulas start the same way — find the mean, subtract it from each value, square the differences, and add them up. They differ only in the divisor:
- Population standard deviation: σ = √[ Σ(xᵢ − μ)² ÷ N ] — divide by N, the full count. Use when your data includes every member of the group (all 30 students in one class).
- Sample standard deviation: s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ] — divide by n − 1. Use when your data is a subset standing in for a larger population (30 students sampled from a whole school).
Dividing by n − 1, known as Bessel’s correction, compensates for the fact that a sample’s spread around its own mean slightly understates the true population spread. Variance is simply the standard deviation squared: σ² or s².
The Empirical Rule: 68–95–99.7
For data that follows a normal (bell-shaped) distribution, the standard deviation slices the data into predictable bands:
- About 68% of values fall within 1 standard deviation of the mean.
- About 95% fall within 2 standard deviations.
- About 99.7% fall within 3 standard deviations.
Example: adult IQ scores are designed with a mean of 100 and a standard deviation of 15. The rule says roughly 68% of people score between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145. This is why a value more than 3 standard deviations from the mean is commonly flagged as an outlier — under normality, fewer than 0.3% of observations land there.
Worked Example: Eight Data Points
Compute the sample standard deviation of 4, 8, 6, 5, 3, 7, 9, 6.
Step 1: Mean. Sum = 48, n = 8, so x̄ = 48 ÷ 8 = 6. Step 2: Squared deviations. (4−6)² = 4, (8−6)² = 4, (6−6)² = 0, (5−6)² = 1, (3−6)² = 9, (7−6)² = 1, (9−6)² = 9, (6−6)² = 0. Step 3: Sum them: 4 + 4 + 0 + 1 + 9 + 1 + 9 + 0 = 28. Step 4: Divide by n − 1 = 7 for the sample variance: 28 ÷ 7 = 4. Step 5: Take the square root: s = √4 = 2.
As a population, you would divide by 8 instead: variance = 3.5 and σ ≈ 1.8708 — always slightly smaller than the sample result.
Frequently Asked Questions
What is the difference between sample and population standard deviation?
Population standard deviation (σ) divides the sum of squared deviations by N and applies when you have data for the entire group. Sample standard deviation (s) divides by n − 1 and applies when your data is a subset used to estimate a larger population. The sample version is always slightly larger.
Why does the sample formula divide by n − 1 instead of n?
Because a sample’s values are measured against their own mean, their spread systematically understates the population’s true spread. Dividing by n − 1 — Bessel’s correction — inflates the result just enough to make the sample variance an unbiased estimate of the population variance.
What does a standard deviation value actually tell you?
It is the typical distance of a data point from the mean, in the same units as the data. Test scores with a mean of 75 and a standard deviation of 5 mostly sit between 70 and 80, while a standard deviation of 15 means scores commonly range from 60 to 90 — far less consistent.
What is the relationship between variance and standard deviation?
Variance is the average of the squared deviations from the mean, and standard deviation is its square root. Variance is expressed in squared units (dollars², points²), which is hard to interpret, so standard deviation is usually reported instead because it shares the data’s original units.
What is the 68-95-99.7 rule?
For normally distributed data, about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. It offers a fast way to judge how unusual a value is: anything beyond 2 standard deviations sits in roughly the most extreme 5% of the distribution.