Standard Deviation Calculator
Calculate population and sample standard deviation, variance, mean, median, mode, quartiles, IQR, and more — with step-by-step solutions and histogram visualization.
2
POP SD (σ)
2.14
SAMPLE SD (s)
5
MEAN (x̄)
Enter Your Data
Separate values with commas, spaces, or new lines. Example: 10, 20, 30, 40, 50
Quick Datasets
Population Std Dev
σ (sigma) — divide by N
2
Variance σ² = 4
Sample Std Dev
s — divide by (N−1)
2.1381
Variance s² = 4.5714
5
Mean (x̄)
4.5
Median
8
N (Count)
7
Range
More Descriptive Statistics
4
Mode
2
Min
9
Max
4
Q1
6
Q3
2
IQR
42.76%
Coefficient of Variation
40
Sum (Σx)
Data Distribution (Histogram)
How your 8 values are spread across 8 equal-width bins.
Understanding Standard Deviation
Standard deviation is the most widely used measure of how spread out a set of numbers is around its average. A small standard deviation tells you the values are bunched tightly near the mean; a large one tells you they're scattered widely. It's built from the variance — the average of each value's squared distance from the mean — which is then square-rooted to bring the units back in line with the original data.
Population vs Sample Standard Deviation
Population SD (σ): σ = √[ Σ(xᵢ − μ)² / N ] — use when you have ALL data points in the population.Sample SD (s): s = √[ Σ(xᵢ − x̄)² / (N−1) ] — use when your data is a SAMPLE used to estimate a larger population.Bessel's correction: dividing by (N−1) instead of N slightly inflates s to correct for a sample's tendency to understate true population spread.Variance = SD² (unsquared units) · Coefficient of Variation (CV) = (SD / Mean) × 100%Rule of thumb: if you can't (or don't need to) collect every member of the group, you're working with a sample — use sample standard deviation. Only use population standard deviation when your data set genuinely is the whole population, with nothing left out.
— FAQ