Measure spread around the mean—population σ or sample s from variance, with empirical-rule context and squared-units interpretation.
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Standard deviation (σ or s) quantifies spread around the mean—how far typical values stray. Test scores 88, 90, 92 have low σ; scores 60, 90, 120 have high σ. σ = √(variance).
Choose sample (n−1) for data sampled from a larger population; population (n) when you have every data point. Using the wrong formula is a common error that produces slightly incorrect variance estimates.
Results
Count: 4
Mean: 5.0000
Variance: 5.0000
Standard deviation: 2.2361
Formula: sigma = sqrt(sum((x - mean)^2) / n)
Mean tells you the center; standard deviation tells you whether data cling tight or sprawl wide.
**Steps:** (1) Find mean μ. (2) Subtract μ from each point, square the gap. (3) Average those squared gaps—that is **variance** σ². (4) Take the square root to return to original units—**standard deviation** σ.
**Population vs sample:** divide by N for population σ; divide by (n−1) for sample s when data are a subset estimating a larger group. Exam problems specify which.
**Empirical rule (normal data):** ~68% within 1σ of mean, ~95% within 2σ, ~99.7% within 3σ. Real skewed data violate this—still useful when distributions are bell-shaped.
Variance is in squared units (dollars²); σ fixes units back to dollars. Square root calculator completes the last step if you already have variance.
Enter data, choose population or sample mode, read σ (or s) and variance—pair with mean/median/mode for full summary.
Concise answers for common searches — definitions, steps, and comparisons.
σ = √(Σ(x−μ)² / N). Square root of mean squared deviation from population mean μ.
Bessel's correction makes sample variance an unbiased estimator of population variance.
Every value equals the mean—no spread at all.
σ = √(variance). Variance is σ². If variance is 16, σ = 4.
Computes mean, variance, and standard deviation from a numeric list with toggle for population (N) vs sample (n−1) denominator.
Formula
Population: σ² = Σ(x_i − μ)² / N; σ = √(σ²). Sample: s² = Σ(x_i − x̄)² / (n−1); s = √(s²).Comma-separated measurements from spreadsheet or lab notes.
Full factory batch → population N. Random sample of students → sample n−1.
Variance is squared units; σ restores original units.
σ is always interpreted relative to the mean—compute both from the same list.
Input
10, 10, 10, 10Output
σ = 0No deviation from mean—perfect constant data.
Input
2, 4, 4, 4, 5, 5, 7, 9Output
μ = 5, σ ≈ 2Classic textbook set—symmetric spread around 5.
Input
1, 2, 3, 4, 5 (sample)Output
s ≈ 1.581 vs population σ ≈ 1.414n−1 denominator inflates sample variance slightly.
Input
10, 11, 12, 13, 50Output
σ large vs cluster aloneSingle 50 pulls mean and σ upward—inspect outliers.
Input
Variance = 25Output
σ = 5Square root step returns to original units.
Input
Normal-like sample, μ=100, σ=15Output
~68% between 85 and 1151σ band illustration for bell-shaped data.
Bolt diameters clustered tightly → low σ passes tolerance; high σ triggers process review.
Daily return standard deviation estimates portfolio risk—higher σ, wilder swings.
Two classes same mean 75%—lower σ class is more uniform; higher σ has wider skill range.
Five titration trials: small σ in mL means consistent technique.
Step-by-step chains that connect related tools for common tasks.
| Symbol | Denominator | When to use |
|---|---|---|
| σ (population) | N | Entire group known |
| s (sample) | n − 1 | Subset estimating population |
| Variance σ² | N or n−1 | Squared units before √ |
| Data pattern | σ tendency | Example |
|---|---|---|
| Identical values | 0 | 5,5,5,5 |
| Tight cluster | Low | 98,99,100,101,102 |
| Wide scatter | High | 10,50,90 |
| With outlier | Inflated | 1,2,3,100 |
76 ± 4 is interpretable; σ alone lacks center.
Tiny samples produce unstable σ—more data tightens estimate.
One glance shows whether σ summarizes spread fairly.
Confirm center measure before interpreting distance from mean.
Sample standard deviation divides by n−1 (Bessel correction) for unbiased variance estimate.
Variance 49 (units²) → σ = 7 in original units.
68–95–99.7 rule assumes approximate normality—income data may fail badly.
Dollars vs meters—standardize or compare coefficient of variation (σ/μ).
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No—square root of variance is always non-negative.
CV = σ/μ—unitless relative spread for comparing unlike scales.
For normal data, about 68% of points fall within μ ± σ.
σ pairs with mean. For median, IQR is the usual spread companion.
σ is in percentage points same as data—interpret relative to mean percent.
Squaring deviations magnifies outliers—σ grows faster than robust measures like IQR.
Mean median mode calculator for center; this tool for spread from the same list.
Your dataset is analyzed locally—numbers are not uploaded.
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Reviewed by EverydayTools Editorial Team on 2026-07-03.
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