Live sigma desk for sample or population SD, with a deviation peel, empirical-rule companion, and z-score strip.
Educational spread literacy only. Not a CAS, exam oracle, or inferential statistics package. Confirm methods with your course when grades matter.
By Ahmet C. Toplutaş·Site owner & editor · Math tools
Last updated May 21, 2026 · Reviewed May 21, 2026 · Planning estimates only - not tax, medical, or investment advice
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Standard deviation measures how far values typically sit from the mean, in the same units as the data. The desk shows s (sample) or σ (population) based on a quiet mode chip.
This page's framing is a sigma desk, with a deviation peel, an empirical-rule companion, and a z-score strip. Population vs sample is a mode chip, not a twin-board hero. It is not mean-median-mode's center trio, not slope's two-point desk, not payment's term rail, and not APR↔APY twin framing.
Decision framing: “How spread out is this list, and where does one probe value sit in SD units?”
Educational spread literacy only. Not a computer-algebra system, exam oracle, or inferential statistics package. Confirm methods with your course when grades matter. See our disclaimers.
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Each value contributes a deviation from the mean and a squared deviation. Summing the squares and dividing by N (population) or n − 1 (sample) yields variance. The square root returns to original units as standard deviation.
Tip: load Outlier pull. Watch the peel highlight the extreme point and how SD jumps.
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For roughly bell-shaped data, about 68% / 95% / 99.7% of observations fall within 1 / 2 / 3 standard deviations of the mean. The companion prints those intervals with an explicit normality caution. Skewed or multimodal lists should not lean on the rule.
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A z-score places one probe value in SD units: z = (x − mean) / SD. Positive means above the mean; negative means below. Use it to ask “how unusual is this score?” without claiming a full hypothesis test.
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Mean, variance, SD, and z.
Example: 2, 4, 4, 4, 5, 5, 7, 9 as a population → mean 5, σ = 2.
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Sibling boards when the job is center, expression eval, or glossary lookup.
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Scenario A: Tight set. Quiz scores clustered near 80. Small s; empirical bands are narrow; a probe of 84 sits only a little above the mean.
Load Spread set: wider s, broader ±1σ band, and a high probe with a larger z.
Scenario B: Population chip. The classic 2, 4, 4, 4, 5, 5, 7, 9 list with ÷ N yields σ = 2 when treated as a full population.
Scenario C: Outlier pull. One extreme value inflates SD and dominates the peel. Compare with mean / median / mode to see how centers move.
Scenario D: Need a center report. Hand off to mean-median-mode when the question is which average to publish, not how wide the spread is.
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Stop here for descriptive spread around a mean. Hand off for center choice, expression evaluation, or formal inferential tests in dedicated statistics software.
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It takes a list of numbers, computes the mean, peels each deviation and squared deviation, then forms variance and standard deviation. Choose sample (÷ n − 1) or population (÷ N) as a mode chip. An empirical-rule companion shows ±1/2/3σ bands under a normality caution, and an optional z-score strip places one probe value in SD units.
Population SD (σ) divides by N. Sample SD (s) divides by n − 1 (Bessel’s correction) when the list is a sample used to estimate a larger population. The board keeps this as a mode chip, not a twin-formula hero.
For each value it shows x − mean and (x − mean)², then sums the squares into variance. Standard deviation is the square root, returning to the original data units.
For roughly bell-shaped data, about 68% / 95% / 99.7% of values fall within 1 / 2 / 3 standard deviations of the mean. The bands are orientation only; skewed or multimodal lists break the rule.
z = (x − mean) / SD places a probe value in standard-deviation units. Positive z sits above the mean; negative z sits below. It needs a non-zero SD.
Mean-median-mode owns center. This page owns spread around the mean. Hand off to mean-median-mode when the question is which average to report.
No. It is educational spread literacy for homework and quick checks. Confirm methods, rounding, and inferential tests with your course or software when grades or research matter.
LIVE SIGMA BOARD
Sigma desk →Scene presets
Comma, space, or semicolon separated
Mode chip
SIGMA DESK
s (sample)
1.4142
n = 6 · mean 80 · Sample (s, ÷ n − 1)
DEVIATION PEEL
Largest |deviations|
EMPIRICAL-RULE COMPANION
Orientation for roughly bell-shaped data only. Skew and multimodality break the rule.
Z-SCORE STRIP
Updates as you type. Share writes board state; typing does not rewrite the address bar.
s
1.4142
mean 80 · var 2 · z 2.8284
Standard deviation measures typical distance from the mean in the same units as the data. Larger SD means more spread; zero means every value is identical.
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