The calculation, without hidden assumptions
Standard deviation is easy to quote and easy to misuse if the denominator is hidden. MAXScanner makes the sample-versus-population choice explicit, shows the mean and the full squared-deviation sum before dividing, reports the matching variance, and then takes the square root. A per-observation table exposes each deviation, squared deviation and z-score, while standard error, coefficient of variation and Tukey outlier flags add the context strong current statistics tools provide without pretending those diagnostics decide what data should be kept.
How to use this calculator
Paste the numeric observations from your worksheet, CSV column or notes. Repeated-value shorthand such as 10:3 is supported when it makes the input easier to audit.
Choose Sample if the data are used to estimate a larger population; choose Population if these values are the complete population being described.
Check the mean, variance and Σ(x−mean)², then expand the deviation table to see which observations contribute most to the spread and how their z-scores compare.
Where people use it
- •Homework and statistics checks with the working visible
- •Quality-control or measurement-spread summaries
- •Comparing sample and population formulas on the same dataset
- •Inspecting deviations, squared deviations and z-scores observation by observation
Example: 2, 4, 4, 4, 5, 5, 7, 9
The mean is 5 and the sum of squared deviations is 32. Population variance is 32/8 = 4 and population SD is 2. Sample variance is 32/7 ≈ 4.5714 and sample SD is about 2.1381.
What the result does not assume
- •Sample versus population is a modeling choice, not a formatting option. The calculator cannot decide whether your observations represent the whole population relevant to your question.
- •A z-score is expressed relative to the selected standard deviation; when the standard deviation is zero, all observations are identical and the calculator reports zero z-scores.
- •Coefficient of variation divides the selected SD by the absolute mean and is unavailable when the mean is zero. It is most interpretable for ratio-scale quantities with a meaningful zero.
Frequently asked questions
Why does sample standard deviation divide by n−1?+
When a sample mean is estimated from the same observations, one degree of freedom is used. Dividing by n−1 is Bessel’s correction and is the standard sample-variance formula.
Is variance just standard deviation squared?+
Yes. Standard deviation is the square root of variance, so variance is expressed in squared units while standard deviation returns to the original measurement unit.
What if all values are the same?+
Every deviation from the mean is zero, so variance and standard deviation are both zero.
Can I use frequency shorthand?+
Yes. For example, 10:3 expands to three observations equal to 10 before the mean and deviations are calculated.
Semantic next steps
Continue the calculation
These links move to a different input, formula or project stage rather than a keyword variation of this page.
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OpenFree publisher widget
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Give readers the live calculator without rebuilding the math. Copy the responsive embed in one click, or customize width, height, border and corner radius before publishing. The branded MAXScanner attribution stays visible.
Calculator embed HTML
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Keep the result connected to the real job
MAXScanner keeps the formula and assumptions beside the result so you can verify the number before using it in a drawing, estimate, specification, worksheet or document. Where a supplier, manufacturer, drawing or applicable standard owns a requirement, that source remains authoritative.