Standard Deviation & Variance Calculator
Calculate sample and population standard deviation (s, σ), variance (s², σ²), and sum of squares with deviation steps.
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Statistical Dispersion: Sample vs. Population Standard Deviation & Variance
The Empirical Rule (68-95-99.7% Rule of Standard Deviations)
Empirical Rule: In normal bell-curve distributions, exactly 68.2% of all observations fall within 1 standard deviation ($pm 1sigma$), and 95.4% fall within 2 standard deviations ($pm 2sigma$).
Standard deviation is a fundamental metric in probability theory and statistics that quantifies the degree of dispersion or variation of a numerical dataset relative to its arithmetic mean. A low standard deviation indicates values cluster tightly near the average, while a high standard deviation reflects widely dispersed data.
1. Sample vs. Population Formulas (Bessel's Correction)
| Metric Type | Formula | Divisor & Rationale |
|---|---|---|
| Sample Standard Deviation ($s$) | $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ | Uses $n - 1$ degrees of freedom (Bessel's correction) to prevent underestimating population spread. |
| Population Standard Deviation ($\sigma$) | $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$ | Divides by $N$ when data encompasses 100% of all members in the entire population. |
🔗 Dispersion & Statistical Modeling
Standard deviation quantifies the spread of data points around the arithmetic mean. Visit our ➗ Math & Statistics Hub for complete data analysis solutions.
Examine central tendency measures or scientific notation formats with the related tools below.
Frequently Asked Questions (Standard Deviation)
Variance is the average of the squared deviations from the mean (expressed in squared units like dollars²). Standard deviation is the square root of variance, restoring values back to original units (like dollars).
No. Because deviations are squared before summing, standard deviation is always $ge 0$. A standard deviation of exactly 0 occurs only when all data points are identical.