Standard Deviation Calculator

Sample vs population: when to use which

The n−1 divisor (Bessel’s correction) produces an unbiased estimator of the population variance from sample data. With n as divisor you systematically underestimate the true population variance. For large n the difference shrinks but it matters at small sample sizes.

Standard deviation intuition

If a set has mean 100 and SD 15, then (assuming roughly normal distribution):

• 68% of values fall within 85-115 (1 SD)
• 95% within 70-130 (2 SD)
• 99.7% within 55-145 (3 SD)

That is the 68-95-99.7 rule, also called the empirical rule. IQ scores, human heights, and many natural measurements follow it closely.

Coefficient of variation

CV = SD / mean. A unitless measure of dispersion, useful when comparing variability across datasets with different means. A CV of 0.1 (10%) means the SD is 10% of the mean, roughly. Not meaningful for data that can cross zero.

Z-scores

For each value x: z = (x − mean) / SD. Tells you how many SDs above or below the mean that value sits. |z| > 2 is often flagged as a possible outlier; |z| > 3 is quite rare in normal data.

Common mistakes

• Using population when you should use sample. Underestimates variability in a sample dataset.
• Mixing mean and SD from different units. Always check the scale.
• Applying normal-distribution rules to non-normal data. Skewed or multimodal data breaks the 68-95-99.7 heuristic. Plot a histogram first.
• Ignoring outliers. One extreme value can triple your SD. Robust alternatives (median absolute deviation, interquartile range) exist for heavy-tailed data.