Dispersion Calculator

What "dispersion" means

Two datasets can share the same average and still be completely different. The sets {50, 50, 50} and {0, 50, 100} both have a mean of 50, but the first is perfectly consistent while the second swings from one extreme to the other. Measures of dispersion (also called spread or variability) quantify that difference: they tell you how far the individual values tend to sit from the center. A mean without a measure of dispersion is only half the story.

The measures this calculator computes

Given a data set x₁, x₂, …, xₙ with mean x̄, this tool reports every standard measure of spread:

• Range = max − min. The simplest measure — the distance between the largest and smallest value. Fast to read, but driven entirely by the two most extreme points.
• Variance. The average of the squared deviations from the mean. Population variance σ² = Σ(xᵢ − μ)² ÷ n. Sample variance s² = Σ(xᵢ − x̄)² ÷ (n − 1). Squaring keeps positive and negative deviations from cancelling and puts extra weight on far-out points.
• Standard deviation = √variance. Because variance is in squared units, the square root brings the measure back into the original units of the data, which is why the standard deviation is the most-quoted measure of spread. Population σ = √σ²; sample s = √s².
• Interquartile range (IQR) = Q3 − Q1. The range of the middle 50% of the data. It ignores the top and bottom quarters, so it is resistant to outliers.
• Mean absolute deviation (MAD) = (1/n) Σ|xᵢ − x̄|. The average distance from the mean using absolute values instead of squares. Less sensitive to extreme values than the standard deviation.
• Coefficient of variation (CV) = s ÷ x̄ × 100%. A unitless, relative measure that expresses the standard deviation as a percentage of the mean, letting you compare the spread of datasets with different units or scales.

Sample vs. population: why n − 1?

When your data is the entire group you care about, divide by n. When your data is a sample drawn from a larger population, divide by n − 1. This is Bessel's correction. Estimating deviations from the sample mean (rather than the true, unknown population mean) systematically underestimates the spread; dividing by n − 1 instead of n corrects that bias so the sample variance is an unbiased estimator of the population variance. The correction matters most for small samples — for a sample of 5 it inflates the variance by 25% — and becomes negligible as n grows large.

A worked example

Take the data set 4, 8, 15, 16, 23, 42 (n = 6). The sum is 108, so the mean is 18. The squared deviations from 18 are 196, 100, 9, 4, 25, and 576, which sum to 910. Population variance = 910 ÷ 6 ≈ 151.67, so σ ≈ 12.32. Sample variance = 910 ÷ 5 = 182, so s ≈ 13.49. The range is 42 − 4 = 38, and the coefficient of variation is 13.49 ÷ 18 × 100% ≈ 74.9%.

Which measure should you report?

For roughly symmetric data without extreme outliers, the standard deviation is the standard choice: it shares the data's units and feeds directly into z-scores, confidence intervals, and the empirical (68–95–99.7) rule. For skewed data or data with outliers, prefer the IQR, which describes the bulk of the distribution without being dragged around by a single extreme value. Use the coefficient of variation when comparing variability across datasets measured on different scales — for example, comparing the consistency of a process that outputs values near 5 with one that outputs values near 5,000.

Frequently Asked Questions

Common Tip Percentages

General Household Budget Ratios (50/30/20 Rule)

Common Measurement Conversions for Everyday Use