Measures of Dispersion
It's not just about the center. Understand how spread out your data is using Range, Variance, and Standard Deviation.
The Range
Simplicity & Limitation
The simplest measure of dispersion. It is the difference between the largest and smallest values.
Not Resistant
The Range is NOT resistant to outliers. A single extreme value can drastically inflate the range, giving a misleading impression of the data's spread.
Standard Deviation
The "Average" Distance
Measures the average distance of each data point from the mean. It is the most common measure of spread.
InteractiveStandard Deviation Calculator
Variance
Variance is simply the square of the standard deviation. While useful for mathematical modeling, Standard Deviation is preferred for interpretation because its units match the original data.
The Empirical Rule
InteractiveThe Empirical Visualizer
Chebyshev’s Inequality
For Any Distribution
Unlike the Empirical Rule, Chebyshev's Inequality applies to ANY shape of distribution (skewed or symmetric).
InteractiveChebyshev's Inequality Calculator
The Formula: For any distribution, at least of observations lie within standard deviations of the mean.
Common values:
At least this percentage of data lies within standard deviations of the mean:
Calculation:
💡 Key Insight: Unlike the Empirical Rule (which only applies to bell-shaped distributions), Chebyshev's Inequality works for any distribution. However, it gives a minimum guarantee, so the actual percentage is often higher.