Range, Variance and Standard Deviation

Range, variance and standard deviation

The range is the easiest spread to compute: R = max − min. Variance takes the squared distances from the mean and averages them. For a whole population you divide by N: σ² = Σ(xᵢ − μ)² / N. For a sample you divide by n − 1 instead, s² = Σ(xᵢ − x̄)² / (n − 1); that's Bessel's correction, and it keeps s² from coming out too small. The standard deviation is just √variance, which puts you back in the original units. Take [4, 8, 15, 16, 23, 42]: the mean lands near 18, the range is 38, the sample s is about 13.3 and σ about 12.1. Quartiles cut the sorted data into pieces — Q1 is the median of the bottom half, Q3 the median of the top half, and IQR = Q3 − Q1 holds up well when outliers are around.

Applications

Six Sigma quality control leans on it, where Cp and Cpk track how much a process wobbles. The same idea shows up in financial risk metrics like volatility σ and VaR, in lab work as measurement error, in psychometrics as test reliability, and in A/B testing, where the effect size you can detect hinges on σ.

FAQ

Why squared deviations? If you add up the raw deviations they cancel out to zero, since the positives and negatives offset each other. Squaring drops the signs and, as a bonus, leans harder on outliers, which is what you want in plenty of models.

n or n − 1? When you have the full population, divide by N. When you only have a sample and want to say something about the population behind it, divide by n − 1 (Bessel). Stick with n and you'll end up underestimating the real variance.

Range or standard deviation? The range is fast, but it only looks at two points, so a single stray value can throw it off badly. Standard deviation reads every data point and is the go-to measure when you're dealing with normal distributions.