mathematics//statistics//variance

2026-09-06

How to quantify dispersion without confusing it with useful information. The **variance** is the mean squared deviation from the mean. For a random variable \(X\) with finite second moment,

How to quantify dispersion without confusing it with useful information. The variance is the mean squared deviation from the mean. For a random variable XXX with finite second moment,

μ=E[X],Var⁡(X)=E[(X−μ)2]=σ2.\mu=\mathbb E[X],\qquad \operatorname{Var}(X)=\mathbb E[(X-\mu)^2]=\sigma^2.μ=E[X],Var(X)=E[(X−μ)2]=σ2.

The standard deviation σ=Var⁡(X)\sigma=\sqrt{\operatorname{Var}(X)}σ=Var(X) returns the units of XXX: for a temperature, variance is in degrees squared and standard deviation in degrees.

Dispersion, concentration and shape. A small variance means little quadratic dispersion around the mean, a large one more. The standard deviation gives a width scale but not the full shape: two distributions can share mean and variance and be very different, symmetric or skewed, unimodal or clustered, with different tails. Extremes weigh a lot because of the square.

Sample version. For data x1,…,xNx_1,\ldots,x_Nx1,…,xN, xˉ=1N∑ixi\bar x=\frac1N\sum_i x_ixˉ=N1∑ixi and s2=1N−1∑i(xi−xˉ)2s^2=\frac1{N-1}\sum_i(x_i-\bar x)^2s2=N−11∑i(xi−xˉ)2. The divisor N−1N-1N−1 gives the unbiased estimate under independent, identically distributed sampling with finite variance; dividing by NNN describes the dispersion of the empirical distribution. Different purposes: keep one convention when comparing formulas.

Variance is not information. PCA maximizes retained variance. Calling it "information" abbreviates the geometric criterion used; it is not a general equality with Shannon entropy, predictive power or relevance for a decision. A noisy sensor has high variance; a small difference between two sensors can reveal an important fault. redundant sensors computes a case where the lowest-variance mode keeps exactly the discrepancy between sensors. There is no universal choice between "much" and "little" variance: ask whether the goal is process variability, measurement uncertainty, compression or diagnosis.

covariance studies joint variation; the covariance matrix generalizes variance to any direction; projection defines what can be lost when compressing.