variance

Topics

• descriptive statistics
• probability

Variance measures the spread of data points around their mean, quantifying how much values in a distribution deviate from the expected value.

For a random variable $X$, variance is defined as:

$$Var(X)=E[(X-E[X]{)}^2]$$

This represents the expected squared deviation from the mean. For discrete random variables with $n$ equally likely outcomes:

$$Var(X)=\frac{1}{n}\sum _{i=1}^n(x_i-\mu {)}^2$$

Key properties:

• Always non-negative ($Var(X)\ge 0$)
• $Var(aX+b)=a^{2}Var(X)$ for constants $a,b$
• For independent variables, $Var(X+Y)=Var(X)+Var(Y)$

Sample Variance

In statistics, sample variance estimates population variance from a finite sample. The unbiased estimator uses bessel’s correction:

$$s^2=\frac{1}{n-1}\sum _{i=1}^n(x_i-\bar{x}{)}^2$$

Where $\bar{x}$ is the sample mean. The $n-1$ denominator corrects bias in the estimation.

In Python’s NumPy:

• np.var() computes population variance (by default)
• Set ddof=1 for sample variance (default ddof=0)

from numpy import array, var
 
M = array([
[1,2,3,4,5,6],
[1,2,3,4,5,6]])
 
col_var = var(M, ddof=1, axis=0)
row_var = var(M, ddof=1, axis=1)

Note

The choice between population and sample variance depends on whether you’re describing the entire population or estimating from a sample.

Related

• variance of a vector