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Statistical Variance

Tool to calculate the variance of a list of values. Variance is a statistical value that measures the dispersion characteristic of a distribution or sample.

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Statistical Variance -

Tag(s) : Statistics

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# Statistical Variance

## Unbiased Variance Calculator

Tool to calculate the variance of a list of values. Variance is a statistical value that measures the dispersion characteristic of a distribution or sample.

### What is the variance? (Definition)

Variance is a measure of the dispersion of a list of values around its mean value. This value, denoted $V$ or $\mathbb{V}$ or $\mathrm{Var}$ or $\sigma^2$ or $s^2$ characterizes the way in which the data $X$ (random variable) are dispersed by measuring the deviations between each value (of the variable) and the mean value (or expected value $\mathbb{E}$).

$$V(X) = \mathbb{E} \left[(X - \mathbb{E}[X])^{2}\right]$$

or

$$V(X) = \mathbb{E} \left[X^{2}\right]-\mathbb{E}[X]^{2}$$

### How to calculate the statistical variance from a list of numbers? (Formula)

A list of numbers $x_i$ of a discrete random variable $X$ whose mean alue is $m$ and with an unknown distribution, has for variance $V$ according to the formula $$V(X)= \frac{1}{n-1} \sum_{i=1}^{n}(x_{i}-m)^2$$

Example: The (unbiased) variance of the set of 3 numbers 1,2,9 with a mean value of 4 is $V = \frac{1}{3-1} \left( (1-4)^2 + (2-4)^2 + (9-4)^2 \right) = 38/2 = 19$

### What is the relation between variance and standard deviation?

The value of the variance is the square of the standard deviation. Knowing the value of the standard deviation $\sigma$, $V$ can be calculated with the relation: $$V(X) = \sigma^{2}(X)$$

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