Stats - Variance and Standard Deviation

Variance: meaning and formulae

What is variance?

A measure of spread that takes all values into account.

What is the mathematical definition of variance, in words?

The average squared distance from the mean.

What is the formula for variance?

\[\sigma^2 = \frac{\sum (x - \bar{x}) ^ 2}{n}\]

\[\sigma^2 = \frac{\sum x^2}{n} - \bar{x}^2\]

What is the definition of the standard deviation?

The average distance from the mean.

What is the formula for standard deviation?

\[\sigma = \sqrt{\sigma^2}\]

What is the formula for the variance for grouped data?

\[\sigma^2 = \frac{\sum f x^2}{\sum f} - \bar{x}^2\]

Is it $\sum f x^2$ or $(\sum f x)^2$?

$\sum f x^2$

What is $S _ {xx}$?

\[\sum(x - \overline{x})^2\]

What is the formula for variance using $S _ {xx}$?

\[\frac{S _ {xx}}{n}\]

When arguing that including a value in a data set will decrease the standard deviation, what do you need to show?

The value will be less than one standard deviation away from the mean.