# Stats - Variance and Standard Deviation

> Source: https://ollybritton.com/notes/a-level/maths/topics/stats/variance-and-standard-deviation/ · Updated: 2020-09-11 · Tags: stats, year-1, school

##### 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??
$$
\sigma^2 = \frac{\sum (x - \bar{x}) ^ 2}{n}
$$

Or

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

##### 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 $S_{xx}$??
$$
\frac{S_{xx}}{n}
$$

### 2022-05-17
##### 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.

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