Formula Review

$i=\left(\frac{k}{100}\right)\left(n+1\right)$

where i = the ranking or position of a data value,

k = the kth percentile,

n = total number of data.

Expression for finding the percentile of a data value: $\left(\frac{x\text{ + }0.5y}{n}\right)$(100)

where x = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile,

y = the number of data values equal to the data value for which you want to find the percentile,

n = total number of data

$\mu =\frac{{\displaystyle \sum fm}}{{\displaystyle \sum f}}$ Where f = interval frequencies and m = interval midpoints.

The arithmetic mean for a sample (denoted by $\overline{x}$) is $\overline{x}=\frac{\text{Sum of all values in the sample}}{\text{Number of values in the sample}}$

The arithmetic mean for a population (denoted by μ) is $\mu =\frac{\text{Sum of all values in the population}}{\text{Number of values in the population}}$

The Geometric Mean: $\tilde{x}={\left(\prod _{i=1}^n{x}_i\right)}^{\frac{1}{n}}=\sqrt[n]{x_1·x_2\mathrm{···}{x}_n}={(x_1·x_2\mathrm{···}{x}_n)}^{\frac{1}{n}}$

Formula for skewness: $a_3=\sum \frac{{(x_i-\overline{x})}^3}{n{s}^3}$
Formula for Coefficient of Variation:$CV=\frac{s}{\overline{x}}·100\text{conditioned upon}\overline{x}\ne 0$

$s_x=\sqrt{\frac{{\displaystyle \sum f{m}^2}}{n}-{\bar{x}}^2}$ where $\begin{array}{l}{s}_x=\text{ sample standard deviation}\\ \bar{x}\text{ = sample mean}\end{array}$

Formulas for Sample Standard Deviation $s=\sqrt{\frac{\Sigma {(x-\bar{x})}^2}{n-1}}$ or $s=\sqrt{\frac{\Sigma f{(x-\bar{x})}^2}{n-1}}$ or $s=\sqrt{\frac{\left({\displaystyle \sum _{i=1}^n}{x}^2\right)-n{\overline{x}}^2}{n-1}}$ For the sample standard deviation, the denominator is n - 1, that is the sample size - 1.

Formulas for Population Standard Deviation $\sigma = \sqrt{\frac{\Sigma {(x-\mu )}^2}{N}}$ or $\sigma = \sqrt{\frac{\Sigma f{(x–\mu )}^2}{N}}$ or $\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N}{x}_i^2}{N}-{\mu }^2}$ For the population standard deviation, the denominator is N, the number of items in the population.