### 2.2 Measures of the Location of the Data

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

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

### 2.3 Measures of the Center of the 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 $\overline{x}\text{}=\text{}\frac{\text{Sumofallvaluesinthesample}}{\text{Numberofvaluesinthesample}}$

The arithmetic mean for a population (denoted by *μ*) is $\mu =\frac{\text{Sumofallvaluesinthepopulation}}{\text{Numberofvaluesinthepopulation}}$

### 2.5 Geometric Mean

The Geometric Mean: $\stackrel{~}{x}={\left(\prod _{i=1}^{n}{x}_{i}\right)}^{\frac{1}{n}}=\sqrt[n]{{x}_{1}\xb7{x}_{2}\mathrm{\xb7\xb7\xb7}{x}_{n}}={({x}_{1}\xb7{x}_{2}\mathrm{\xb7\xb7\xb7}{x}_{n})}^{\frac{1}{n}}$

### 2.6 Skewness and the Mean, Median, and Mode

Formula for skewness: ${a}_{3}=\sum \frac{{({x}_{i}-\overline{x})}^{3}}{n{s}^{3}}$

Formula for Coefficient of Variation:$CV=\frac{s}{\overline{x}}\xb7100\phantom{\rule{0.2em}{0ex}}\text{conditioned upon}\phantom{\rule{0.2em}{0ex}}\overline{x}\ne 0$

### 2.7 Measures of the Spread of the Data

${s}_{x}=\sqrt{\frac{{\displaystyle \sum f{m}^{2}}}{n}-{\stackrel{\u2013}{x}}^{2}}$ where $\begin{array}{l}{s}_{x}=\text{samplestandarddeviation}\\ \stackrel{\u2013}{x}\text{=samplemean}\end{array}$

Formulas for Sample Standard Deviation
$s=\sqrt{\frac{\Sigma {(x-\stackrel{\u2013}{x})}^{2}}{n-1}}$ or $s=\sqrt{\frac{\Sigma f{(x-\stackrel{\u2013}{x})}^{2}}{n-1}}$
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 $\sigma =\sqrt{\frac{\Sigma f{(x\u2013\mu )}^{2}}{N}}$ 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.