2.2 Measures of the Location of the Data

i=(k100)(n+1)

$�=\left(\frac{�}{100}\right)\left(�+1\right)$ 

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: (x + 0.5yn)

$\left(\frac{�\text{ + }0.5�}{�}\right)$(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

μ=∑fm∑f

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

The arithmetic mean for a sample (denoted by x¯

$\overline{�}$) is x¯ = Sum of all values in the sampleNumber of values in the sample

$\overline{�}=\frac{\text{Sum of all values in the sample}}{\text{Number of values in the sample}}$ 

The arithmetic mean for a population (denoted by μ) is μ=Sum of all values in the populationNumber of values in the population

$�=\frac{\text{Sum of all values in the population}}{\text{Number of values in the population}}$ 

2.5 Geometric Mean

The Geometric Mean: x˜=(∏i=1nxi)1n=x1⋅x2⋅⋅⋅xn−−−−−−−−−√n=(x1⋅x2⋅⋅⋅xn)1n

$\widetilde{�}={\left(\prod _{�=1}^{�}{�}_{�}\right)}^{\frac{1}{�}}=\sqrt[�]{{�}_1·{�}_2\mathrm{···}{�}_{�}}={({�}_1·{�}_2\mathrm{···}{�}_{�})}^{\frac{1}{�}}$ 

2.6 Skewness and the Mean, Median, and Mode

Formula for skewness: a3=∑(xi−x¯)3ns3

${�}_3=\sum \frac{{({�}_{�}-\overline{�})}^3}{�{�}^3}$
Formula for Coefficient of Variation:CV=sx¯⋅100conditioned uponx¯≠0

$��=\frac{�}{\overline{�}}·100\text{conditioned upon}\overline{�}\ne 0$ 

2.7 Measures of the Spread of the Data

sx=∑fm2n−x––2−−−−−−−−−−√

${�}_{�}=\sqrt{\frac{{\displaystyle \sum �{�}^2}}{�}-{\overline{�}}^2}$ where sx= sample standard deviationx–– = sample mean

$\begin{array}{l}{�}_{�}=\text{ sample standard deviation}\\ \overline{�}\text{ = sample mean}\end{array}$ 

Formulas for Sample Standard Deviation s=Σ(x−x–)2n−1−−−−−−√

$�=\sqrt{\frac{�{(�-\overline{�})}^2}{�-1}}$ or s=Σf(x−x–)2n−1−−−−−−−√

$�=\sqrt{\frac{��{(�-\overline{�})}^2}{�-1}}$ or s=(∑i=1nx2)−nx¯2n−1−−−−−−−−−⎷

$�=\sqrt{\frac{\left({\displaystyle \sum _{�=1}^{�}}{�}^2\right)–�{\overline{�}}^2}{�–1}}$ For the sample standard deviation, the denominator is n – 1, that is the sample size – 1.

Formulas for Population Standard Deviation σ = Σ(x−μ)2N−−−−−−√

$� = \sqrt{\frac{�{(�-�)}^2}{�}}$ or σ = Σf(x–μ)2N−−−−−−√

$� = \sqrt{\frac{��{(�–�)}^2}{�}}$ or σ=∑i=1Nx2iN−μ2−−−−−−−−⎷

$�=\sqrt{\frac{{\displaystyle \sum _{�=1}^{�}}{�}_{�}^2}{�}–{�}^2}$ For the population standard deviation, the denominator is N, the number of items in the population.