2.11: Formula Review

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2.3: Measures of the Location of the Data

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

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

$k=\text { the kth percentile, }$

$n=$ total number of data.

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

$n=\text { total number of data }$

2.4: Measures of the Center of the Data

$\mu=\dfrac{\sum f m}{\sum f}$ Where $f=$ interval frequencies and $m=$ interval midpoints.

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

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

2.6: Geometric Mean

The Geometric Mean: $\tilde{x}=\left(\prod_{i=1}^n x_i\right)^{\dfrac{1}{n}}=\sqrt[n]{x_1 \cdot x_2 \cdots x_n}=\left(x_1 \cdot x_2 \cdots x_n\right)^{\dfrac{1}{n}}$

2.7: Skewness and the Mean, Median, and Mode

Formula for skewness: $a_3=\sum \dfrac{\left(x_i-\bar{x}\right)^3}{n s^3}$

Formula for Coefficient of Variation: $C V=\dfrac{s}{\bar{x}} \cdot 100$ conditioned upon $\bar{x} \neq 0$ $\begin{array}{l} \end{array}$
$\begin{array}{l} s_{x} = sample standard deviation \\ \bar{x} = sample mean \end{array}$

$s_x=\sqrt{\dfrac{\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 $s=\sqrt{\dfrac{\Sigma(x-\bar{x})^2}{n-1}}$ or $s=\sqrt{\dfrac{\Sigma f(x-\bar{x})^2}{n-1}}$ or $s=\sqrt{\dfrac{\left(\sum_{i=1}^n x^2\right)-n \bar{x}^2}{n-1}}$ For the sample standard deviation, the denominator is $\boldsymbol{n}-\mathbf{1}$, that is the sample size -1 .$

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

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2.3: Measures of the Location of the Data

2.4: Measures of the Center of the Data

2.6: Geometric Mean

2.7: Skewness and the Mean, Median, and Mode

2.3: Measures of the Location of the Data

2.4: Measures of the Center of the Data

2.6: Geometric Mean

2.7: Skewness and the Mean, Median, and Mode

2.8: Measures of the Spread of the Data