2.11: Formula Review

2.3 Measures of the Location of the Data

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

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: \(\left(\frac{x+0.5 y}{n}\right)(100)\) 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,
\(n=\) total number of data

2.4 Measures of the Center of the Data

\(\mu=\frac{\sum f m}{\sum f}\) Where \(f=\) interval frequencies and \(m=\) interval midpoints.

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

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

2.6 Geometric Mean

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

2.7 Skewness and the Mean, Median, and Mode

Formula for skewness: \(a_3=\sum \frac{\left(x_i-\bar{x}\right)^3}{n s^3}\)
Formula for Coefficient of Variation: \(C V=\frac{s}{\bar{x}} \cdot 100\) conditioned upon \(\bar{x} \neq 0\)

2.8 Measures of the Spread of the Data

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

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

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