2.2 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 \(k\)th 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
2.3 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 \(\overline{x}\)) 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 \(\boldsymbol{\mu}=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}\)
2.5 Geometric Mean
The Geometric Mean: \(\overline{x}=\left(\prod_{i=1}^{n} x_{i}\right)^{\frac{1}{n}}=\sqrt[n]{x_{1} \cdot x_{2} \cdots x_{n}}=\left(x_{1} \cdot x_{2} \cdots x_{n}\right)^{\frac{1}{n}}\)
Formula for skewness: \(a_{3}=\sum \frac{\left(x_{i}-\overline{x}\right)^{3}}{n s^{2}}\)
Formula for Coefficient of Variation:\(C V=\frac{s}{\overline{x}} \cdot 100 \text { conditioned upon } \overline{x} \neq 0\)
2.7 Measures of the Spread of the Data
\(s_{x}=\sqrt{\frac{\sum f m^{2}}{n}-\overline{x}^{2}} \text { where } \) \(\begin{array}{l}{s_{x}=\text { sample standard deviation }} \\ {\overline{x}=\text { sample mean }}\end{array}\)
Formulas for Sample Standard Deviation \(s=\sqrt{\frac{\Sigma(x-\overline{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\Sigma f(x-\overline{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\left(\sum_{t=1}^{n} x^{2}\right)-n x^{2}}{n-1}}\) For the sample standard deviation, the denominator is n - 1, that is the sample size - 1.
Formulas for Population Standard Deviation \(\sigma=\sqrt{\frac{\Sigma(x-\mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\Sigma f(x \mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\sum_{i=1}^{N} x_{i}^{2}}{N}-\mu^{2} F}\) For the population standard deviation, the denominator is N, the number of items in the population.
