2.12: Formula Review
- Page ID
- 45661
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.