2.10: Chapter 2 Formula Review
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2.2 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 \({\mu}=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}\)
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 \(s=\sqrt{\frac{\Sigma({x}_i-\overline{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\Sigma {f}_i({x}_i - \overline{x})^{2}}{n-1}} \)
For the sample standard deviation, the denominator is n - 1, that is the sample size - 1.
Formulas for Population Standard Deviation \({\sigma}=\sqrt{\frac{\Sigma({x}_i - \mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\Sigma {f}_i({x}_i - \mu)^{2}}{N}} \)
For the population standard deviation, the denominator is N, the number of items in the population.