Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

2.11: Formula Review

Last updated

Mar 17, 2025
Save as PDF
- 2.10: Review
- 2.12: Practice

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

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.

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

Support Center

How can we help?