Basic Technical Mathematics, 11th Edition

CHAPTER 22 KEY FORMULAS AND EQUATIONS

Mean	$$\stackrel{\hspace{0.17em}¯\hspace{0.17em}}{x}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{x_1{f}_1\hspace{0.17em}+\hspace{0.17em}{x}_2{f}_2\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\cdots \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{x}_n{f}_n}{f_1\hspace{0.17em}+\hspace{0.17em}{f}_2\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\cdots \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{f}_n}}$$(22.1)
Standard deviation	$$s\hspace{0.17em}=\hspace{0.17em}\sqrt{{\displaystyle \frac{{\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}{(x\hspace{0.17em}-\hspace{0.17em}\stackrel{\hspace{0.17em}¯\hspace{0.17em}}{x})}^2}{n\hspace{0.17em}-\hspace{0.17em}1}}}$$(22.2) $$s\hspace{0.17em}=\hspace{0.17em}\sqrt{{\displaystyle \frac{n\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}{x}^2\right)\hspace{0.17em}-\hspace{0.17em}{\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}x\right)}^2}{n(n\hspace{0.17em}-\hspace{0.17em}1)}}}$$(22.3)
Normal distribution	$$y\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{e^{\hspace{0.17em}-\hspace{0.17em}{(x\hspace{0.17em}-\hspace{0.17em}\mu )}^2\hspace{0.17em}/\hspace{0.17em}2{\sigma }^2}}{\sigma \sqrt{2\pi }}}$$(22.4)
Standard (z) score	$$z\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{x\hspace{0.17em}-\hspace{0.17em}\mu }{\sigma }}$$(22.5)
Standard error of $\stackrel{\hspace{0.17em}¯\hspace{0.17em}}{\mathit{x}}$	$${\sigma }_{\stackrel{\hspace{0.17em}¯\hspace{0.17em}}{x}}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{\sigma }{\sqrt{n}}}$$(22.6)
Least-squares lines	$$y\hspace{0.17em}=\hspace{0.17em}mx\hspace{0.17em}+\hspace{0.17em}b$$(22.7) $$m\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{n{\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}xy\hspace{0.17em}-\hspace{0.17em}\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}x\right)\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}y\right)}{n{\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}{x}^2\hspace{0.17em}-\hspace{0.17em}{\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}x\right)}^2}}$$(22.8) $$b\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}{x}^2\right)\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}y\right)\hspace{0.17em}-\hspace{0.17em}\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}xy\right)\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}x\right)}{n{\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}{x}^2\hspace{0.17em}-\hspace{0.17em}{\left({\displaystyle \hspace{0.17em}\sum \hspace{0.17em}}x\right)}^2}}$$(22.9)