# 11.1: B | Mathematical Phrases, Symbols, and Formulas

## English Phrases Written Mathematically

When the English says: Interpret this as:
$$X$$ is at least 4. $$X \geq 4$$
The minimum of $$X$$ is 4. $$X \geq 4$$
$$X$$ is no less than 4. $$X \geq 4$$
$$X$$ is greater than or equal to 4. $$X \geq 4$$
$$X$$ is at most 4. $$X \leq 4$$
The maximum of $$X$$ is 4. $$X \leq 4$$
$$X$$ is no more than 4. $$X \leq 4$$
$$X$$ is less than or equal to 4. $$X \leq 4$$
$$X$$ does not exceed 4. $$X \leq 4$$
$$X$$ is greater than 4. $$X > 4$$
$$X$$ is more than 4. $$X > 4$$
$$X$$ exceeds 4. $$X > 4$$
$$X$$ is less than 4. $$X < 4$$
There are fewer $$X$$ than 4. $$X < 4$$
$$X$$ is 4. $$X = 4$$
$$X$$ is equal to 4. $$X = 4$$
$$X$$ is the same as 4. $$X = 4$$
$$X$$ is not 4. $$X \neq 4$$
$$X$$ is not equal to 4. $$X \neq 4$$
$$X$$ is not the same as 4. $$X \neq 4$$
$$X$$ is different than 4. $$X \neq 4$$
Table B1

## Symbols and Their Meanings

Chapter (1st used) Symbol Spoken Meaning
Sampling and Data $$\sqrt{ }$$ The square root of same
Sampling and Data $$\pi$$ Pi 3.14159… (a specific number)
Descriptive Statistics $$Q_1$$ Quartile one the first quartile
Descriptive Statistics $$Q_2$$ Quartile two the second quartile
Descriptive Statistics $$Q_3$$ Quartile three the third quartile
Descriptive Statistics $$IQR$$ interquartile range $$Q_3 – Q_1 = IQR$$
Descriptive Statistics $$\overline X$$ $$x$$-bar sample mean
Descriptive Statistics $$\mu$$ mu population mean
Descriptive Statistics $$s$$ s sample standard deviation
Descriptive Statistics $$s^2$$ $$s$$ squared sample variance
Descriptive Statistics $$\sigma$$ sigma population standard deviation
Descriptive Statistics $$\sigma^2$$ sigma squared population variance
Descriptive Statistics $$\Sigma$$ capital sigma sum
Probability Topics $$\{ \}$$ brackets set notation
Probability Topics $$S$$ S sample space
Probability Topics $$A$$ Event A event A
Probability Topics $$P(A)$$ probability of A probability of A occurring
Probability Topics $$P(A|B)$$ probability of A given B prob. of A occurring given B has occurred
Probability Topics $$P(A\cup B)$$ prob. of A or B prob. of A or B or both occurring
Probability Topics $$P(A\cap B)$$ prob. of A and B prob. of both A and B occurring (same time)
Probability Topics $$A^{\prime}$$ A-prime, complement of A complement of A, not A
Probability Topics $$P(A^{\prime})$$ prob. of complement of A same
Probability Topics $$G_1$$ green on first pick same
Probability Topics $$P(G_1)$$ prob. of green on first pick same
Discrete Random Variables $$PDF$$ prob. density function same
Discrete Random Variables $$X$$ X the random variable X
Discrete Random Variables $$X \sim$$ the distribution of X same
Discrete Random Variables $$\geq$$ greater than or equal to same
Discrete Random Variables $$\leq$$ less than or equal to same
Discrete Random Variables $$=$$ equal to same
Discrete Random Variables $$\neq$$ not equal to same
Continuous Random Variables $$f(x)$$ f of x function of x
Continuous Random Variables $$pdf$$ prob. density function same
Continuous Random Variables $$U$$ uniform distribution same
Continuous Random Variables $$Exp$$ exponential distribution same
Continuous Random Variables $$f(x) =$$ f of $$X$$ equals same
Continuous Random Variables $$m$$ m decay rate (for exp. dist.)
The Normal Distribution $$N$$ normal distribution same
The Normal Distribution $$z$$ z-score same
The Normal Distribution $$Z$$ standard normal dist. same
The Central Limit Theorem $$\overline X$$ X-bar the random variable X-bar
The Central Limit Theorem $$\mu_{\overline{x}}$$ mean of X-bars the average of X-bars
The Central Limit Theorem $$\sigma_{\overline{x}}$$ standard deviation of X-bars same
Confidence Intervals $$CL$$ confidence level same
Confidence Intervals $$CI$$ confidence interval same
Confidence Intervals $$EBM$$ error bound for a mean same
Confidence Intervals $$EBP$$ error bound for a proportion same
Confidence Intervals $$t$$ Student's t-distribution same
Confidence Intervals $$df$$ degrees of freedom same
Confidence Intervals $$t_{\frac{\alpha}{2}}$$ student t with α/2 area in right tail same
Confidence Intervals $$p^{\prime}$$ p-prime sample proportion of success
Confidence Intervals $$q^{\prime}$$ q-prime sample proportion of failure
Hypothesis Testing $$H_0$$ H-naught, H-sub 0 null hypothesis
Hypothesis Testing $$H_a$$ H-a, H-sub a alternate hypothesis
Hypothesis Testing $$H_1$$ H-1, H-sub 1 alternate hypothesis
Hypothesis Testing $$\alpha$$ alpha probability of Type I error
Hypothesis Testing $$\beta$$ beta probability of Type II error
Hypothesis Testing $$\overline{X 1}-\overline{X 2}$$ X1-bar minus X2-bar difference in sample means
Hypothesis Testing $$\mu_{1}-\mu_{2}$$ mu-1 minus mu-2 difference in population means
Hypothesis Testing $$P_{1}^{\prime}-P_{2}^{\prime}$$ P1-prime minus P2-prime difference in sample proportions
Hypothesis Testing $$p_{1}-p_{2}$$ p1 minus p2 difference in population proportions
Chi-Square Distribution $$X^2$$ Ky-square Chi-square
Chi-Square Distribution $$O$$ Observed Observed frequency
Chi-Square Distribution $$E$$ Expected Expected frequency
Linear Regression and Correlation $$y = a + bx$$ y equals a plus b-x equation of a straight line
Linear Regression and Correlation $$\hat y$$ y-hat estimated value of y
Linear Regression and Correlation $$r$$ sample correlation coefficient same
Linear Regression and Correlation $$\varepsilon$$ error term for a regression line same
Linear Regression and Correlation $$SSE$$ Sum of Squared Errors same
F-Distribution and ANOVA $$F$$ F-ratio F-ratio
Table B2 Symbols and their Meanings

## Formulas

 Symbols you must know Population Sample $$N$$ Size $$n$$ $$\mu$$ Mean $$\overline x$$ $$\sigma^2$$ Variance $$s^2$$ $$\sigma$$ Standard deviation $$s$$ $$p$$ Proportion $$p^{\prime}$$ Single data set formulae Population Sample $$\mu=E(x)=\frac{1}{N} \sum_{i=1}^{N}\left(x_{i}\right)$$ Arithmetic mean $$\overline{x}=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}\right)$$ Geometric mean $$\tilde{x}=\left(\prod_{i=1}^{n} X_{i}\right)^{\frac{1}{n}}$$ $$Q_{3}=\frac{3(n+1)}{4}, Q_{1}=\frac{(n+1)}{4}$$ Inter-quartile range $$I Q R=Q_{3}-Q_{1}$$ $$Q_{3}=\frac{3(n+1)}{4}, Q_{1}=\frac{(n+1)}{4}$$ $$\sigma^{2}=\frac{1}{N} \sum_{i=1}^{N}\left(x_{i}-\mu\right)^{2}$$ Variance $$s^{2}=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}$$ Single data set formulae Population Sample $$\mu=E(x)=\frac{1}{N} \sum_{i=1}^{N}\left(m_{i} \cdot f_{i}\right)$$ Arithmetic mean $$\overline{x}=\frac{1}{n} \sum_{i=1}^{n}\left(m_{i} \cdot f_{i}\right)$$ Geometric mean $$\tilde{x}=\left(\prod_{i=1}^{n} X_{i}\right)^{\frac{1}{n}}$$ $$\sigma^{2}=\frac{1}{N} \sum_{i=1}^{N}\left(m_{i}-\mu\right)^{2} \cdot f_{i}$$ Variance $$s^{2}=\frac{1}{n} \sum_{i=1}^{n}\left(m_{i}-\overline{x}\right)^{2} \cdot f_{i}$$ $$C V=\frac{\sigma}{\mu} \cdot 100$$ Coefficient of variation $$C V=\frac{s}{\overline{x}} \cdot 100$$
 Basic probability rules $$P(A \cap B)=P(A | B) \cdot P(B)$$ Multiplication rule $$P(A \cup B)=P(A)+P(B)-P(A \cap B)$$ Addition rule $$P(A \cap B)=P(A) \cdot P(B) \text { or } P(A | B)=P(A)$$ Independence test Hypergeometric distribution formulae $$n C x=\left(\begin{array}{c}{n} \\ {x}\end{array}\right)=\frac{n !}{x !(n-x) !}$$ Combinatorial equation $$P(x)=\frac{\left(\begin{array}{c}{A} \\ {x}\end{array}\right)\left(\begin{array}{c}{N-A} \\ {n-x}\end{array}\right)}{\left(\begin{array}{c}{N} \\ {n}\end{array}\right)}$$ Probability equation $$E(X)=\mu=n p$$ Mean $$\sigma^{2}=\left(\frac{N-n}{N-1}\right) n p(q)$$ Variance Binomial distribution formulae $$P(x)=\frac{n !}{x !(n-x) !} p^{x}(q)^{n-x}$$ Probability density function $$E(X)=\mu=n p$$ Arithmetic mean $$\sigma^{2}=n p(q)$$ Variance Geometric distribution formulae $$P(X=x)=(1-p)^{x-1}(p)$$ Probability when $$x$$ is the first success. Probability when $$x$$ is the number of failures before first success $$P(X=x)=(1-p)^{x}(p)$$ $$\mu=\frac{1}{p}$$ Mean Mean $$\mu=\frac{1-p}{p}$$ $$\sigma^{2}=\frac{(1-p)}{p^{2}}$$ Variance Variance $$\sigma^{2}=\frac{(1-p)}{p^{2}}$$ Poisson distribution formulae $$P(x)=\frac{e^{-\mu_{\mu} x}}{x !}$$ Probability equation $$E(X)=\mu$$ Mean $$\sigma^{2}=\mu$$ Variance Uniform distribution formulae $$f(x)=\frac{1}{b-a} \text { for } a \leq x \leq b$$ PDF $$E(X)=\mu=\frac{a+b}{2}$$ Mean $$\sigma^{2}=\frac{(b-a)^{2}}{12}$$ Variance Exponential distribution formulae $$P(X \leq x)=1-e^{-m x}$$ Cumulative probability $$E(X)=\mu=\frac{1}{m} \text { or } m=\frac{1}{\mu}$$ Mean and decay factor $$\sigma^{2}=\frac{1}{m^{2}}=\mu^{2}$$ Variance
 The following page of formulae requires the use of the "$$Z$$", "$$t$$", "$$\chi^2$$" or "$$F$$" tables. $$Z=\frac{x-\mu}{\sigma}$$ Z-transformation for normal distribution $$Z=\frac{x-n p^{\prime}}{\sqrt{n p^{\prime}\left(q^{\prime}\right)}}$$ Normal approximation to the binomial Probability (ignores subscripts) Hypothesis testing Confidence intervals [bracketed symbols equal margin of error] (subscripts denote locations on respective distribution tables) $$Z_{c}=\frac{\overline{x}-\mu_{0}}{\frac{\sigma}{\sqrt{n}}}$$ Interval for the population mean when sigma is known $$\overline{x} \pm\left[Z_{(\alpha / 2)} \frac{\sigma}{\sqrt{n}}\right]$$ $$Z_{c}=\frac{\overline{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}$$ Interval for the population mean when sigma is unknown but $$n>30$$ $$\overline{x} \pm\left[Z_{(\alpha / 2)} \frac{s}{\sqrt{n}}\right]$$ $$t_{c}=\frac{\overline{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}$$ Interval for the population mean when sigma is unknown but $$n<30$$ $$\overline{x} \pm\left[t_{(n-1),(\alpha / 2)} \frac{s}{\sqrt{n}}\right]$$ $$Z_{c}=\frac{p^{\prime}-p_{0}}{\sqrt{\frac{p_{0} q_{0}}{n}}}$$ Interval for the population proportion $$p^{\prime} \pm\left[Z_{(\alpha / 2)} \sqrt{\frac{p^{\prime} q^{\prime}}{n}}\right]$$ $$t_{c}=\frac{\overline{d}-\delta_{0}}{s_{d}}$$ Interval for difference between two means with matched pairs $$\overline{d} \pm\left[t_{(n-1),(\alpha / 2)} \frac{s_{d}}{\sqrt{n}}\right]$$ where $$s_d$$ is the deviation of the differences $$Z_{c}=\frac{\left(\overline{x_{1}}-\overline{x_{2}}\right)-\delta_{0}}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$$ Interval for difference between two means when sigmas are known $$\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[Z_{(\alpha / 2)} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}\right]$$ $$t_{c}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\delta_{0}}{\sqrt{\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)}}$$ Interval for difference between two means with equal variances when sigmas are unknown $$\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[t_{d f,(\alpha / 2)} \sqrt{\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)}\right] \text { where } d f=\frac{\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)^{2}}{\left(\frac{1}{n_{1}-1}\right)\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}\right)+\left(\frac{1}{n_{2}-1}\right)\left(\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)}$$ $$Z_{c}=\frac{\left(p_{1}^{\prime}-p_{2}^{\prime}\right)-\delta_{0}}{\sqrt{\frac{p_{1}^{\prime}\left(q_{1}^{\prime}\right)}{n_{1}}+\frac{p_{2}^{\prime}\left(q_{2}^{\prime}\right)}{n_{2}}}}$$ Interval for difference between two population proportions $$\left(p_{1}^{\prime}-p_{2}^{\prime}\right) \pm\left[Z_{(\alpha / 2)} \sqrt{\frac{p_{1}^{\prime}\left(q_{1}^{\prime}\right)}{n_{1}}+\frac{p_{2}^{\prime}\left(q_{2}^{\prime}\right)}{n_{2}}}\right]$$ $$\chi_{c}^{2}=\frac{(n-1) s^{2}}{\sigma_{0}^{2}}$$ Tests for $$GOF$$, Independence, and Homogeneity $$\chi_{c}^{2}=\sum \frac{(O-E)^{2}}{E}$$where$$O =$$ observed values and $$E =$$ expected values $$F_{c}=\frac{s_{1}^{2}}{s_{2}^{2}}$$ Where $$s_{1}^{2}$$ is the sample variance which is the larger of the two sample variances The next 3 formule are for determining sample size with confidence intervals. (note: $$E$$ represents the margin of error) $$n=\frac{Z^{2}\left(\frac{a}{2}\right)^{\sigma^{2}}}{E^{2}}$$ Use when sigma is known $$E=\overline{x}-\mu$$ $$n=\frac{Z^{2}\left(\frac{a}{2}\right)^{(0.25)}}{E^{2}}$$ Use when $$p^{\prime}$$ is unknown $$E=p^{\prime}-p$$ $$n=\frac{Z^{2}\left(\frac{a}{2}\right)^{\left[p^{\prime}\left(q^{\prime}\right)\right]}}{E^{2}}$$ Use when p'p′ is uknown $$E=p^{\prime}-p$$
 Simple linear regression formulae for $$y=a+b(x)$$ $$r=\frac{\Sigma[(x-\overline{x})(y-\overline{y})]}{\sqrt{\Sigma(x-\overline{x})^{2} * \Sigma(y-\overline{y})^{2}}}=\frac{S_{x y}}{S_{x} S_{y}}=\sqrt{\frac{S S R}{S S T}}$$ Correlation coefficient $$b=\frac{\Sigma[(x-\overline{x})(y-\overline{y})]}{\Sigma(x-\overline{x})^{2}}=\frac{S_{x y}}{S S_{x}}=r_{y, x}\left(\frac{s_{y}}{s_{x}}\right)$$ Coefficient $$b$$ (slope) $$a=\overline{y}-b(\overline{x})$$ $$y$$-intercept $$s_{e}^{2}=\frac{\Sigma\left(y_{i}-\hat{y}_{i}\right)^{2}}{n-k}=\frac{\sum_{i=1}^{n} e_{i}^{2}}{n-k}$$ Estimate of the error variance $$S_{b}=\frac{s_{e}^{2}}{\sqrt{\left(x_{i}-\overline{x}\right)^{2}}}=\frac{s_{e}^{2}}{(n-1) s_{x}^{2}}$$ Standard error for coefficient $$b$$ $$t_{c}=\frac{b-\beta_{0}}{s_b}$$ Hypothesis test for coefficient $$\beta$$ $$b \pm\left[t_{n-2, \alpha / 2} S_{b}\right]$$ Interval for coefficient $$\beta$$ $$\hat{y} \pm\left[t_{\alpha / 2} * s_{e}\left(\sqrt{\frac{1}{n}+\frac{\left(x_{p}-\overline{x}\right)^{2}}{s_{x}}}\right)\right]$$ Interval for expected value of $$y$$ $$\hat{y} \pm\left[t_{\alpha / 2} * s_{e}\left(\sqrt{1+\frac{1}{n}+\frac{\left(x_{p}-\overline{x}\right)^{2}}{s_{x}}}\right)\right]$$ Prediction interval for an individual $$y$$ ANOVA formulae $$S S R=\sum_{i=1}^{n}\left(\hat{y}_{i}-\overline{y}\right)^{2}$$ Sum of squares regression $$S S E=\sum_{i=1}^{n}\left(\hat{y}_{i}-\overline{y}_{i}\right)^{2}$$ Sum of squares error $$S S T=\sum_{i=1}^{n}\left(y_{i}-\overline{y}\right)^{2}$$ Sum of squares total $$R^{2}=\frac{S S R}{S S T}$$ Coefficient of determination
 The following is the breakdown of a one-way ANOVA table for linear regression. Source of variation Sum of squares Degrees of freedom Mean squares $$F$$-ratio Regression $$SSR$$ $$1$$ or $$k−1$$ $$M S R=\frac{S S R}{d f_{R}}$$ $$F=\frac{M S R}{M S E}$$ Error $$SSE$$ $$n-k$$ $$M S E=\frac{S S E}{d f_{E}}$$ Total $$SST$$ $$n−1$$