Alexander Holmes; Barbara Illowsky; Susan Dean

English Phrases Written Mathematically

When the English says:	Interpret this as:
X is at least 4.	X ≥ 4
The minimum of X is 4.	X ≥ 4
X is no less than 4.	X ≥ 4
X is greater than or equal to 4.	X ≥ 4
X is at most 4.	X ≤ 4
The maximum of X is 4.	X ≤ 4
X is no more than 4.	X ≤ 4
X is less than or equal to 4.	X ≤ 4
X does not exceed 4.	X ≤ 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 ≠ 4
X is not equal to 4.	X ≠ 4
X is not the same as 4.	X ≠ 4
X is different than 4.	X ≠ 4

Table B1

Symbols and Their Meanings

Chapter (1st used)	Symbol	Spoken	Meaning
Sampling and Data	$\sqrt{\begin{matrix} \end{matrix}}$	The square root of	same
Sampling and Data	$π$	Pi	3.14159… (a specific number)
Descriptive Statistics	Q₁	Quartile one	the first quartile
Descriptive Statistics	Q₂	Quartile two	the second quartile
Descriptive Statistics	Q₃	Quartile three	the third quartile
Descriptive Statistics	IQR	interquartile range	Q₃ – Q₁ = IQR
Descriptive Statistics	$\bar{x}$	x-bar	sample mean
Descriptive Statistics	$μ$	mu	population mean
Descriptive Statistics	s	s	sample standard deviation
Descriptive Statistics	$s^{2}$	s squared	sample variance
Descriptive Statistics	$σ$	sigma	population standard deviation
Descriptive Statistics	$σ^{2}$	sigma squared	population variance
Descriptive Statistics	$Σ$	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′	A-prime, complement of A	complement of A, not A
Probability Topics	P(A')	prob. of complement of A	same
Probability Topics	G₁	green on first pick	same
Probability Topics	P(G₁)	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 ~	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	≠	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	$\bar{X}$	X-bar	the random variable X-bar
The Central Limit Theorem	$μ_{\bar{x}}$	mean of X-bars	the average of X-bars
The Central Limit Theorem	$σ_{\bar{x}}$	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	$t_{\frac{α}{2}}$	student t with α/2 area in right tail	same
Confidence Intervals	$p'$	p-prime	sample proportion of success
Confidence Intervals	$q'$	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	$α$	alpha	probability of Type I error
Hypothesis Testing	$β$	beta	probability of Type II error
Hypothesis Testing	$\bar{X 1} - \bar{X 2}$	X1-bar minus X2-bar	difference in sample means
Hypothesis Testing	$μ_{1} - μ_{2}$	mu-1 minus mu-2	difference in population means
Hypothesis Testing	${P^{'}}_{1} - {P^{'}}_{2}$	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	$Χ^{2}$	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	$ε$	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	$n$
$μ$	Mean	$\overline{x}$
$σ^{2}$	Variance	$s^{2}$
$σ$	Standard deviation	$s$
$p$	Proportion	$p ′$
Single data set formulae
Population		Sample
$μ = E (x) = \frac{1}{N} \sum_{i = 1}^{N} (x_{i})$	Arithmetic mean	$\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} (x_{i})$
	Geometric mean	$\tilde{x} = {(\prod_{i = 1}^{n} X_{i})}^{\frac{1}{n}}$
$Q_{3} = \frac{3 (n + 1)}{4}$ , $Q_{1} = \frac{(n + 1)}{4}$	Inter-quartile range $I Q R = Q_{3} - Q_{1}$	$Q_{3} = \frac{3 (n + 1)}{4}$ , $Q_{1} = \frac{(n + 1)}{4}$
$σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} {(x_{i} - μ)}^{2}$	Variance	$s^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \overline{x})}^{2}$
Single data set formulae
Population		Sample
$μ = E (x) = \frac{1}{N} \sum_{i = 1}^{N} (m_{i} \cdot f_{i})$	Arithmetic mean	$\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} (m_{i} \cdot f_{i})$
	Geometric mean	$\tilde{x} = {(\prod_{i = 1}^{n} X_{i})}^{\frac{1}{n}}$
$σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} {(m_{i} - μ)}^{2} \cdot f_{i}$	Variance	$s^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(m_{i} - \overline{x})}^{2} \cdot f_{i}$
$C V = \frac{σ}{μ} \cdot 100$	Coefficient of variation	$C V = \frac{s}{\overline{x}} \cdot 100$

Table B3

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
$P (A \cap B) = P (A) \cdot P (B)$ or $P (A \| B) = P (A)$			Independence test
Hypergeometric distribution formulae
$n C x = (\binom{n}{x}) = \frac{n!}{x! (n - x)!}$		Combinatorial equation
$P (x) = \frac{(\binom{A}{x}) (\binom{N - A}{n - x})}{(\binom{N}{n})}$		Probability equation
$E (X) = μ = n p$		Mean
$σ^{2} = (\frac{N - n}{N - 1}) n p (q)$		Variance
Binomial distribution formulae
$P (x) = \frac{n!}{x! (n - x)!} p^{x} {(q)}^{n - x}$		Probability density function
$E (X) = μ = n p$		Arithmetic mean
$σ^{2} = n p (q)$		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	$P (X = x) = {(1 - p)}^{x} (p)$
$μ = \frac{1}{p}$	Mean	Mean	$μ = \frac{1 - p}{p}$
$σ^{2} = \frac{(1 - p)}{p^{2}}$	Variance	Variance	$σ^{2} = \frac{(1 - p)}{p^{2}}$
Poisson distribution formulae
$P (x) = \frac{e^{- μ} μ^{x}}{x!}$		Probability equation
$E (X) = μ$		Mean
$σ^{2} = μ$		Variance
Uniform distribution formulae
$f (x) = \frac{1}{b - a}$ for $a \leq x \leq b$		PDF
$E (X) = μ = \frac{a + b}{2}$		Mean
$σ^{2} = \frac{{(b - a)}^{2}}{12}$		Variance
Exponential distribution formulae
$P (X \leq x) = 1 - e^{- m x}$		Cumulative probability
$E (X) = μ = \frac{1}{m}$ or $m = \frac{1}{μ}$		Mean and decay factor
$σ^{2} = \frac{1}{m^{2}} = μ^{2}$		Variance

Table B4

The following page of formulae requires the use of the " $Z$ ", " $t$ ", " $χ^{2}$ " or " $F$ " tables.
$Z = \frac{x - μ}{σ}$	Z-transformation for normal distribution
$Z = \frac{x - n p'}{\sqrt{n p' (q')}}$	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)
$Z_{c} = \frac{\bar{x} - μ_{0}}{\frac{σ}{\sqrt{n}}}$	Interval for the population mean when sigma is known $\bar{x} \pm [Z_{(α / 2)} \frac{σ}{\sqrt{n}}]$
$Z_{c} = \frac{\bar{x} - μ_{0}}{\frac{s}{\sqrt{n}}}$	Interval for the population mean when sigma is unknown but $n > 30$ $\bar{x} \pm [Z_{(α / 2)} \frac{s}{\sqrt{n}}]$
$t_{c} = \frac{\bar{x} - μ_{0}}{\frac{s}{\sqrt{n}}}$	Interval for the population mean when sigma is unknown but $n < 30$ $\bar{x} \pm [t_{(n - 1), (α / 2)} \frac{s}{\sqrt{n}}]$
$Z_{c} = \frac{p' - p_{0}}{\sqrt{\frac{p_{0} q_{0}}{n}}}$	Interval for the population proportion $p' \pm [Z_{(α / 2)} \sqrt{\frac{p' q'}{n}}]$
$t_{c} = \frac{\bar{d} - δ_{0}}{\frac{s_{d}}{\sqrt{n}}}$	Interval for difference between two means with matched pairs $\bar{d} \pm [t_{(n - 1), (α / 2)} \frac{s_{d}}{\sqrt{n}}]$ where $s_{d}$ is the deviation of the differences
$Z_{c} = \frac{(\bar{x_{1}} - \bar{x_{2}}) - δ_{0}}{\sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}}$	Interval for difference between two means when sigmas are known $(\bar{x_{1}} - \bar{x_{2}}) \pm [Z_{(α / 2)} \sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}]$
$t_{c} = \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - δ_{0}}{\sqrt{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}}$	Interval for difference between two means with equal variances when sigmas are unknown $({\bar{x}}_{1} - {\bar{x}}_{2}) \pm [t_{d f, (α / 2)} \sqrt{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}]$ where $d f = \frac{{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}^{2}}{(\frac{1}{n_{1} - 1}) (\frac{{(s_{1})}^{2}}{n_{1}}) + (\frac{1}{n_{2} - 1}) (\frac{{(s_{2})}^{2}}{n_{2}})}$
$Z_{c} = \frac{({p'}_{1} - {p'}_{2}) - δ_{0}}{\sqrt{\frac{{p'}_{1} ({q'}_{1})}{n_{1}} + \frac{{p'}_{2} ({q'}_{2})}{n_{2}}}}$	Interval for difference between two population proportions $({p'}_{1} - {p'}_{2}) \pm [Z_{(α / 2)} \sqrt{\frac{{p'}_{1} ({q'}_{1})}{n_{1}} + \frac{{p'}_{2} ({q'}_{2})}{n_{2}}}]$
$χ_{c}^{2} = \frac{(n - 1) s^{2}}{σ_{0}^{2}}$	Tests for GOF, Independence, and Homogeneity $χ_{c}^{2} = Σ \frac{{(O - E)}^{2}}{E}$ 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 formulae are for determining sample size with confidence intervals. (note: E represents the margin of error)
$n = \frac{Z_{(\frac{a}{2})}^{2} σ^{2}}{E^{2}}$ Use when sigma is known $E = \bar{x} - μ$	$n = \frac{Z_{(\frac{a}{2})}^{2} (0.25)}{E^{2}}$ Use when $p'$ is unknown $E = p' - p$	$n = \frac{Z_{(\frac{a}{2})}^{2} [p' (q')]}{E^{2}}$ Use when $p'$ is unknown $E = p' - p$

Table B5

Simple linear regression formulae for $y = a + b (x)$
$r = \frac{Σ [(x - \bar{x}) (y - \bar{y})]}{\sqrt{Σ {(x - \bar{x})}^{2} * Σ {(y - \bar{y})}^{2}}} = \frac{S_{x y}}{S_{x} S_{y}} = \sqrt{\frac{S S R}{S S T}}$	Correlation coefficient
$b = \frac{Σ [(x - \bar{x}) (y - \bar{y})]}{Σ {(x - \bar{x})}^{2}} = \frac{S_{x y}}{{S S}_{x}} = r_{y, x} (\frac{s_{y}}{s_{x}})$	Coefficient b (slope)
$a = \bar{y} - b (\bar{x})$	y-intercept
$s_{e}^{2} = \frac{Σ {(y_{i} - {\hat{y}}_{i})}^{2}}{n - k} = \frac{Σ_{i = 1}^{n} e_{i}^{2}}{n - k}$	Estimate of the error variance
$S_{b} = \frac{s_{e}^{2}}{\sqrt{{(x_{i} - \bar{x})}^{2}}} = \frac{s_{e}^{2}}{(n - 1) s_{x}^{2}}$	Standard error for coefficient b
$t_{c} = \frac{b - β_{0}}{s_{b}}$	Hypothesis test for coefficient β
$b \pm [t_{n - 2, α / 2} S_{b}]$	Interval for coefficient β
$\hat{y} \pm [t_{α / 2} * s_{e} (\sqrt{\frac{1}{n} + \frac{{(x_{p} - \bar{x})}^{2}}{s_{x}}})]$	Interval for expected value of y
$\hat{y} \pm [t_{α / 2} * s_{e} (\sqrt{1 + \frac{1}{n} + \frac{{(x_{p} - \bar{x})}^{2}}{s_{x}}})]$	Prediction interval for an individual y
ANOVA formulae
$S S R = Σ_{i = 1}^{n} {({\hat{y}}_{i} - \bar{y})}^{2}$	Sum of squares regression
$S S E = Σ_{i = 1}^{n} {({\hat{y}}_{i} - {\bar{y}}_{i})}^{2}$	Sum of squares error
$S S T = Σ_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}$	Sum of squares total
$R^{2} = \frac{S S R}{S S T}$	Coefficient of determination

Table B6

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	$S S R$	$1$ or $k - 1$	$M S R = \frac{S S R}{d f_{R}}$	$F = \frac{M S R}{M S E}$
Error	$S S E$	$n - k$	$M S E = \frac{S S E}{d f_{E}}$
Total	$S S T$	$n - 1$

Table B7

B | Mathematical Phrases, Symbols, and Formulas

English Phrases Written Mathematically

Symbols and Their Meanings

Formulas