Contents

Mathematical Phrases, Symbols, and Formulas

Appendix F Mathematical Phrases, Symbols, and Formulas

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 F1

Formulas

Formula 1: Factorial $n! = n (n - 1) (n - 2) . . . (1) n! = n (n - 1) (n - 2) . . . (1)$

$0! = 1 0! = 1$

Formula 2: Combinations $(\begin{array}{l} n \\ r \end{array}) = \frac{n!}{(n - r)! r!} (\begin{array}{l} n \\ r \end{array}) = \frac{n!}{(n - r)! r!}$

Formula 3: Binomial Distribution $X ~ B (n, p) X$

$P (X = x) = (\begin{matrix} n \\ x \end{matrix}) p^{x} q^{n - x} P (X = x) = (\begin{matrix} n \\ x \end{matrix}) p^{x} q^{n - x}$ , for $x = 0, 1, 2, . . ., n x = 0, 1, 2, . . ., n$

Formula 4: Geometric Distribution $X ~ G (p) X$

$P (X = x) = q^{x - 1} p P (X = x) = q^{x - 1} p$ , for $x = 1, 2, 3, . . . x = 1, 2, 3, . . .$

Formula 5: Hypergeometric Distribution $X ~ H (r, b, n) X$

$P (X = x) = (\frac{(\binom{r}{x}) (\binom{b}{n - x})}{(\binom{r + b}{n})}) P (X = x) = (\frac{(\binom{r}{x}) (\binom{b}{n - x})}{(\binom{r + b}{n})})$

Formula 6: Poisson Distribution $X ~ P (μ) X$

$P (X = x) = \frac{μ^{x} e^{- μ}}{x!} P (X = x) = \frac{μ^{x} e^{- μ}}{x!}$

Formula 7: Uniform Distribution $X ~ U (a, b) X$

$f (X) = \frac{1}{b - a} f (X) = \frac{1}{b - a}$ , $a < x < b a < x < b$

Formula 8: Exponential Distribution $X ~ E x p (m) X$

$f (x) = m e^{- m x} m > 0, x \geq 0 f (x) = m e^{- m x} m > 0, x \geq 0$

Formula 9: Normal Distribution $X ~ N (μ, σ^{2}) X$

$f (x) = \frac{1}{σ \sqrt{2 π}} e^{\frac{{- (x - μ)}^{2}}{{2 σ}^{2}}} f (x) = \frac{1}{σ \sqrt{2 π}} e^{\frac{{- (x - μ)}^{2}}{{2 σ}^{2}}}$ , $- \infty < x < \infty$

Formula 10: Gamma Function $Γ (z) = {\int^{}}_{\infty}^{0} x^{z - 1} e^{- x} d x Γ (z) = {\int^{}}_{\infty}^{0} x^{z - 1} e^{- x} d x$ $z > 0 z > 0$

$Γ (\frac{1}{2}) = \sqrt{π} Γ (\frac{1}{2}) = \sqrt{π}$

$Γ (m + 1) = m! Γ (m + 1) = m!$ for $m m$ , a nonnegative integer

otherwise: $Γ (a + 1) = a Γ (a) Γ (a + 1) = a Γ (a)$

Formula 11: Student's t-distribution $X ~ t_{d f} X$

$f (x) = \frac{{(1 + \frac{x^{2}}{n})}^{\frac{- (n + 1)}{2}} Γ (\frac{n + 1}{2})}{\sqrt{nπ} Γ (\frac{n}{2})} f (x) = \frac{{(1 + \frac{x^{2}}{n})}^{\frac{- (n + 1)}{2}} Γ (\frac{n + 1}{2})}{\sqrt{nπ} Γ (\frac{n}{2})}$

$X = \frac{Z}{\sqrt{\frac{Y}{n}}} X = \frac{Z}{\sqrt{\frac{Y}{n}}}$

$Z ~ N (0, 1), Y ~ Χ_{d f}^{2} Z$ , $n n$ = degrees of freedom

Formula 12: Chi-Square Distribution $X ~ Χ_{d f}^{2} X$

$f (x) = \frac{x^{\frac{n - 2}{2}} e^{\frac{- x}{2}}}{2^{\frac{n}{2}} Γ (\frac{n}{2})} f (x) = \frac{x^{\frac{n - 2}{2}} e^{\frac{- x}{2}}}{2^{\frac{n}{2}} Γ (\frac{n}{2})}$ , $x > 0 x > 0$ , $n n$ = positive integer and degrees of freedom

Formula 13: F Distribution $X ~ F_{d f (n), d f (d)} X$

$d f (n) = d f (n)$ degrees of freedom for the numerator

$d f (d) = d f (d)$ degrees of freedom for the denominator

$f (x) = \frac{Γ (\frac{u + v}{2})}{Γ (\frac{u}{2}) Γ (\frac{v}{2})} {(\frac{u}{v})}^{\frac{u}{2}} x^{(\frac{u}{2} - 1)} [1 + (\frac{u}{v}) x^{- 0.5 (u + v)}] f (x) = \frac{Γ (\frac{u + v}{2})}{Γ (\frac{u}{2}) Γ (\frac{v}{2})} {(\frac{u}{v})}^{\frac{u}{2}} x^{(\frac{u}{2} - 1)} [1 + (\frac{u}{v}) x^{- 0.5 (u + v)}]$

$X = \frac{Y_{u}}{W_{v}} X = \frac{Y_{u}}{W_{v}}$ , $Y Y$ , $W W$ are chi-square

Symbols and Their Meanings

Chapter (1st used)	Symbol	Spoken	Meaning
Sampling and Data	$\sqrt{\begin{matrix} \end{matrix}} \sqrt{\begin{matrix} \end{matrix}}$	The square root of	same
Sampling and Data	$π π$	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	$\bar{x} \bar{x}$	x-bar	sample mean
Descriptive Statistics	$μ μ$	mu	population mean
Descriptive Statistics	s s_x sx	s	sample standard deviation
Descriptive Statistics	$s^{2} s^{2}$ $s_{x}^{2} s_{x}^{2}$	s squared	sample variance
Descriptive Statistics	$σ σ$ $σ_{x} σ_{x}$ σx	sigma	population standard deviation
Descriptive Statistics	$σ^{2} σ^{2}$ $σ_{x}^{2} σ_{x}^{2}$	sigma squared	population variance
Descriptive Statistics	$Σ Σ$	capital sigma	sum
Probability Topics	${} {}$	brackets	set notation
Probability Topics	$S S$	S	sample space
Probability Topics	$A A$	Event A	event A
Probability Topics	$P (A) P (A)$	probability of A	probability of A occurring
Probability Topics	$P (A \| B) P (A \| B)$	probability of A given B	prob. of A occurring given B has occurred
Probability Topics	$P (A OR B) P (A OR B)$	prob. of A or B	prob. of A or B or both occurring
Probability Topics	$P (A AND B) P (A AND B)$	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. distribution function	same
Discrete Random Variables	X	X	the random variable X
Discrete Random Variables	X ~	the distribution of X	same
Discrete Random Variables	B	binomial distribution	same
Discrete Random Variables	G	geometric distribution	same
Discrete Random Variables	H	hypergeometric dist.	same
Discrete Random Variables	P	Poisson dist.	same
Discrete Random Variables	$λ λ$	Lambda	average of Poisson distribution
Discrete Random Variables	$\geq \geq$	greater than or equal to	same
Discrete Random Variables	$\leq \leq$	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	k	k	critical value
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	CLT	Central Limit Theorem	same
The Central Limit Theorem	$\bar{X} \bar{X}$	X-bar	the random variable X-bar
The Central Limit Theorem	$μ_{x} μ_{x}$	mean of X	the average of X
The Central Limit Theorem	$μ_{\bar{x}} μ_{\bar{x}}$	mean of X-bar	the average of X-bar
The Central Limit Theorem	$σ_{x} σ_{x}$	standard deviation of X	same
The Central Limit Theorem	$σ_{\bar{x}} σ_{\bar{x}}$	standard deviation of X-bar	same
The Central Limit Theorem	$Σ X Σ X$	sum of X	same
The Central Limit Theorem	$Σ x Σ x$	sum of x	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	$t_{\frac{α}{2}} t_{\frac{α}{2}}$	student t with a/2 area in right tail	same
Confidence Intervals	$p' p'$ ; $\hat{p} \hat{p}$	p-prime; p-hat	sample proportion of success
Confidence Intervals	$q' q'$ ; $\hat{q} \hat{q}$	q-prime; q-hat	sample proportion of failure
Hypothesis Testing	$H_{0} H_{0}$	H-naught, H-sub 0	null hypothesis
Hypothesis Testing	$H_{a} H_{a}$	H-a, H-sub a	alternate hypothesis
Hypothesis Testing	$H_{1} H_{1}$	H-1, H-sub 1	alternate hypothesis
Hypothesis Testing	$α α$	alpha	probability of Type I error
Hypothesis Testing	$β β$	beta	probability of Type II error
Hypothesis Testing	$\bar{X 1} - \bar{X 2} \bar{X 1} - \bar{X 2}$	X1-bar minus X2-bar	difference in sample means
Hypothesis Testing	$μ_{1} - μ_{2} μ_{1} - μ_{2}$	mu-1 minus mu-2	difference in population means
Hypothesis Testing	${P^{'}}_{1} - {P^{'}}_{2} {P^{'}}_{1} - {P^{'}}_{2}$	P1-prime minus P2-prime	difference in sample proportions
Hypothesis Testing	$p_{1} - p_{2} p_{1} - p_{2}$	p1 minus p2	difference in population proportions
Chi-Square Distribution	$Χ^{2} Χ^{2}$	Ky-square	Chi-square
Chi-Square Distribution	$O O$	Observed	Observed frequency
Chi-Square Distribution	$E E$	Expected	Expected frequency
Linear Regression and Correlation	y = a + bx	y equals a plus b-x	equation of a line
Linear Regression and Correlation	$\hat{y} \hat{y}$	y-hat	estimated value of y
Linear Regression and Correlation	$r r$	correlation coefficient	same
Linear Regression and Correlation	$ε ε$	error	same
Linear Regression and Correlation	SSE	Sum of Squared Errors	same
Linear Regression and Correlation	1.9s	1.9 times s	cut-off value for outliers
F-Distribution and ANOVA	F	F-ratio	F-ratio

Table F2 Symbols and their Meanings