11.2: B - Mathematical Phrases, Symbols, and Formulas

English Phrases Written Mathematically

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

 

Symbols and Their Meanings

Chapter (1st used)	Symbol	Spoken	Meaning
Sampling and Data	\(\sqrt{ } \)	The square root of	same
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 \(B\)	probability of \(A\) occurring given \(B\) has occurred
Probability Topics	\(P(A\cup B)\)	probability of \(A\) or \(B\)	probability of \(A\) or \(B\) or both occurring
Probability Topics	\(P(A\cap B)\)	probability of \(A\) and \(B\)	probability of both \(A\) and \(B\) occurring (same time)
Probability Topics	\(A^{\prime}\)	\(A\)-prime; complement of \(A\)	complement of \(A\); not \(A\)
Probability Topics	\(P(A^{\prime})\)	probability of the complement of \(A\)	same
Probability Topics	\(G_1\)	green on first pick	same
Probability Topics	\(P(G_1)\)	probability of green on first pick	same
The Normal Distribution	\(N\)	normal distribution	same
The Normal Distribution	\(z\)	\(z\)-score	same
The Normal Distribution	\(Z\)	standard normal distribution	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's \(t\) with \(\alpha\)/2 area in each tail	same
Confidence Intervals	\(P^{\prime}\)	\(P\)-prime	sample proportion of success or interest
Hypothesis Testing	\(H_0\)	\(H\)-naught, \(H\)-sub-0	null hypothesis
Hypothesis Testing	\(H_a\)	\(H\)-a, \(H\)-sub a	alternative (or research) hypothesis
Hypothesis Testing	\(H_1\)	\(H\)-1, \(H\)-sub 1	alternative (or research) hypothesis
Hypothesis Testing	\(\alpha\)	alpha	probability of Type I error
Hypothesis Testing	\(\beta\)	beta	probability of Type II error
Hypothesis Testing	\(\overline{x}_1-\overline{x}_2\)	\(x\)1-bar minus \(x\)2-bar	difference in sample means
Hypothesis Testing	\(\mu_{1}-\mu_{2}\)	mu-1 minus mu-2	difference in population means
Hypothesis Testing	\(P_{1}^{\prime}-P_{2}^{\prime}\)	\(P\)1-prime minus \(P\)2-prime	difference in sample proportions
Hypothesis Testing	\(P_{1}-P_{2}\)	\(P\)1 minus \(P\)2	difference in population proportions
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

 

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
\(Q_{3}=\frac{3(N+1)}{4}, Q_{1}=\frac{(N+1)}{4}\)	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	\(s^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}\)
\(\sigma^{2}=\frac{1}{N} \sum_{i=1}^{N}\left(x_{i}-\mu\right)^{2} \cdot f_{i}\)	Variance	\(s^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2} \cdot f_{i}\)

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

 

The following formulae require the use of the \(z\), \(t\), or \(F\) tables.
\(z=\frac{x-\mu}{\sigma}\)	z-transformation for normal distribution
Test statistics	Confidence intervals [bracketed symbols equal margin of error] (subscripts denote locations on respective distribution tables)
\(z_{obs}=\frac{\overline{x}-\mu_{0}}{\frac{\sigma}{\sqrt{n}}}\)	Interval for the population mean when sigma is known \(\overline{x} \pm\left[z_{(\alpha / 2)} \frac{\sigma}{\sqrt{n}}\right]\)
\(z_{obs}=\frac{\overline{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}\)	Interval for the population mean when sigma is unknown and \(n > 100\) \(\overline{x} \pm\left[z_{(\alpha / 2)} \frac{s}{\sqrt{n}}\right]\)
\(t_{obs}=\frac{\overline{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}\)	Interval for the population mean when sigma is unknown and \(n < 100\) \(\overline{x} \pm\left[t_{(n-1),(\alpha / 2)} \frac{s}{\sqrt{n}}\right]\)
\(z_{obs}=\frac{P^{\prime}-P_0}{\sqrt{\frac{P_0 (1-P_0)}{n}}}\)	Interval for the population proportion \(P^{\prime} \pm\left[z_{(\alpha / 2)} \sqrt{\frac{P^{\prime} \left(1-P^{\prime}\right)}{n}}\right]\)
\(t_{obs}=\frac{\bar{x}_d-\mu_{d}}{\frac{s_{d}}{\sqrt{n}}}\)	Interval for difference between two means with matched pairs \(\bar{x}_d \pm\left[t_{(n-1),(\alpha / 2)} \frac{s_{d}}{\sqrt{n}}\right]\) where \(s_d\) is the deviation of the differences
\(z_{obs}=\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left({\mu_{1}}-{\mu_{2}}\right)}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\)	Interval for difference between two independent means when \(n > 100\) \(\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[z_{(\alpha / 2)} \sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}\right]\)
\(z_{obs}=\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left({\mu_{1}}-{\mu_{2}}\right)}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\)	Interval for difference between two independent means when \(n < 100\) \(\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[t_{(n_1+n_2-2),(\alpha / 2)} \sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}\right]\)
	Interval for difference between two population proportions \(\left(P_{1}^{\prime}-P_{2}^{\prime}\right) \pm\left[z_{(\alpha / 2)} \sqrt{\frac{P_{1}^{\prime}\left(1-P_{1}^{\prime}\right)}{n_{1}}+\frac{P_{2}^{\prime}\left(1-P_{2}^{\prime}\right)}{n_{2}}}\right]\)

Simple linear regression formulae for \(Y=a+b(X)\)
\[r_{X Y}=\frac{\sum (X_{i}-\overline{X})*(Y_{i}-\overline{Y})}{\sqrt{\sum (X_{i}-\overline{X})^{2}*\sum (Y_{i}-\overline{Y})^{2}}}\nonumber\] \[r_{X Y}=\frac{\sum X_{i} Y_{i}-\frac{\left(\sum X_{i}\right)\left(\sum Y_{i}\right)}{n}}{\sqrt{\left[\sum X_{i}^{2}-\frac{\left(\sum X_{i}\right)^{2}}{n}\right]*\left[\sum Y_{i}^{2}-\frac{\left(\sum Y_{i}\right)^{2}}{n}\right]}}\nonumber\]	Correlation coefficient
\[b_{1}=\frac{\Sigma(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{\Sigma(X_{i}-\overline{X})^{2}}\nonumber\] \[b_{1}=\frac{\sum X_{i} Y_{i}-\frac{\left(\sum X_{i}\right)\left(\sum Y_{i}\right)}{n}}{\sum X_{i}^{2}-\frac{\left(\sum X_{i}\right)^{2}}{n}}\nonumber\] \[b_{1}=r_{X Y}\left(\frac{s_{Y}}{s_{X}}\right)\nonumber\]	Coefficient \(b\) (or \(b_1\), slope)
\[b_{0}=\overline{Y}-b_{1} \overline{X}\nonumber\]	\(Y\)-intercept (\(a\), or \(b_0\))
\(s_{e}^{2}=\frac{\Sigma\left(Y_{i}-\hat{Y}_{i}\right)^{2}}{n-k}=\frac{\sum_{i=1}^{n} e_{i}^{2}}{n-k}\)	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 \(b\)
\(t_{obs}=\frac{b-\beta_{0}}{s_b}\)	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
\(SS_R=n_1\left(\bar{x}_{1}-\bar{x}\right)^2+\cdots+n_g\left(\bar{x}_{g}-\bar{x}\right)^2\)	Sum of squares regression
\(SS_E=\left(n_1-1\right)s_1^2+\cdots+\left(n_g-1\right)s_g^2\)	Sum of squares error
\(SS_T=SS_R + SS_E\)	Sum of squares total
\(R^{2}=\frac{SS_R}{SS_T}\)	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	\(n_1\left(\bar{x}_{1}-\bar{x}\right)^2+\cdots+n_g\left(\bar{x}_{g}-\bar{x}\right)^2\)	\(1\) or \(g−1\)	\(M S R=\frac{S S_R}{d f_{R}}\)	\(F=\frac{M S_R}{M S_E}\)
Error	\(\left(n_1-1\right)s_1^2+\cdots+\left(n_g-1\right)s_g^2\)	\(n-g\)	\(M S E=\frac{S S_E}{d f_{E}}\)
Total	\(SS_R + SS_E\)	\(n−1\)