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14.2: B | Mathematical Phrases, Symbols, and Formulas

  • Page ID
    4632
  • 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 \(C V=\frac{s}{\overline{x}} \cdot 100\)

    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
    \(P(A \cap B)=P(A) \cdot P(B) \text { or } P(A | B)=P(A)\) 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

    Table B4

    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
    \(E=p^{\prime}-p\)

    Table B5

    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

    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 \(SSR\) \(1\) 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 \(SST\) \(n−1\)    

    Table B7