14.0: B | Mathematical Phrases, Symbols, and Formulas
- Page ID
- 4632
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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. | \(X \neq 4\) |
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 |
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 | |||
| \(f(x)=\frac{1}{b-a} \text { for } a \leq x \leq b\) | |||
| \(E(X)=\mu=\frac{a+b}{2}\) | 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 | \(SST\) | \(n−1\) | ||