Mathematical Phrases, Symbols, and Formulas
[latexpage]
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 |
Formulas
Formula 1: Factorial
\(n!=n\left(n-1\right)\left(n-2\right)…\left(1\right)\text{}\)
\(0!=1\text{}\)
Formula 2: Combinations
\(\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{\left(n-r\right)!r!}\)
Formula 3: Binomial Distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}B\left(n,p\right)\)
\(P\left(X=x\right)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}{q}^{n-x}\), for \(x=0,1,2,…,n\)
Formula 4: Geometric Distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}G\left(p\right)\)
\(P\left(X=x\right)={q}^{x-1}p\), for \(x=1,2,3,…\)
Formula 5: Hypergeometric Distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}H\left(r,b,n\right)\)
\(P\text{(}X=x\text{)}=\left(\frac{\left(\genfrac{}{}{0}{}{r}{x}\right)\left(\genfrac{}{}{0}{}{b}{n-x}\right)}{\left(\genfrac{}{}{0}{}{r+b}{n}\right)}\right)\)
Formula 6: Poisson Distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}P\left(\mu \right)\)
\(P\text{(}X=x\text{)}=\frac{{\mu }^{x}{e}^{-\mu }}{x!}\)
Formula 7: Uniform Distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}U\left(a,b\right)\)
\(f\left(X\right)=\frac{1}{b-a}\), \(a<x<b\)
Formula 8: Exponential Distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}Exp\left(m\right)\)
\(f\left(x\right)=m{e}^{-mx}m>0,x\ge 0\)
Formula 9: Normal Distribution\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}N\left(\mu ,{\sigma }^{2}\right)\)
\(f\text{(}x\text{)}=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{{-\left(x-\mu \right)}^{2}}{{2\sigma }^{2}}}\) , \(\phantom{\rule{12pt}{0ex}}–\infty <x<\infty \)
Formula 10: Gamma Function
\(\Gamma \left(z\right)=\underset{\infty }{\overset{0}{{\int }^{\text{}}}}{x}^{z-1}{e}^{-x}dx\)\(z>0\)
\(\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }\)
\(\Gamma \left(m+1\right)=m!\) for \(m\), a nonnegative integer
otherwise: \(\Gamma \left(a+1\right)=a\Gamma \left(a\right)\)
Formula 11: Student’s t-distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{t}_{df}\)
\(f\text{(}x\text{)}=\frac{{\left(1+\frac{{x}^{2}}{n}\right)}^{\frac{-\left(n+1\right)}{2}}\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\mathrm{n\pi }}\Gamma \left(\frac{n}{2}\right)}\)
\(X=\frac{Z}{\sqrt{\frac{Y}{n}}}\)
\(Z\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}N\left(0,1\right),\phantom{\rule{2px}{0ex}}Y\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{Χ}_{df}^{2}\), \(n\) = degrees of freedom
Formula 12: Chi-Square Distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{Χ}_{df}^{2}\)
\(f\text{(}x\text{)}=\frac{{x}^{\frac{n-2}{2}}{e}^{\frac{-x}{2}}}{{2}^{\frac{n}{2}}\Gamma \left(\frac{n}{2}\right)}\), \(x>0\) , \(n\) = positive integer and degrees of freedom
Formula 13: F Distribution
\(X\phantom{\rule{2px}{0ex}}~\phantom{\rule{2px}{0ex}}{F}_{df\left(n\right),df\left(d\right)}\)
\(df\left(n\right)\phantom{\rule{2px}{0ex}}=\phantom{\rule{2px}{0ex}}\)degrees of freedom for the numerator
\(df\left(d\right)\phantom{\rule{2px}{0ex}}=\phantom{\rule{2px}{0ex}}\)degrees of freedom for the denominator
\(f\left(x\right)=\frac{\Gamma \left(\frac{u+v}{2}\right)}{\Gamma \left(\frac{u}{2}\right)\Gamma \left(\frac{v}{2}\right)}{\left(\frac{u}{v}\right)}^{\frac{u}{2}}{x}^{\left(\frac{u}{2}-1\right)}\left[1+\left(\frac{u}{v}\right){x}^{-0.5\left(u+v\right)}\right]\)
\(X=\frac{{Y}_{u}}{{W}_{v}}\), \(Y\), \(W\) are chi-square
Symbols and Their Meanings
| Chapter (1st used) | Symbol | Spoken | Meaning |
|---|---|---|---|
| Sampling and Data | \(\sqrt{\begin{array}{c}\text{ }\\ \text{ }\end{array}}\) | The square root of | same |
| Sampling and Data | \(\pi \) | 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 | \(\overline{x}\) | x-bar | sample mean |
| Descriptive Statistics | \(\mu \) | mu | population mean |
| Descriptive Statistics | ssxsx | s | sample standard deviation |
| Descriptive Statistics | \({s}^{2}\)\({s}_{x}^{2}\) | s squared | sample variance |
| Descriptive Statistics | \(\sigma \)\({\sigma }_{x}\)σx | sigma | population standard deviation |
| Descriptive Statistics | \({\sigma }^{2}\)\({\sigma }_{x}^{2}\) | sigma squared | population variance |
| Descriptive Statistics | \(\Sigma \) | capital sigma | sum |
| Probability Topics | \(\left\{\right\}\) | brackets | set notation |
| Probability Topics | \(S\) | S | sample space |
| Probability Topics | \(A\) | Event A | event A |
| Probability Topics | \(P\left(A\right)\) | probability of A | probability of A occurring |
| Probability Topics | \(P\left(\mathit{\text{A}}\text{|}\mathit{\text{B}}\right)\) | probability of A given B | prob. of A occurring given B has occurred |
| Probability Topics | \(P\left(A\text{ OR }B\right)\) | prob. of A or B | prob. of A or B or both occurring |
| Probability Topics | \(P\left(A\text{ AND }B\right)\) | 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 | 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 | \(\lambda \) | Lambda | average of Poisson distribution |
| Discrete Random Variables | \(\ge \) | greater than or equal to | same |
| Discrete Random Variables | \(\le \) | 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 | 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 | \(\overline{X}\) | X-bar | the random variable X-bar |
| The Central Limit Theorem | \({\mu }_{x}\) | mean of X | the average of X |
| The Central Limit Theorem | \({\mu }_{\overline{x}}\) | mean of X-bar | the average of X-bar |
| The Central Limit Theorem | \({\sigma }_{x}\) | standard deviation of X | same |
| The Central Limit Theorem | \({\sigma }_{\overline{x}}\) | standard deviation of X-bar | same |
| The Central Limit Theorem | \(\Sigma X\) | sum of X | same |
| The Central Limit Theorem | \(\Sigma x\) | 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 | \({t}_{\frac{\alpha }{2}}\) | student t with a/2 area in right tail | same |
| Confidence Intervals | \(p\prime \); \(\stackrel{^}{p}\) | p-prime; p-hat | sample proportion of success |
| Confidence Intervals | \(q\prime \); \(\stackrel{^}{q}\) | q-prime; q-hat | 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 | \(\overline{X1}-\overline{X2}\) | 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 | \({{P}^{\prime }}_{1}-{{P}^{\prime }}_{2}\) | 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 line |
| Linear Regression and Correlation | \(\stackrel{^}{y}\) | y-hat | estimated value of y |
| Linear Regression and Correlation | \(r\) | correlation coefficient | same |
| Linear Regression and Correlation | \(\epsilon \) | 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 |
