Appendices
Appendix F: Mathematical Phrases, Symbols, and Formulas
Barbara Illowsky & OpenStax et al.
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: Factorialn!=n(n−1)(n−2)...(1)
0!=1
Formula 2: Combinations(nr)=n!(n−r)!r!
Formula 3: Binomial DistributionX~B(n,p)
P(X=x)=(nx)pxqn−x, for x=0,1,2,...,n
Formula 4: Geometric DistributionX~G(p)
P(X=x)=qx−1p, for x=1,2,3,...
Formula 5: Hypergeometric DistributionX~H(r,b,n)
P(X=x)=((rx)(bn−x)(r+bn))
Formula 6: Poisson DistributionX~P(μ)
P(X=x)=μxe−μx!
Formula 7: Uniform DistributionX~U(a,b)
f(X)=1b−a, a<x<b
Formula 8: Exponential DistributionX~Exp(m)
f(x)=me−mxm>0,x≥0
Formula 9: Normal DistributionX~N(μ,σ2)
f(x)=1σ2π√e−(x−μ)22σ2 , –∞<x<∞
Formula 10: Gamma FunctionΓ(z)=∫∞0xz−1e−xdx z>0
Γ(12)=π‾‾√
Γ(m+1)=m! for m, a nonnegative integer
otherwise: Γ(a+1)=aΓ(a)
Formula 11: Student’s t-distributionX~tdf
f(x)=(1+x2n)−(n+1)2Γ(n+12)nπ√Γ(n2)
X=ZYn√
Z~N(0,1),Y~Χ2df, n = degrees of freedom
Formula 12: Chi-Square DistributionX~Χ2df
f(x)=xn−22e−x22n2Γ(n2), x>0 , n = positive integer and degrees of freedom
Formula 13: F DistributionX~Fdf(n),df(d)
df(n)=degrees of freedom for the numerator
df(d)=degrees of freedom for the denominator
f(x)=Γ(u+v2)Γ(u2)Γ(v2)(uv)u2x(u2−1)[1+(uv)x−0.5(u+v)]
X=YuWv, Y, W are chi-square
Symbols and Their Meanings
| Chapter (1st used) | Symbol | Spoken | Meaning |
|---|---|---|---|
| Sampling and Data | ‾‾‾‾‾√ | The square root of | same |
| Sampling and Data | 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 | x⎯⎯ | x-bar | sample mean |
| Descriptive Statistics | μ | mu | population mean |
| Descriptive Statistics | s sx sx | s | sample standard deviation |
| Descriptive Statistics | s2 s2x | s squared | sample variance |
| Descriptive Statistics | σ σx σx | sigma | population standard deviation |
| Descriptive Statistics | σ2 σ2x | sigma squared | population variance |
| Descriptive Statistics | Σ | 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 OR B) | prob. of A or B | prob. of A or B or both occurring |
| Probability Topics | 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 | 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 | Lambda | average of Poisson distribution |
| Discrete Random Variables | ≥ | greater than or equal to | same |
| Discrete Random Variables | ≤ | 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 | X⎯⎯⎯ | X-bar | the random variable X-bar |
| The Central Limit Theorem | μx | mean of X | the average of X |
| The Central Limit Theorem | μx⎯⎯ | mean of X-bar | the average of X-bar |
| The Central Limit Theorem | σx | standard deviation ofX | same |
| The Central Limit Theorem | σx⎯⎯ | standard deviation ofX-bar | same |
| The Central Limit Theorem | ΣX | sum of X | same |
| The Central Limit Theorem | Σx | 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 | tα2 | student t with a/2 area in right tail | same |
| Confidence Intervals | p′; pˆ | p-prime; p-hat | sample proportion of success |
| Confidence Intervals | q′; qˆ | q-prime; q-hat | sample proportion of failure |
| Hypothesis Testing | H0 | H-naught, H-sub 0 | null hypothesis |
| Hypothesis Testing | Ha | H-a, H-sub a | alternate hypothesis |
| Hypothesis Testing | H1 | H-1, H-sub 1 | alternate hypothesis |
| Hypothesis Testing | α | alpha | probability of Type I error |
| Hypothesis Testing | β | beta | probability of Type II error |
| Hypothesis Testing | X1⎯⎯⎯⎯⎯−X2⎯⎯⎯⎯⎯ | X1-bar minus X2-bar | difference in sample means |
| Hypothesis Testing | μ1−μ2 | mu-1 minus mu-2 | difference in population means |
| Hypothesis Testing | P′1−P′2 | P1-prime minus P2-prime | difference in sample proportions |
| Hypothesis Testing | p1−p2 | p1 minus p2 | difference in population proportions |
| Chi-Square Distribution | Χ2 | 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 | yˆ | y-hat | estimated value of y |
| Linear Regression and Correlation | r | correlation coefficient | same |
| Linear Regression and Correlation | 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 |