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 | 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 | x⎯⎯ | x-bar | sample mean |

Descriptive Statistics | μ | mu | population mean |

Descriptive Statistics | s s _{x}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 | G_{1} |
green on first pick | same |

Probability Topics | P(G_{1}) |
prob. of green on first pick | same |

Discrete Random Variables | PDF |
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 | pdf |
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 |