# 0.6 Mathematical phrases, symbols, and formulas

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## 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

## 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

## 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

## 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

Symbols and their meanings
Chapter (1st used) Symbol Spoken Meaning
Sampling and Data 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 x sx 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 prob. of A or B prob. of A or B or both occurring
Probability Topics 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 $\lambda$ 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 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 $\overline{X}$ 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 $\overline{X1}-\overline{X2}$ X 1-bar minus X 2-bar difference in sample means
Hypothesis Testing ${\mu }_{1}-{\mu }_{2}$ mu -1 minus mu -2 difference in population means
Hypothesis Testing ${{P}^{\prime }}_{1}-{{P}^{\prime }}_{2}$ P 1-prime minus P 2-prime difference in sample proportions
Hypothesis Testing ${p}_{1}-{p}_{2}$ p 1 minus p 2 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 1.9 s 1.9 times s cut-off value for outliers
F -Distribution and ANOVA F F -ratio F -ratio

pls I need understand this statistics very will is giving me problem
Sixty-four third year high school students were given a standardized reading comprehension test. The mean and standard deviation obtained were 52.27 and 8.24, respectively. Is the mean significantly different from the population mean of 50? Use the 5% level of significance.
No
Ariel
how do I find the modal class
look for the highest occuring number in the class
Kusi
the probability of an event occuring is defined as?
The probability of an even occurring is expected event÷ event being cancelled or event occurring / event not occurring
Gokuna
what is simple bar chat
Simple Bar Chart is a Diagram which shows the data values in form of horizontal bars. It shows categories along y-axis and values along x-axis. The x-axis displays above the bars and y-axis displays on left of the bars with the bars extending to the right side according to their values.
statistics is percentage only
the first word is chance for that we use percentages
it is not at all that statistics is a percentage only
Shambhavi
I need more examples
how to calculate sample needed
mole of sample/mole ratio or Va Vb
Gokuna
how to I solve for arithmetic mean
Yeah. for you to say.
James
yes
niharu
how do I solve for arithmetic mean
niharu
add all the data and divide by the number of data sets. For example, if test scores were 70, 60, 70, 80 the total is 280 and the total data sets referred to as N is 4. Therfore the mean or arthritmatic average is 70. I hope this helps.
Jim
*Tan A - Tan B = sin(A-B)/CosA CosB ... *2sinQ/Cos 3Q = tan 3Q - tan Q
standard error of sample
what is subjective probability
how to calculate the Steadman rank correlation
David
what is sampling? i want to know about the definition of sampling.
what is sample...?
In terms of Statistics or Research , It is a subset of population for measurement.
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