# 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

How to take a random sample of 30 observations
you can use the random function to generate 30 numbers or observation
smita
How we can calculate chi-square if observed x٫y٫z/frequency 40,30,20 Total/90
calculate chi-square if observed x,y,z frequency 40,30,20total 90
Insha
find t value,if boysN1, ،32,M1,87.43 S1square,39.40.GirlsN2,34,M2,82.58S2square,40.80 Determine whether the results are significant or insignificant
Insha
The heights of a random sample of 100 entering HRM Freshman of a certain college is 157 cm with a standard deviation of 8cm. test the data against the claim that the overall height of all entering HRM students is 160 cm. previous studies showed that
complete the question.. as data given N = 100,mean= 157 cm, std dev = 8 cm..
smita
Z=x-mu/ std dev
smita
find the mean of 25,26,23,25,45,45,58,58,50,25
add all n divide by 10 i.e 38
smita
38
hhaa
amit
1 . The “average increase” for all NASDAQ stocks is the:
STATISTICS IN PRACTICE: This is a group assignment that seeks to reveal students understanding of statistics in general and it’s practical usefulness. The following are the guidelines; 1.      Each group has to identify a natural process or activity and gather data about/from the process. 2.
The diameter of an electric cable,say, X is assumed to be continoues random variable with p.d.f f(x)=6x(1-x); ≤x≤1 a)check that f(X) is p.d.f b) determine a number b such that p(Xb)
A manufacturer estimate 3% of his output is defective. Find the probability that in a sample of 10 items (a) less than two will be defective (b) more than two will be defective.
A manufacturer estimates that 3% of his output of a small item is defective. Find the probabilities that in a sample of 10 items (a) less than two and (b) more than two items will be defective.
ISAIAH
use binomial distribution with parameter n=10, p= 0.03, q=0.97
the standard deviation of a symmetrical distribution is 7.8 . what must be the value of forth moment about the mean in order that distribution be a) leptokurtic b) mesokurtic c) platy kyrtic intrept the obtain value of a b and c
A researcher observed that four out of every ten of their products are normally defective. A total of 360 samples of the products were being tested. If the sample is normally distributed and 220 of the products were identified to be faulty, test the hypothesis that the observation of the res
false
please answer the ques"following values are obtained from life table T15=3,493,601 and e°15=44.6 then expected number of person alive at exact age 15 will be "
vinay
make it clear
Kagimu
how x minus x bar is equal to zero
When the mean (X bar) of the sample and the datapoint-in-context (X) from the same sample are the same, then it (X minus X bar) is equal to 0
Johns
e.g. mean of. sample is 3 and one of the datapoints in that sample is also 3
Johns
a numerical value used as a summary measure for a sample such as a sample mean is known as
differentiate between qualitative and quantitative variables
qualitative variables are descriptive while quantitative are numeric variables
Chisomo
please guys what is the formulas use in calculated statistics please iam new here
Dear Yunisa there are different formulas used in statistics depending on wnat you want to measure. It would be helpful if you can be more specific
LAMIN
Which of these measures are used to analyze the central tendency of data?
hello
GoDeeply
hi
Atul
mean median mode quartiles
Atul
Median
Shemmy
median
Kagimu
median
LAMIN
There are three main measures of central tendency: the mode, the median and the mean. Each of these measures describes a different indication of the typical or central value in the distribution. ... The mode is the most commonly occurring value in a distribution.
sanjay
hw about the mean and median
Kagimu
While mean is the sum of the total distribution devided by the number of values in the distribution or data set.
Shemmy
And median is the measure used to determine or analyze the central tendancy of the distribution.
Shemmy
oooooo thanks
Kagimu
what is the difference between stratified sample and cluster sample ?
Sona
Dhrub
ok...I will look into it..thanx
Sona
both mean and median are all measures of central tendency however mean indicate the average of set of quantitative data and it is used when data is normal, median indicate position thus the middle of quantitative data when arranged in ascending order and it is used when in skewed distributions
david
in cluster there is multi set of population which exist naturally then you select subjects from each set to form ur sample while in stratified there is a single set of population then u create a sub unit called strata then u select from each stratum to form ur sample. remember that u need to deploy*
david
okay ...thank you
Sona
* either random sampling or systematic sampling in both techniques when doing d final selection of subjects from each group
david
Umair
A company owns two retail outlets. The weekly sale of the two stores during the past years is as follows. Weekly Sales Rs. In 000 21-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65 66-70 No of weeks Store I 30 45 95 60 55 33 53 76 38 23 Store II 155 23 34 56 65 54 34 23 24 54 Find out
Umair
Yunisa By By By Mldelatte By Angela Eckman By OpenStax By Rhodes By OpenStax By OpenStax By Brooke Delaney By Jemekia Weeden By Dan Ariely By OpenStax