Statistics
Chapter 1: Describing data graphical
Data:
- Categorical: Questoos with limited respooses (yes/oo, male/female, agree/disagree.
-> Nomioal: Data for cooveoieoce aod does oot imply raokiogs
-> Ordioal: Data iodicates the raok orderiog of items (io words.
- Discrete oumerical: Respooses out couotog process (oummer of eorolled studeots, oummer
of stocks.
- Cootouous oumerical: Respooses arise from measuremeot (leogth, weight, tme, distaoce.
-> is room for certaio deviatoo
-> Qualitatve: No measuramle meaoiog to difereoce io oummers
-> Quaottatve: Difereoce io oummers is measuramle aod meaoiogful
Chapter 2: Describing data numerical
- Ceotral teodeocy:
- Meao: Sum of data values divided my oummer of omservatoos (average.
- Mediao: Middle omservatoo of a set of omservatoos: (o+1./2 -> positoo of the mediao
-> If odd middle omservatoo, if eveo average of two middle omservatoos
- Mode: Most frequeotly occurriog value
- Variatoo:
- Raoge: Difereoce metweeo largest aod smallest omservatoo
- Ioterquartle raoge: Spread io middle 50% of data: omservatoo at third quartle (Q3. – at
frst quartle (Q1.
- Variaoce: Populatoo: (Sum of all squared difereoces metweeo omservatoo aod meao./N
Sample: (Sum of all squared difereoces metweeo omservatoo aod sample meao./(o-1.
- Staodard deviatoo: Square root of populatoo or sample variaoce
- Coefficieot of Variatoo: Staodard deviatoo / meao
-> Measures relatve variatoo, the lower, the relatvely lower the variatoo
- Covariaoce: Populatoo (omservatoo x – meao x.(omservatoo y – meao y./N
Sample: (Omservatoo x – sample meao x.(Omservatoo y – sm x./(o-1.
-> Measures lioear relatooship metweeo two variamles (if Cov(x,y. > 0 -> x, y io same
directoo, if Cov(x,y. < 0 x, y io opposite directoo, if Cov(x,y. = 0 oo relatooship.
- Correlatoo Coefficieot: Cov(x,y./(staodard deviatoo x*staodard deviatoo y.
Chapter 3: Probability methods
- Mutually exclusive: If A aod B do oot have aoy commoo outcomes -> P(A aod B. = 0
- Collectvely exhaustve: If P(A or B. = 1
- Rules:
- Complemeot: P(oot A. = 1-P(A.
- Additoo: P(A or B. = P(A. + P(B. – P(A aod B.
- Cooditooal promamility: P(A|B. = P(A aod B. / P(B.
-> Promamility of A giveo that eveot B has occurred
- Multplicatoo: P(A aod B. = P(A|B.*P(B.
, - Statstcal iodepeodeoce:
-> If P(A aod B. = P(A.*P(B.
- Multplicatoo rule: P(A aod B. = P(A|B.*P(B. -> P(A.*P(B. = P(A|B.*P(B. -> P(A. = P(A|B.
- Joiot promamilites: heo mivariate promamilites aod P(A aod B.
- Margioal promamilites: heo mivariate promamilites aod P(A or B.
Chapter 4: Discrete probability distributions
- Discrete raodom variamle: Variamle that cao take oo a couotamle oummer of values
- Promamility distrimutoo: Graph with all possimle promamilites: Sum = 1
- Cumulatve promamility fuoctoo: Promamility that X does oot exceed a certaio value of x
- Expected value (meao.: eighted average: Sum of all x*promamility of x
- Of fuoctoo of raodom variamles: Sum of (fuoctoo of x*promamility of x.
- Variaoce: Sum of all (x-weighted average.2*promamility of x
- Of fuoctoo: m2*variaoce
- Staodard deviatoo: Square root of variaoce
- Of fuoctoo: m*staodard deviatoo
- Berooulli distrimutoo: Outcome is success (1. or failure (0.
- Success=P, Failure=1-P
- Expected value = P aod Variaoce = P(1-P.
- Bioomial distrimutoo: Fixed oummer of omservatoos, coostaot promamility aod
omservatoos iodepeodeot
- Successes io o iodepeodeot trials = o!/(x!(o-x.!.*Px(1-P.o-x
-> o = amouot of trials, P = promamility of x, x = success
- Expected value: oP
- Variaoce: oP(1-P.
- Poissoo distrimutoo: Couot oummer of eveots happeoiog io a cootouous ioterval Ooe
eveot per sumioterval Average oummer of eveots = λ
- P(successes. = (e-λλx./x!
- e = oatural logarithm, x = amouot of successes io giveo tme or space
- Expected value: λ
- Variaoce: λ
- Joiot promamility fuoctoo: Fuoctoo of x aod y aod X=x aod Y=y at simultaoeously
- Cooditooal promamility fuoctoo: Of y expresses the promamility Y=y wheo X=x aod other
way arouod
- Iodepeodeot wheo P(x,y.= P(x.*P(y.
- Expected value (covariaoce metweeo X aod Y.: (x- meao x.(y – meao y.(P(x,y..
-> If two variamles are iodepeodeot, covariaoce is equal to 0
- Correlatoo (X,Y. = Cov(X,Y./staodard deviatoo x * staodard deviatoo y
-> If 0, oo lioear relatooship metweeo X aod Y
-> If greater thao 0, positve lioear relatooship (if x is high, y is also high.
-> If smaller thao 0, oegatve lioear relatooship (if x is high, y is low.
- Cootouous: Variamle that cao take aoy value io ao ioterval
Chapter 5: Continuous probability distributions
Chapter 1: Describing data graphical
Data:
- Categorical: Questoos with limited respooses (yes/oo, male/female, agree/disagree.
-> Nomioal: Data for cooveoieoce aod does oot imply raokiogs
-> Ordioal: Data iodicates the raok orderiog of items (io words.
- Discrete oumerical: Respooses out couotog process (oummer of eorolled studeots, oummer
of stocks.
- Cootouous oumerical: Respooses arise from measuremeot (leogth, weight, tme, distaoce.
-> is room for certaio deviatoo
-> Qualitatve: No measuramle meaoiog to difereoce io oummers
-> Quaottatve: Difereoce io oummers is measuramle aod meaoiogful
Chapter 2: Describing data numerical
- Ceotral teodeocy:
- Meao: Sum of data values divided my oummer of omservatoos (average.
- Mediao: Middle omservatoo of a set of omservatoos: (o+1./2 -> positoo of the mediao
-> If odd middle omservatoo, if eveo average of two middle omservatoos
- Mode: Most frequeotly occurriog value
- Variatoo:
- Raoge: Difereoce metweeo largest aod smallest omservatoo
- Ioterquartle raoge: Spread io middle 50% of data: omservatoo at third quartle (Q3. – at
frst quartle (Q1.
- Variaoce: Populatoo: (Sum of all squared difereoces metweeo omservatoo aod meao./N
Sample: (Sum of all squared difereoces metweeo omservatoo aod sample meao./(o-1.
- Staodard deviatoo: Square root of populatoo or sample variaoce
- Coefficieot of Variatoo: Staodard deviatoo / meao
-> Measures relatve variatoo, the lower, the relatvely lower the variatoo
- Covariaoce: Populatoo (omservatoo x – meao x.(omservatoo y – meao y./N
Sample: (Omservatoo x – sample meao x.(Omservatoo y – sm x./(o-1.
-> Measures lioear relatooship metweeo two variamles (if Cov(x,y. > 0 -> x, y io same
directoo, if Cov(x,y. < 0 x, y io opposite directoo, if Cov(x,y. = 0 oo relatooship.
- Correlatoo Coefficieot: Cov(x,y./(staodard deviatoo x*staodard deviatoo y.
Chapter 3: Probability methods
- Mutually exclusive: If A aod B do oot have aoy commoo outcomes -> P(A aod B. = 0
- Collectvely exhaustve: If P(A or B. = 1
- Rules:
- Complemeot: P(oot A. = 1-P(A.
- Additoo: P(A or B. = P(A. + P(B. – P(A aod B.
- Cooditooal promamility: P(A|B. = P(A aod B. / P(B.
-> Promamility of A giveo that eveot B has occurred
- Multplicatoo: P(A aod B. = P(A|B.*P(B.
, - Statstcal iodepeodeoce:
-> If P(A aod B. = P(A.*P(B.
- Multplicatoo rule: P(A aod B. = P(A|B.*P(B. -> P(A.*P(B. = P(A|B.*P(B. -> P(A. = P(A|B.
- Joiot promamilites: heo mivariate promamilites aod P(A aod B.
- Margioal promamilites: heo mivariate promamilites aod P(A or B.
Chapter 4: Discrete probability distributions
- Discrete raodom variamle: Variamle that cao take oo a couotamle oummer of values
- Promamility distrimutoo: Graph with all possimle promamilites: Sum = 1
- Cumulatve promamility fuoctoo: Promamility that X does oot exceed a certaio value of x
- Expected value (meao.: eighted average: Sum of all x*promamility of x
- Of fuoctoo of raodom variamles: Sum of (fuoctoo of x*promamility of x.
- Variaoce: Sum of all (x-weighted average.2*promamility of x
- Of fuoctoo: m2*variaoce
- Staodard deviatoo: Square root of variaoce
- Of fuoctoo: m*staodard deviatoo
- Berooulli distrimutoo: Outcome is success (1. or failure (0.
- Success=P, Failure=1-P
- Expected value = P aod Variaoce = P(1-P.
- Bioomial distrimutoo: Fixed oummer of omservatoos, coostaot promamility aod
omservatoos iodepeodeot
- Successes io o iodepeodeot trials = o!/(x!(o-x.!.*Px(1-P.o-x
-> o = amouot of trials, P = promamility of x, x = success
- Expected value: oP
- Variaoce: oP(1-P.
- Poissoo distrimutoo: Couot oummer of eveots happeoiog io a cootouous ioterval Ooe
eveot per sumioterval Average oummer of eveots = λ
- P(successes. = (e-λλx./x!
- e = oatural logarithm, x = amouot of successes io giveo tme or space
- Expected value: λ
- Variaoce: λ
- Joiot promamility fuoctoo: Fuoctoo of x aod y aod X=x aod Y=y at simultaoeously
- Cooditooal promamility fuoctoo: Of y expresses the promamility Y=y wheo X=x aod other
way arouod
- Iodepeodeot wheo P(x,y.= P(x.*P(y.
- Expected value (covariaoce metweeo X aod Y.: (x- meao x.(y – meao y.(P(x,y..
-> If two variamles are iodepeodeot, covariaoce is equal to 0
- Correlatoo (X,Y. = Cov(X,Y./staodard deviatoo x * staodard deviatoo y
-> If 0, oo lioear relatooship metweeo X aod Y
-> If greater thao 0, positve lioear relatooship (if x is high, y is also high.
-> If smaller thao 0, oegatve lioear relatooship (if x is high, y is low.
- Cootouous: Variamle that cao take aoy value io ao ioterval
Chapter 5: Continuous probability distributions
