Descriptive statistics Variance = ! ! (! ! − ! ) ! = ! ! [ ! ! ! − ! ! ! ] k=n for population k=n-1 for sample Coeff. of variation = !.!. ! * V p/100 = X k when k > np/100 if it is not an integer; V p/100 = (X np/100 + X np/100+1 )/2 when np/100 is an integer Probability - Relative Risk = !" (!|!) !" (!|! ) - Bayes rules: Pr ! ! = !" ! ! !" (!) !" ! ! !" ! ![!!!" ! ! ]!" (! ) Given +ve Given -ve +ve -ve Disease PV + 1 – PV - Given disease Sensitivity False –ve (1-SS) No disease 1 – PV + PV - Given no disease False +ve (1-SF) Specificity Probability distribution - ! = ! ! ; ! ! = !"# ! = !{ ! − ! ! } ! = ! ! ! − [! ! ] ! (for all X) - ! = !" ! ; ! ! = [(! − !) ! ! ! ] = [! ! ! ! ] − ! ! Discrete X - ! = !" ! !" ; ! ! = [(! − !) ! ! ! ]!" = [! ! ! ! ]!" − ! ! Continuous X Binomial (limited by n) Pr ! = ! = ! ! ! ! ! (1 − !) !!! !" !"# Poisson (not limited by n) Pr(X = x) = ! !! ! ! !! where ! = !" ! ! Normal [for N(0,1 2 )] Normalization: ! = !!! ! ! ! = 1 ! 2! ! ! ! !! ! (!!!) ! [!(!; 0,1) = 1 2! ! ! ! ! ! ! ] Φ ! = Pr ! ≤ ! = ! !; 0,1 !" ! !! Φ ! ! = Pr ! ≤ ! ! = ! ! ! ! * Φ −! = 1 − Φ(!) ; Pr ! ≤ ! ≤ ! = Pr ! ! ≤ Z ≤ ! ! = Φ !!! ! − Φ( !!! ! ) * Pr(X ≤ a) = Pr(X < a) for continuous random variables only (as Pr(X = a) =0) * Approximation = equalize E(X) and Var(X) of different distribution Poisson approximation to binomial when np < 5 (remember to check values) Normal approximation to binomial when npq ≥ 5 (remember continuity correction) Pr ! ≤ ! ≤ ! ≈ Pr (! − 0.5 < ! < ! + 0.5) ; Pr ! = ! ≈ Pr (! − 0.5 < ! < ! + 0.5) special cases: !" ! = 0 ≈ !" ! < 0.5 ; !"(! = !) ≈ Pr (Y > n − 0.5) Relationships between random variables - ! !" = !"#$(! = ! ∩ ! = !) ; E(XY) = E(X)E(Y) if X,Y are independent - linear combination (l.c.) for all Xs : ! ! = ! ! !(! ! ) = ! ! ! ! - !"# !, ! = ! ! − ! ! ! − ! ! = ! !" − ! ! ! ! !"# !, ! = !"# ! ; !"# !, −! = −!"# ! ; !"# !, ! = 0 if X,Y are independent !"# !, !" + !" = !"#$ !, ! + !"#$(!, !) - !"## !, ! = ! !" = !"#(!,!) ! ! ! ! - l.c. for all Xs : !"# ! = ! ! ! !"# ! ! + 2 ! ! ! ! !"# ! ! , ! ! = ! ! ! ! ! ! + 2 ! ! ! ! ! ! ! ! !"##(! ! , ! ! ) - (sample covariance) ! !" = ! !!! (! ! − ! )(! ! − ! ) ! !!! ! !! = ! !!! (! ! − ! ) ! = ! ! (sample variance) when X = Y sample corr. coeff. = ! !" = ! !" ! !! ! !! Point estimation - Choice of estimator: ! ! = ! for unbiased; ! ! < !(! ∗ ) for minimum variance - ! ! = ! !!! (! ! − ! ) ! ! !!! is estimator for pop. variance σ 2 - ! is the best estimator for pop. mean μ for N.D. standard error = ! ! = !"# ! = !/ ! !/ ! is estimator for standard error - ! is the best estimator for pop. prop. p for N.D. standard error = ! ! = !"# ! = !"/! ! ! /! is estimator for standard error - ! ~! !, ! ! ! !ℎ!" !~! !, ! ! ; ! ≈ ! !, ! ! ! !ℎ!" ! ≥ 30 (central limit theorem) Sampling distribution Sampling distribution