Parameters 
Formula 
Population mean 
μ = ( Σ Xi ) / N 
Population standard deviation 
σ = sqrt [ Σ ( Xi  μ )2 / N ] 
Population variance 
σ2 = Σ ( Xi – μ )2 / N 
Variance of population proportion 
σP2 = PQ / n 
Standardized score 
Z = (X – μ) / σ 
Population correlation coefficient 
ρ = [ 1 / N ] * Σ { [ (Xi  μX) / σx ] * [ (Yi  μY) / σy ] } 
Statistics 
Formula 
Sample mean 
x = ( Σ xi ) / n 
Sample standard deviation 
s = sqrt [ Σ ( xi  x )2 / ( n  1 ) ] 
Sample variance 
s2 = Σ ( xi – x )2 / ( n – 1 ) 
Variance of sample proportion 
sp2 = pq / (n – 1) 
Pooled sample proportion 
p = (p1 * n1 + p2 * n2) / (n1 + n2) 
Pooled sample standard deviation 
sp = sqrt [ (n1  1) * s12 + (n2  1) * s22 ] / (n1 + n2 – 2) ] 
Sample correlation coefficient 
r = [ 1 / (n  1) ] * Σ { [ (xi  x) / sx ] * [ (yi  y) / sy ] } 
Correlation 

Pearson productmoment correlation 
r = Σ (xy) / sqrt [ ( Σ x2 ) * ( Σ y2 ) ] 
Linear correlation (sample data) 
r = [ 1 / (n  1) ] * Σ { [ (xi  x) / sx ] * [ (yi  y) / sy ] } 
Linear correlation (population data) 
ρ = [ 1 / N ] * Σ { [ (Xi  μX) / σx ] * [ (Yi  μY) / σy ] } 
Simple Linear Regression 

Simple linear regression line 
ŷ = b0 + b1x 
Regression coefficient 
b1 = Σ [ (xi  x) (yi  y) ] / Σ [ (xi  x)2] 
Regression slope intercept 
b0 = y – b1 * x 
Regression coefficient 
b1 = r * (sy / sx) 
Standard error of regression slope = 
sb1 = sqrt [ Σ(yi  ŷi)2 / (n  2) ] / sqrt [ Σ(xi  x)2 ] 
Counting 

n factorial 
n! = n * (n1) * (n – 2) * . . . * 3 * 2 * 1. By convention, 0! = 1. 
Permutations of n things, taken r at a time 
nPr = n! / (n – r)! 
Combinations of n things, taken r at a time 
nCr = n! / r!(n – r)! = nPr / r! 
Probability 
Formula 
Rule of addition 
P(A ∪ B) = P(A) + P(B) – P(A ∩ B) 
Rule of multiplicatio 
P(A ∩ B) = P(A) P(BA) 
Rule of subtraction 
P(A’) = 1 – P(A) 
Rule of multiplication 
P(A ∩ B) = P(A) P(BA) 
Rule of subtraction 
P(A’) = 1 – P(A) 
Random Variables 

Expected value of X 
E(X) = μx = Σ [ xi * P(xi) ] 
Variance of X 
Var(X) = σ2 = Σ [ xi  E(x) ]2 * P(xi) = Σ [ xi  μx ]2 * P(xi) 
Normal random variable 
zscore = z = (X – μ)/σ 
Chisquare statistic 
Χ2 = [ ( n  1 ) * s2 ] / σ2 
f statistic 
f = [ s12/σ12 ] / [ s22/σ22 ] 
Expected value of sum of random variables 
E(X + Y) = E(X) + E(Y) 
Expected value of difference between random variables 
E(X – Y) = E(X) – E(Y) 
Variance of the sum of independent random variables 
Var(X + Y) = Var(X) + Var(Y) 
Variance of the difference between independent random variables 
Var(X – Y) = Var(X) + Var(Y) 
Sampling Distributions 

Mean of sampling distribution of the mean 
μx = μ 
Mean of sampling distribution of the proportion 
μp = P 
Standard deviation of proportion 
σp = sqrt[ P * (1  P)/n ] = sqrt( PQ / n ) 
Standard deviation of the mean 
σx = σ/sqrt(n) 
Standard deviation of difference of sample means 
σd = sqrt[ (σ12 / n1) + (σ22 / n2) ] 
Standard deviation of difference of sample proportions 
σd = sqrt{ [P1(1  P1) / n1] + [P2(1  P2) / n2] } 
Standard Error 

Standard error of proportion 
SEp = sp = sqrt[ p * (1  p)/n ] = sqrt( pq / n ) 
Standard error of difference for proportions 
SEp = sp = sqrt{ p * ( 1 – p ) * [ (1/n1) + (1/n2) ] } 
Standard error of the mean 
SEx = sx = s/sqrt(n) 
Standard error of difference of sample means 
SEd = sd = sqrt[ (s12 / n1) + (s22 / n2) ] 
Standard error of difference of paired sample means 
SEd = sd = { sqrt [ (Σ(di  d)2 / (n  1) ] } / sqrt(n) 
Pooled sample standard error 
spooled = sqrt [ (n1  1) * s12 + (n2  1) * s22 ] / (n1 + n2 – 2) ] 
Standard error of difference of sample proportions 
sd = sqrt{ [p1(1  p1) / n1] + [p2(1  p2) / n2] } 
Discrete Probability Distribution 

Binomial formula: 
P(X = x) = b(x; n, P) = nCx * Px * (1 – P)n – x = nCx * Px * Qn – x 
Mean of binomial distribution 
μx = n * P 
Variance of binomial distribution 
σx2 = n * P * ( 1 – P ) 
Negative Binomial formula: 
P(X = x) = b*(x; r, P) = x1Cr1 * Pr * (1 – P)x – r 
Mean of negative binomial distribution 
μx = rQ / P 
Variance of negative binomial distribution 
σx2 = r * Q / P2 
Geometric formula: 
P(X = x) = g(x; P) = P * Qx – 1 
Mean of geometric distribution 
μx = Q / P 
Variance of geometric distribution 
σx2 = Q / P2 
Hypergeometric formula: 
P(X = x) = h(x; N, n, k) = [ kCx ] [ NkCnx ] / [ NCn ] 
Mean of hypergeometric distribution 
μx = n * k / N 
Variance of hypergeometric distribution 
σx2 = n * k * ( N – k ) * ( N – n ) / [ N2 * ( N  1 ) ] 
Poisson formula: 
P(x; μ) = (eμ) (μx) / x! 
Mean of Poisson distribution 
μx = μ 
Variance of Poisson distribution 
σx2 = μ 
Multinomial formula: 
P = [ n! / ( n1! * n2! * ... nk! ) ] * ( p1n1 * p2n2 * . . . * pknk ) 
Linear Transformations 

Mean of a linear transformation 
E(Y) = Y = aX + b. 
Variance of a linear transformation 
Var(Y) = a2 * Var(X). 
Standardized score 
z = (x – μx) / σx. 
tscore 
t = (x – μx) / [ s/sqrt(n) ]. 
Estimation 

Confidence interval: 
Sample statistic + Critical value * Standard error of statistic 
Margin of error 
(Critical value) * (Standard deviation of statistic) 
Margin of error 
(Critical value) * (Standard error of statistic) 
Hypothesis Testing 

Standardized test statistic 
(Statistic – Parameter) / (Standard deviation of statistic) 
Onesample ztest for proportions: 
zscore = z = (p – P0) / sqrt( p * q / n ) 
Twosample ztest for proportions: 
zscore = z = z = [ (p1  p2)  d ] / SE 
Onesample ttest for means: 
tscore = t = (x – μ) / SE 
Twosample ttest for means: 
tscore = t = [ (x1  x2)  d ] / SE 
Matchedsample ttest for means: 
tscore = t = [ (x1  x2)  D ] / SE = (d – D) / SE 
Chisquare test statistic 
Χ2 = Σ[ (Observed  Expected)2 / Expected ] 
Degrees of Freedom 

Onesample ttest: 
DF = n – 1 
Twosample ttest: DF 
(s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1  1) ] + [ (s22 / n2)2 / (n2  1) ] } 
Twosample ttest, pooled standard error: 
DF = n1 + n2 – 2 
Simple linear regression, test slope: 
DF = n – 2 
Chisquare goodness of fit test: 
DF = k – 1 
Chisquare test for homogeneity: 
DF = (r – 1) * (c – 1) 
Chisquare test for independence: 
DF = (r – 1) * (c – 1) 
Sample Size 

Mean (simple random sampling): 
n = { z2 * σ2 * [ N / (N  1) ] } / { ME2 + [ z2 * σ2 / (N  1) ] } 
Proportion (simple random sampling): 
n = [ ( z2 * p * q ) + ME2 ] / [ ME2 + z2 * p * q / N ] 
Proportionate stratified sampling: 
nh = ( Nh / N ) * n 
Neyman allocation (stratified sampling): 
nh = n * ( Nh * σh ) / [ Σ ( Ni * σi ) ] 
Optimum allocation (stratified sampling): 
nh = n * [ ( Nh * σh ) / sqrt( ch ) ] / [ Σ ( Ni * σi ) / sqrt( ci ) ] 