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 product-moment 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 * (n-1) * (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(B|A)
Rule of subtraction P(A’) = 1 – P(A)
Rule of multiplication P(A ∩ B) = P(A) P(B|A)
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 z-score = z = (X – μ)/σ
Chi-square 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) = x-1Cr-1 * 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 ] [ N-kCn-x ] / [ 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) = (e-μ) (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.
t-score 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)
One-sample z-test for proportions: z-score = z = (p – P0) / sqrt( p * q / n )
Two-sample z-test for proportions: z-score = z = z = [ (p1 – p2) – d ] / SE
One-sample t-test for means: t-score = t = (x – μ) / SE
Two-sample t-test for means: t-score = t = [ (x1 – x2) – d ] / SE
Matched-sample t-test for means: t-score = t = [ (x1 – x2) – D ] / SE = (d – D) / SE
Chi-square test statistic Χ2 = Σ[ (Observed – Expected)2 / Expected ]
Degrees of Freedom  
One-sample t-test: DF = n – 1
Two-sample t-test: DF (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 – 1) ] + [ (s22 / n2)2 / (n2 – 1) ] }
Two-sample t-test, pooled standard error: DF = n1 + n2 – 2
Simple linear regression, test slope: DF = n – 2
Chi-square goodness of fit test: DF = k – 1
Chi-square test for homogeneity: DF = (r – 1) * (c – 1)
Chi-square 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 ) ]
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