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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