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# Statistics Tutorial: Important Statistics Formulas

Population mean = μ = ( Σ Xi ) / N

Population standard deviation = σ = √ [ 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) / N

Population correlation coefficient = ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] }

## Statistics

Unless otherwise noted, these formulas assume simple random sampling.

Sample mean = x = ( Σ xi ) / n

Sample standard deviation = s = √[ 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 = √ [ (n1 - 1) * s12 + (n2 - 1) * s22 ] / (n1 + n2 - 2) ]

Sample correlation coefficient = r = [ 1 / (n - 1) ] * Σ { [ (xi - x) / sx ] * [ (yi - y) / sy ] }

## 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 = √ [ Σ(yi - ŷi)2 / (n - 2) ] / √ [ 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

Rule of addition: P(A B) = P(A) + P(B) - P(A B)

Rule of multiplication: P(A B) = P(A) P(B|A)

Rule of subtraction: P(A') = 1 - P(A)

## Random Variables

In the following formulas, X and Y are random variables, and a and b are constants.

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 = [ s1212 ] / [ s2222 ]

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) = E(X) + E(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 = √[ P * (1 - P)/n ] = √( PQ / n )

Standard deviation of the mean = σx = σ/√(n)

Standard deviation of difference of sample means = σd = √[ (σ12 / n1) + (σ22 / n2) ]

Standard deviation of difference of sample proportions = σd = √{ [P1(1 - P1) / n1] + [P2(1 - P2) / n2] }

## Standard Error

Standard error of proportion = SEp = sp = √[ p * (1 - p)/n ] = √( pq / n )

Standard error of difference for proportions = SEp = sp = √{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

Standard error of the mean = SEx = sx = s/√(n)

Standard error of difference of sample means = SEd = sd = √[ (s12 / n1) + (s22 / n2) ]

Standard error of difference of paired sample means = SEd = sd = { √ [ (Σ(di - d)2 / (n - 1) ] } / √(n)

Pooled sample standard error = spooled = √ [ (n1 - 1) * s12 + (n2 - 1) * s22 ] / (n1 + n2 - 2) ]

Standard error of difference of sample proportions = sd = √{ [p1(1 - p1) / n1] + [p2(1 - p2) / n2] }

## Discrete Probability Distributions

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) (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

For the following formulas, assume that Y is a linear transformation of the random variable X, defined by the equation: Y = aX + b.

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/√(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) / √( 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

The correct formula for degrees of freedom (DF) depends on the situation (the nature of the test statistic, the number of samples, underlying assumptions, etc.).

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

Below, the first two formulas find the smallest sample sizes required to achieve a fixed margin of error, using simple random sampling. The third formula assigns sample to strata, based on a proportionate design. The fourth formula, Neyman allocation, uses stratified sampling to minimize variance, given a fixed sample size. And the last formula, optimum allocation, uses stratified sampling to minimize variance, given a fixed budget.

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 ) / √( ch ) ] / [ Σ ( Ni * σi ) / √( ci ) ]

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