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Statistics Formulas Sample Mean x x i n Population Mean x xi N Interquartile Range IQR Q3 Q1 Sample Variance Sample Standard Deviation x x 2 s 2 sx n 1 Population Variance x i i n 1 Population Standard Deviation 2 2 x x 2 i x 2 x x N i x N Coefficient of Variation Standard Deviation CV 100 % Mean Z-Score Zi xi x x or Z i i s Sample Covariance sxy Population Covariance x xy y i xy i n 1 xi x yi y N Pearson Product Moment Correlation Coefficient (Pearson R): Sample Data rxy s xy sx s y Population Data xy xy x y -1- Statistics Formulas Weighted Mean xw w x w i i i Sample Mean for Grouped Data xg fM i g i n Sample Variance for Grouped Data s g2 f M i i Population Mean for Grouped Data xg 2 g Sample Standard Deviation for Grouped Data sg f i M i xg n 1 N Population Variance for Grouped Data 2 n 1 fi M i f M i g 2 i N Population Standard Deviation for Grouped Data 2 g -2- f i M i g 2 N Statistics Formulas Counting Rule for Combinations n Cr n! r!(n r )! Counting Rule for Permutations n Pr n! (n r )! Computing Probability Using the Complement P( A) 1 P( AC ) Addition Law P( A B) P( A) P( B) P( A B) Conditional Probability P( A B) or P( A | B) P( B) Multiplication Law P( A B) P( B) P( A | B) P( B | A) or P( A B) P( A) P( A B) P( A) P( B | A) Multiplication Law for Independent Events P( A B) P( A) P( B) Expected Value of a Discrete Random Variable E( x) x f ( x) Variance of a Discrete Random Variable 2 Var ( x) 2 x f ( x) Binomial Probability Function f ( x) n! p x (1 p) ( n x ) x!(n x)! Expected Value and Variance for the Binomial Distribution E ( x) np Var ( x) 2 np(1 p) Poisson Probability Function f ( x) xe x! where f (x) = the probability of x occurrences in an interval = the expected value or mean number of occurrences in an interval e = 2.718 -3- Statistics Formulas Sampling & Sampling Distributions Standard Deviation of For a Finite Population x x (Standard Error) For an Infinite Population N n N 1 n x n Sampling Proportions Standard Deviation of p̂ For a Finite Population pˆ N n N 1 (Standard Error) For an Infinite Population pˆ qˆ n Interval Estimate of a Population Mean: xE Where Where Known Unknown E Z n Interval Estimate of a Population Mean: xE pˆ pˆ qˆ n s E t n Sample Size for an Interval Estimate of a Population Mean Z 2 2 n E2 Interval Estimate of a Population Proportion pˆ Z pˆ qˆ n Sample Size for an Interval Estimate of a Population Proportion Z 2 p *q * n E2 -4- Statistics Formulas Sample Mean x x i n Sample Variance Sample Standard Deviation x x x x 2 s 2 2 i sx n 1 i n 1 Sampling Proportions Standard Deviation of For a Finite Population N n N 1 pˆ p̂ (Standard Error) For an Infinite Population pˆ 1 pˆ n pˆ Interval Estimate of a Population Mean: xE Where Where Known E Z n Interval Estimate of a Population Mean: xE pˆ 1 pˆ n x Z n Unknown s E t x n s x t x n Interval Estimate of a Population Proportion p 1 p n pZ Interval Estimate of the Difference Between Two Population Proportions p1 p2 Z p1 1 p1 p2 1 p2 n1 n2 Z n Sample Size for an Interval Estimate of Population Proportions and Normal Distributions 2 * * Z pq n E2 2 2 E2 -5- 2 x Statistics Formulas Z-Value Formulas for Normal and “Approximately Normal” Distributions – Calculations of Test Statistics Z x x t Known x 0 Unknown Sx n n Test Statistic for Hypothesis Tests About Two Independent Proportions p p 1 1 p 1 p n n Z 1 2 1 2 Standard Error of x1 x2 x x 1 12 2 n1 22 n2 Degrees of Freedom for the t Distribution Using Two Independent Random Samples 2 s12 s22 n1 n2 d. f . 2 2 2 2 1 s 1 s 1 2 n1 1 n1 n2 1 n2 Test Statistic for Hypothesis Tests About Two Independent Means; Population Standard Deviations Unknown t x x D 1 2 0 s12 s22 n1 n2 Test Statistic for Hypothesis Tests Involving Matched Samples t d d sd n -6- Statistics Formulas Mean Difference Involving Matched (or Dependent) Samples d d i n Standard Deviation Notation Used for Matched Samples d d 2 sd i n 1 Interval Estimate of Means of Matched Samples d t / 2 S dn Test Statistic for Hypothesis Tests About a Population Variance 2 n 1s 2 2 d. f . n 1 2 0 Test Statistic for Hypothesis Tests About Two Population Variances when 12 22 s12 F 2 s2 df numerator n1 1 and df denominator n2 1 Interval Estimate of the Difference Between Two Population Means: σ1 and σ2 Known and Unknown x1 x2 Z / 2 12 n1 22 n2 (σ known) x1 x2 t / 2 S12 S 22 n1 n2 (σ unknown) Pooled Sample Standard Deviation (Assumptions that σ1 and σ2 are equal and populations are approximately normal) & Test Statistic for Comparison of Independent Samples Sp n1 1S12 n2 1S22 n1 n2 2 d . f . n1 n2 2 -7- t x x D 1 Sp 2 1 1 n1 n2 0 Statistics Formulas Chi-Square Goodness of Fit Test Statistic 2 f o f e 2 fe ; d . f . (c 1) Chi-Square Test for Independence Test Statistic 2 i f oij f eij 2 f eij ; d . f . (c 1)(r 1) j Testing for the Equality of k Population Means—ANOVA Sample Mean for Treatment j nj xj x ij i 1 nj Sample Variance for Treatment j x nj s 2j ij x j i 1 2 n j 1 Overall Sample Mean (Grand Mean) k x nj x j 1 i 1 nT ij x k Mean Square Due to Treatments MSTR Sum of Squares Due to Treatments k SSTR k 1 SSTR n j x j x j 1 Mean Square Due to Error SSE - 8 -nT k MSE 2 Statistics Formulas Sum of Squares Due to Error SSE n j 1s 2j k j 1 Test Statistic for the Equality of k Population Means F MSTR MSE Total Sum of Squares nj k SST xij x j 1 i 1 Partitioning of Sum of Squares 2 SST SSTR SSE Multiple Comparison Procedures Test Statistic for Fisher’s LSD Procedure t xi x j MSE 1 ni n1j Fisher’s LSD LSD t / 2 MSE Completely Randomized Designs Mean Square Due to Treatments n x k MSTR j j 1 j 1 ni n1j x k 1 Mean Square Due to Error n k MSE j 1 j 1s 2j nT k -9- 2 Statistics Formulas F Test Statistic F MSTR MSE Randomized Block Designs Total Sum of Squares b k 2 SST xij x i 1 j 1 Sum of Squares Due to Treatments k SSTR b x j x j 1 2 Sum of Squares Due to Blocks b SSBL k x i x i 1 Sum of Squares Due to Error 2 SSE SST SSTR SSBL Factorial Experiments Total Sum of Squares a b r SST xijk x i 1 j 1 k 1 Sum of Squares for Factor A a SSA br x i x i 1 2 Sum of Squares for Factor B b SSB ar x j x j 1 - 10 - 2 2 Statistics Formulas Sum of Squares for Interaction a b SSAB r x ij x i x j x i 1 j 1 Sum of Squares for Error SSE SST SSA SSB SSAB Simple Linear Regression Formulas Simple Linear Regression Model Simple Linear Regression Equation y 0 1 x E y 0 1 x Estimated Simple Linear Regression Equation yˆ b0 b1 x Least Squares Criterion min 2 ˆ y y i i Slope and y-Intercept for the Estimated Regression Equation b1 x x y y x x i i 2 i b0 y b1 x Total Sum of Squares SST yi y Sum of Squares Due to Error 2 SSE yi yˆ i 2 Sum of Squares Due to Regression SSR yˆi y - 11 - 2 2 Statistics Formulas Total Sum of Squares for Regression SST SSR SSE Coefficient of Determination r2 SSR SST Sample Correlation Coefficient rxy sign of b1 Coefficien t of Determinat ion sign of b1 r 2 Mean Square Error (Estimate of σ2 ) s 2 MSE SSE n2 Standard Error of the Estimate s 2 MSE Standard Deviation of b1 b 1 SSE n2 x x 2 i Estimated Standard Deviation of b1 sb1 s x x 2 i t Test Statistic b1 t sb1 Mean Square Regression MSR SSR # independen t variable s - 12 - Statistics Formulas F Test Statistic F Estimated Standard Deviation of MSR MSE ŷ p x x 1 n x x 2 s yˆ p s Confidence Interval for p 2 i E y p yˆ p t / 2 s yˆ p Estimated Standard Deviation of an Individual Value x x 1 1 n x x 2 sind s p 2 i Prediction Interval for yp yˆ p t / 2 sind Residual for Observation i yi yˆ p Standard Deviation of the i th Residual s yi yˆi s 1 hi Standardized Residual for Observation i Leverage of Observation i ˆi yi y s yi yˆ i 2 xi x 1 hi n xi x - 13 - 2 Statistics Formulas Nonparametric Method Formulas Mann-Whitney-Wilcoxon Test (Large Sample) Mean : T 1 2 n1 n1 n2 1 Standard Deviation: T n n n1 n2 1 1 12 1 2 Kruskall-Wallis Test Statistic k Ri2 12 W 3 nT 1 nT nT 1 i 1 ni Spearman Rank –Correlation Coefficient rs 1 6 di2 n n2 1 - 14 -