Statistical Symbols and Formulas Taken from: http://www2.uta.edu/ssw/Basham/Statistical Symbols and Formulas.doc Statistical Symbols The summation sign = Mean = M t distribution for rejecting or accepting the null hypothesis = t Beta weight or beta coefficient = Z Score = z Chi square = 2 Coefficient of variation = V Standardized scores, and Z distribution = Z Degrees of freedom = d Correlation = r Frequency = Point Biserial Correlation = rpb Gamma (also G) = Statistical significance = p Kendall’s tau-b = b Statistical significance = Sig. Lambda ( also called the Guttman coefficient of predictability) = The 95% Confidence Interval = 95%CI T-test = t Level of significance = Pearson’s correlation coefficient = r Population mean score = One Way ANOVA or Comparison Test of Between Group Differences = F Correlation Coefficient for ANOVA (3+ categories) = eta Population size = N Population variance = 2 ANOVA test of significance of difference among means = Tukey-b procedure Rho = Sample size = n Multiple correlation coefficient symbol ( 2 or more I.V.). = R Sample variance = s2 In SPSS Program Standard deviation or standard error of sample = s Beta = (Standardized regression slope). Slope based on standard scores = average change in D.V. in S.D. units, associated with 1 S.D. increase in I.V. (1 point increase in I.V. = 1 S.D.) Standard Deviation = SD Standard Error of the mean = SE B = (Unstandardized regression slope). Slope based on raw scores = Indicates 1 Statistical Symbols and Formulas average change in D.V. for 1 point increase in I.V. = Beta weight or beta coefficient Variance Measures in a Population (Simple Distribution and Frequency Distribution) R2 change = tells whether or not variables entered at that point add anything over and above variables that have been added previously. Population Standard Deviation ___ = 2 , or Variance = X2 X X2 Covariance (of x and y) = XY (Simple Distribution) Statistical Formulas Population Variance Measures of Central Tendency 2 = 1( X2 – ( X )2 N–1 (Simple Distribution) (Frequency Distribution) Mean = = N Population Mean = X X Population Variance 1 n Xi n i 1 2 = 1 ( fX2 – ( fX )2 N N–1 (Frequency Distribution) Variance = Mean = = f N Sum of squared distributions from the mean for all cases (number of cases –1) or, Mdn = simple distribution = center score odd population Mdn = simple distribution = Even population 2 center scores divided by 2 (X-M)2 N–1 Note: col:[X M X-M (X-M)2] (X-M)2 N–1 or, Mdn = frequency distribution = LRL – (PN-CFL .h) FI X 1 n 2 1 n X i X i n i 1 n i 1 2 Mode = Most Frequent Score 2 Statistical Symbols and Formulas Variance Measures in a Sample (Simple Distribution and Frequency Distribution) Standard Error of The Mean Formula: SEM = SD N–1 Population Standard Deviation ___ s = s2 Confidence Interval at 95% Formula: (Simple Distribution) Calculating Percent Formula: Sample Variance 3 x X = 3x100 = 300/6 = x = 50 6 100 s2 = 1( x2 – ( x )2 n–1 (Frequency Distribution) M + (1.96)(SEM) Z Statistic Formula: Z=M- m Sample Variance s2 = 1 ( fx2 – ( fx )2 n n–1 m = SD N –1 t-test Statistic Formula: Statistical Average Formula: M + 1 = Mean + 1 Standard Deviation t = M1 – M2 SE diff Raw Score Transformation Formula: Confidence Interval for t-test Formula: Z Standard Scores CI = M1 – M2 + (?) (SE diff) scores – mean = X - M standard deviation Point Biserial Correlation Coefficient: ________________ Probability Density Formula: M + 1SD = 68% M + 1.96SD = 95% Coefficient of Variation Formula: Coefficient of Variation = SD x 100 Mean rpb = _____t2_______ t2 + ( n1 + n2 – 2) Note: Used for t test for independent groups Eta Correlation Coefficient Formula: (for ANOVA 3+ Categories) eta2 = SS Between / SS Total, or between = eta2 total ____ 3 Statistical Symbols and Formulas eta = eta2 Formula for Characterizing a Straight Line: D Index Calculation Formula: (Measure of effect size) y = a + Bx Difference between 2 group means Avg. SD of the 2 groups y = Predicted value of Dependent Variable (D.V.) a = Intercept value of D.V. when the Independent Variable (I.V.) = 0 B = Slope = Average change of D.V. associated with a 1 point increase in the I.V. x = Value of I. V. Note: Used in Meta Analysis (or SD of Control group) Note: less accurate Formula for Calculating Covariance of two variables (x and y): Covariance = XY Residual = for a particular person their actual value minus their predicted value. Therefore, XY 1 n 1 n n X Y X Yi i i i n i 1 n i 1 i 1 The coefficient of correlation of X an Y is then stated as: XY Then to get a better measure of correlation calculate: Formula for Calculating a Residual: Residual = D.V. – y Residual = actual value of D.V. – predicted value. Conversion Formula for Standard Scores: Bivariate Value of I.V. _ -Mean of I.V. or, z = x - x S.D. of I.V. S.D. x 1 n n X iYi X i Yi n i 1 i 1 i 1 n XY 2 2 n 2 1 n n 2 1 n X i X i Yi Yi n i 1 i 1 n i 1 i 1 Formula for Calculating Covariance Matrix: 2 C X XY a 2x2 Formula for a Bivariate Normal Distribution: 4 0 0 XY Y2 Formula for Calculating the Eigenvalues of the Covariance Matrix: p det I C Note: Does not change shape of distribution. XY 2 X XY Y2 e ( x 2 y2 ) dx dy 1 Formula for Calculating the Slope of a Regression Line: ˆ X iYi nX Y X 2 i nX 2 4 Statistical Symbols and Formulas Multivariate Interaction Effects Multivariate ANCOVA (Computation Formula: formula for adjusted means): Computation formula for data points = Y=a+X1(pre)B(pre) + X2(group)B(group) Y=a+X1B1 + X2B2 + X12B12 Example: Yd=a+Xagbag + Xincbinc + X(ag)(inc)b(ag)(inc) Constant = a Unit of associated change =1, or 0 =X Predicted value of 1st slope = bag Predicted value of 2nd slope = binc Predicted value of score = Y Spearman Correlation Formula: rs = 1- 6D2 n(n2 – 1) Pearson Correlation Formula: r = _SP_ SSxSSy Intercept Constant = a Descriptive mean of pretest condition = X1 Unit of associated change =Experimental group =1, or Control group =0 =X2 Predicted value of 1st slope = B(pre) In Bivariate regression only: ( see SPSS symbol definitions below). Beta = r B, or Beta = r Note: r B Predicted value of 2nd slope = B (post) Predicted value of score = Y adjusted experimental mean = Y adjusted control mean = __ __ where SP = (X – X)(Y – Y) = Logistic Regression Odds Ratio XY - (X) (Y) n Formula: 100 (OR – 1) = % Change Partial Correlation Ratio Formula: r2 = a a+d (i.e. 100 (.5 – 1) = 50% decrease in odds of participation) Chi-Square Statistic Formula: Part Correlation Ratio Formula: r2 = a a+b+c+d 2 = ( o - e)2 e 5 Statistical Symbols and Formulas To be continued … 6