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Bivariate Data Analysis: Summation and Sum of Squares Calculations

This document outlines the process for calculating key summations and the sum of squares in bivariate data analysis using a dataset of tattoos (x) and piercings (y) recorded at a concert. It includes detailed steps for deriving the five essential summations: Σx, Σy, Σx², Σxy, and Σy², and the calculation of the sums of squares SS(x), SS(y), and SS(xy). Understanding these calculations is crucial for further statistical analysis and interpretations in the context of bivariate relationships.

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Bivariate Data Analysis: Summation and Sum of Squares Calculations

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  1. Bivariate Data ~ Preliminary Calculations x y 2 5 8 7 5 6 3 4 6 8 • Given: At a concert, the number of tattoos, x, and the number of piercings, y, that a person had was recorded: • Find: a)The 5 summations: x, y, x2, xy, and y2 b)The 3 sum of squares: SS(x), SS(y), SS(xy)

  2. Finding Summations x y 2x5=10 52 = 25 22 = 4 2 5 8x7=56 72 = 49 82 = 64 8 7 2 5 5 2 25 5 2 62 = 36 5x6=30 52 = 25 5 6 8 8 7 7 8 7 87 3x4=12 42 = 16 32 = 9 3 4 5 6 6 5 5 56 6 6x8=48 82 = 64 62 = 36 6 8 3 4 4 4 3 34 3 68 6 8 6 6 8 8 • Use a table format to find the extensions for each pair of data and the5 summations: y2 xy x2 x = 24 y = 30 x2 = 138 y2 = 190 xy = 156 1. Find the x and y by totaling the x and y columns 2. For x2, multiply each x by itself and total the column 3. For y2, multiply each y by itself and total the column 4. For xy, multiply each pair of x and y values and total the column Note: Save these 5 summations for future formula work

  3. The SS Formulas ~ Knowing the Parts (x)2 n x x2 - SS(x)= SS(x) x2 n • SS(x) is the “sum of squares for x”, a frequent factor in bivariate data analysis • x2 is the “sum of squared x’s”, the sum of all x-squared data • x is the “sum of x”, the sum of all x data • n is the “sample size”, the number of data

  4. The SS Formulas ~ Knowing the Parts (y)2 n y y2 - SS(y)= SS(y) y2 n • SS(y) is the “sum of squares for y”, a frequent factor in bivariate data analysis • y2 is the “sum of squared y’s”, the sum of all y-squared data • y is the “sum of y”, the sum of all y data • n is the “sample size”, the number of data

  5. The SS Formulas ~ Knowing the Parts (x)(y) n xy - SS(xy)= (x)(y) xy SS(xy) n • SS(xy) is the “sum of squares for xy”, a frequent factor in bivariate data analysis • xy is the “sum of xy”, the sum of all xy products • (x)(y) is the product of the two summations, x and y • n is the “sample size”, the number of data

  6. Finding the 3 Sum of Squares x = 24 y = 30 n = 5 x = 24 y = 30 x2= 138 y2= 190 xy = 156 n = 5 n = 5 (24)2 5 24 30 5 (x)2 n SS(x) = x2 - = = 190 138 138 - 138 - 115.2 = 22.8 5 (30)2 5 (y)2 n (y)2 n y2 - SS(y) = y2 - = = 190 - 190 - 180 = 10 (24)(30) 5 (24)(30) (x)(y) n xy- SS(xy) = = = 156 156 - 190 - 180 = 10 5 (x)(y) n xy- (x)2 n x2 - • The summations from the table: x = 24 y = 30 x2= 138 y2= 190 xy = 156 n = 5 Note: Do not round the SS values, round after next calculation

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