 # Cbs 221 Correlation and Regression

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Covenant University, Ota,

College of Development Studies,

CBS 221: Statistics for Business and Social Sciences II

2013/2014 Omega Semester

Week 12 Topic: Correlation and Regression

1. Objectives

By the end of this set of lectures the students should be able to:

1. Explain the concepts of Correlation and Regression
2. Solve problems on Correlation and Regression

INTRODUCTION

Correlation and regression analyses are vital tools for finding out the nature of relationships between variables. In correlation analysis, variables are studied simultaneously to identify how they are related while in regression analysis, a single variable is made the focus of the study while the remaining variables are studied for possible impacts on the variable of focus.

CORRELATION ANALYSIS

This means co-relationship between variables and it starts from finding out how variables are varying together which is termed covariance. The covariance of X and Y can be expressed as:

[pic 1] where n-1 = degree of freedom

The covariance of X and Y measures the linear dependence between X and Y. Thus, if X and Y are independent then Covxy = 0. Note that correlation is not synonymous causation, e.g. when a lecturer reports a high correlation between cultism and failure, it does not necessarily mean that cultism is the cause of failure in examinations. The correlation coefficient r ranges between -1 and +1 (i.e. -1< r < +1). Positive when variables are related, negative when they are not related. When the value is above 0.5 it is high, below 0.5 it is low and when it is 0.5 it is moderate. The correlation coefficient can be computed using the Karl Pearson’s Product Moment Correlation coefficient given as:

[pic 2]     OR

[pic 3]

Example 1:

 X 10 21 25 34 40 42 53 52 60 63 Y 11 12 12 15 22 25 20 20 32 31

The Spearman’s rank correlation coefficient is the one popular method of computing rank correlation coefficient and is derived from the Pearson’s correlation coefficient. The formula is given as:

[pic 4]

Where D is the difference in ranks between corresponding values of X and Y while N is the number of pairs of variables.

Solution:

 X Y [pic 5] [pic 6] [pic 7] [pic 8] [pic 9] 10 11 -30 -9 270 900 81 21 12 -19 -8 152 361 64 25 12 -15 -8 120 225 64 34 15 -6 -5 30 36 25 40 22 0 2 0 0 4 42 25 2 5 10 4 25 53 20 13 0 0 169 0 52 20 12 0 0 144 0 60 32 20 12 240 400 144 63 31 23 11 253 529 121 400 200 1075 2768 528

[pic 10];    [pic 11];    r = [pic 12]

Example 2:

Two judges were asked to rank 10 beauty competitors in order of their beauty. Their rankings are as follows:

 Competitor A B C D E F G H I J Rank by Judge X 2 3 7 1 8 5 10 6 9 4 Rank by Judge Y 1 4 8 2 7 6 9 5 10 3

Calculate the Spearman’s rank correlation coefficient and comment on the degree of agreement between the judges.

Example 3:

Nine students, A, B, C, D, E, F, G, H, I are ranked in terms of their beauty and character. The most beautiful and most well behaved are ranked as follows:

...

...

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