Bivariate Statistics Y Nominal X Ordinal Interval Nominal
Bivariate Statistics Y Nominal X Ordinal Interval Nominal Ordinal Interval 2 Rank-sum Kruskal-Wallis H t-test ANOVA Spearman rs (rho) Pearson r Regression
November 2, 2009 Sir Francis Galton Karl Pearson http: //www. york. ac. uk/depts/maths/histstat/people/
Source: Raymond Fancher, Pioneers of Psychology. Norton, 1979.
A correlation coefficient is a numerical expression of the degree of relationship between two continuous variables.
Pearson’s r -1 r +1 -1 +1
Sample _ C XC sc n Sample _ D XD sd n Population Sample _ B µ Sample _ E XE se n n XB sb Sample _ A XA sa n
Sample. C r. XY Sample. D Population r. XY _ E Sample r. XY Sample. B r. XY Sample. A r. XY
Pearson’s r -1 r +1 -1 +1 Pearson’s r is a function of the sum of the cross-product of z-scores for x and y.
Pearson’s r r= zxzy N
Sample. C r. XY Sample. D Population r. XY _ E Sample r. XY Sample. B r. XY Sample. A r. XY
The familiar t distribution, at N-2 degrees of freedom, can be used to test the probability that the statistic r was drawn from a population with = 0 H 0 : XY = 0 H 1 : XY 0 where t= r N-2 1 - r 2
Some uses of r • Association of two variables • Reliability estimates • Validity estimates
Factors that affect r Non-linearity Restriction of range / variability Outliers Reliability of measure / measurement error
Pearson’s r -1 r +1 -1 +1 Pearson’s r can also be interpreted as how far the scores of Y individuals tend to deviate from the mean of X when they are expressed in standard deviation units.
Pearson’s r -1 r +1 -1 +1 Pearson’s r can also be interpreted as the expected value of z. Y given a value of z. X. tend to deviate from the mean of X when they are expressed in standard deviation units. The expected value of z. Y is z. X*r If you are predicting z. Y from z. X where there is a perfect correlation (r=1. 0), then z. Y=z. X. . If the correlation is r=. 5, then z. Y=. 5 z. X.
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