1. Statistics 11 Correlations Definitions: A correlation is measure of association between two quantitative variables with respect to a single individual A correlation coefficient is a descriptive statistic that quantifies the degree of the association between two variables

2. Statistics 12 Types of Correlations: Positive: high values of one variable are associated with high values of the other variable Negative: high values of one variable are associated with low values of the other variable Zero: values of one variable are not associated with the values of the other variable Perfect: each value of one variable is associated with only a single value of the other variable and plot a straight line

3. Statistics 13 The same set of n = 6 pairs of scores (X and Y values) is shown in a table and in a scatterplot. Notice that the scatterplot allows you to see the relationship between X and Y.

4. Statistics 14 Examples of positive and negative relationships. (a) Beer sales are positively related to temperature. (b) Coffee sales are negatively related to temperature.

5. Statistics 15 (a) shows a strong positive relationship, approximately +0.90; (b) shows a relatively weak negative correlation, approximately –0.40; (c) shows a perfect negative correlation, –1.00; (d) shows no linear trend, 0.00.

6. Statistics 16 Calculating the Correlation

7. Statistics 17 Coefficient of Determination After calculating a Pearson Product-moment Correlation Coefficient you can go a step further by calculating the Coefficient of Determination which indicates how much of the variability in one variable is proportional to the variability in the other variable.

8. Statistics 18 Definition: The coefficient of determination is a measure of the proportion of variance that can be accounted for in one variable because of its association with another variable Calculation: Square the Pearson Product-moment Correlation Coefficient Coefficient of Determination = r ²

9. Statistics 19 Example: Given a correlation of r = +0.5 between IQ and reading speed The coefficient of determination (r ² ) says that 25% of the variation in the reading speed of your subjects is related to the variability in their individual IQ's Which also means that 75% of the variation in reading speed of your subjects is related to some other factor(s), i.e. 1 - r ²

10. Statistics 110 Final Words Correlations DO NOT EVER indicate causation The Pearson Product-moment Correlation Coefficient requires the measurement of two quantitative variables on each individual The Pearson Product-moment Correlation Coefficient is only applicable to LINEAR relations

11. Statistics 111 Parametric Statistical Tests Population parameters are specified –Shape: i.e. normal –Variance: i.e. equal Interval scale of measurement NonParametric Statistical Tests Do not specify the parameters of population Most require only an ordinal scale of measurement

12. Statistics 112 Statistical Tests DataDesignOne-SampleTwo-SampleK-Sample Nominal BetweenChi-Square One Sample Test Chi-Square two sample test Within Ordinal Between Runs Test Mann-Whitney U Test Kruska- Wallis one- way Anova WithinWilcoxon matched-pairs signed ranks Friedman two-way Anova

13. Statistics 113 Functional and rational of test Calculation and interpretation –General –Specific example

14. Statistics 114 Function Assume a data set that can be arranged in mutually exclusive categories Question is whether the number of occurrences in each category is different than what would be expected by chance if the null hypothesis were true For example: –Modes of play in children –Opinions about gun control

15. Statistics 115 Chi square one sample test will allow you to determine whether your observations are different than would be expected by chance Are there more aggressive children in this sample than what one would expect from a random sample from the population Are there fewer people in this sample in favor of gun control than would be expected from a random sample from the population

16. Statistics 116 Rational and Method Basically, one sample chi square test compares Observed frequencies –number of observed occurrences within each category Expected frequencies –number of occurrences within each category expected by chance if null hypothesis is true

17. Statistics 117 The formula to accomplish this comparison: Where: –O = observed frequencies –E = expected frequencies

18. Statistics 118 The logic of the test is then simple If the differences between O and E are small, chi- square will be small If the differences between O and E are large, chi- square will be large And if chi-square is large enough –Your conclusion will be that the observed frequencies are such that your sample does not come from the population from which the null hypothesis was derived –i.e., you reject the null hypothesis

19. Statistics 119 Evaluate chi-square using the chi-square distribution for your chosen alpha level Using degrees of freedom equal to (k-1) Reject, at your alpha level, if observed chi- square is greater than tabled value

20. Statistics 120 Question: Does "post position" (1-8) make and difference in the outcome of horse racing? Data: Observe 144 races and record starting post position of winner

21. Statistics 121

22. Statistics 122 Summary Calculate chi square Determine df, i.e. k-1 Evaluate null hypothesis with regard to chi square table Interpretation:

23. Statistics 123 The End