1. Statistical Evaluation of Data Chapter 14 George S. Robinson, Jr., Ph.D. Department of Psychology North Carolina A&T State University

2. Two General Categories of Statistical Techniques • Descriptive statistics • are methods that help the researchers to organize, summarize, and simplify the results • Inferential statistics • are methods that use the results obtained from samples to help make generalizations about populations

3. Statistics Terminology • Statistic • a summary value that describes a sample • (e.g., mean of the sample, standard deviation of the sample) • Parameter • a summary value that describes a population • (e.g., mean of the population, standard deviation of the population)

4. Descriptive Statistics • Frequency distribution • frequency of the number of cases in each category • Frequency distribution table • Frequency distribution graphs • histogram • polygon • bar graph

5. Frequency distribution table

6. Frequency distribution graphs - Histogram

7. Frequency distribution graphs - Bar graph

8. Frequency distribution graphs - Pie chart

9. Frequency distribution table - 2

10. Frequency distribution graphs - Histogram - 2

11. Frequency distribution graphs - Bar graphs - 2

12. Frequency distribution graphs - Pie chart

13. Measures of Central Tendency • Central tendency • a statistical measure that identifies a single score that defines the center of a distribution • mean • median • mode • bimodal • multimodal

14. Measures of Variability • Variability • a measure of the spread of scores in a distribution • variance • the average squared distance from the mean • standard deviation • the square root of the variance; the average distance from the mean

15. Selecting a Descriptive Statistical Procedure

16. Using Graphs to Summarize Data • line graph • bar graph • histogram • pie chart • scatter plot

17. Stacked Bar Graph

18. Stacked Bar Graph - 2

19. Line Graph

20. Correlation • Correlation • describes the degree of the relationship and direction between two variables • the direction of the relationship • positive • negative • form of the correlation • Pearson r (interval or ratio data) • Spearman r (ordinal data) • strength of the relationship • -1 to +1

21. Correlation Matrix

22. Correlation - Scatter Plot

23. Inferential Statistics • Inferential statistics • Making inferences (generalizations) from a sample to a population • Sampling error • Naturally occurring difference (error) between a sample statistic and the corresponding population parameter

24. Hypothesis Tests • Hypothesis test • A procedure that determines whether the sample data provide enough evidence to conclude that the original hypothesis is correct

25. Five Elements of a Hypothesis Test • The Null Hypothesis • States that there is no difference, no effect, or no relationship • The Sample Statistic • Data from the study (e.g., sample means) • The Standard Error • The average difference between a sample statistic and a corresponding population parameter • The Test Statistic • A summary value that measures the degree to which the sample data are in accord with the null hypothesis • Test statistic = sample statistic / standard error  large value (greater than one) leads to rejecting the null hypothesis • The Alpha Level (level of significance) • The maximum probability that the results were due to chance (e.g., alpha level of 0.05 means there is a 0.05 probability the sample results are due to chance; or there is a 95% probability the results are not due to chance

26. Reporting Results from a Hypothesis Test • Statistically significant (significant results) • Extremely unlikely that the results were due to chance • P-values • p < .05 (less than) • p = .032 (actual value) • Reject the null hypothesis if the p-value is less than the previously specified alpha level • Results are not due to chance • Report the results as significant • Report the actual p-value • Fail to reject the null hypothesis if the p-value is greater than the previously specified alpha level • Results are probably due to chance • Report the results as not significant\ • Report the actual p-value

27. Errors in Hypothesis Testing • Type I errors • Occurs when the researcher finds a significant result when actually there is no effect in the population • The researcher selected an unusual sample and incorrectly concludes that there is a significant effect • The p-value represents the probability of a Type I error • Type II errors • Occurs when the researcher does not find a significant result when actually there is a real effect in the population • The effect was too small to show up in the sample • 1 – p-value represents the probability of a Type II error

28. Measures of Effect Size • Simply report the results • Cohen’s d - (used with t-test) = sample mean difference / sample standard deviation • 0 < d < 0.2 small effect (mean difference less than 0.2 standard deviation • 0.2 < d < 0.8 medium effect (mean difference around 0.5 standard deviation) • d > 0.8 large effect (mean difference more than 0.8 standard deviation) • r2 – correlation (correlation squared) • Phi-squared (F2) and Cramer’s V (chi-square) • Eta-squared - ANOVA

29. Examples of Hypothesis Tests • Tests for mean differences • Two-group between-subjects test • Independent-measures t test • H0 = there is no difference in the number of life-time sexual partners between males and females • H1 = there is a difference between the number of life-time sexual partners between males and females • t(468) = 4.88, p = .000 468 = degrees of freedom (df); 4.88 = t value; .000 = significance level (2-tailed) • Repeated-measures t test • reported the same

30. Examples of Hypothesis Tests – cont. • More than two groups single-factor analysis of variance (one-way analysis of variance) • One-way ANOVA • H0: there is no difference in the number of hours watching BET between freshmen, sophomores, juniors, seniors, graduate students, or others • H1: there is a difference in the number of hours watching BET between freshmen, sophomores, juniors, seniors, graduate students, or others • F(5, 480) = 2.82, p = .016

31. Examples of Hypothesis Tests – cont. • Correlation (Spearman) • H0: there is no relationship between the risk of female unprotected oral sex and male unprotected oral sex • H1: there is a relationship between the risk of female unprotected oral sex and male unprotected oral sex • rs = 0.75, N = 469, p = .000 • The same for Pearson correlation except use r

32. Examples of Hypothesis Tests – cont. • Test for proportions • Chi-square test for independence (Crosstabs) • H0: there is no relationship between gender and music preference • H1: there is a relationship between gender and music preference • X2(8, N = 487) = 78.85, p = .000