Descriptive Statistics: Analyzing Data and Relationships

1. 14 Descriptive Statistics

2. Descriptive Statistics • The goal of descriptive statistics is to describe sample data • Can be contrasted with inferential statistics where the goal is to make inferences about populations from sample data

3. Frequency Distributions • A listing of values in a data set along with their frequency

5. Graphic Representations of Data • Bar graph • used with categorical variables • height of bar represents frequency of category • bars should not touch • Histogram • used with quantitative variables • no space between bars

6. Figure 14.2 A bar graph of undergraduate major.

7. Figure 14.3 Histogram of starting salary.

8. Graphic Representations of Data (cont'd) • Line graphs • also used with quantitative variables • particularly useful for interpreting interactions • Scatterplots • depicts relationship between two quantitative variables

9. Figure 14.5 Line graph of results from pretest–posttest control group design studying effectiveness of social skills treatment.

10. Figure 14.6 A scatterplot of starting salary by college GPA.

11. Measures of Central Tendency • Provide a single value that is typical of the distribution of scores • mode • most frequently occurring value • least useful measure of central tendency • median • middle score when numbers are in ascending or descending order

12. Measures of Central Tendency (cont'd) • Provide a single value that is typical of the distribution of scores • Mean • arithmetic average • most commonly used measure of central tendency

13. Measures of Variability • Provides a numerical value indicating the amount of variation in a group of scores • range • highest score minus lowest score • rarely used as a measure of variability • variance • average deviation of the data values from their mean in squared units

14. Measures of Variability (cont'd) • Provides a numerical value indicating the amount of variation in a group of scores • standard deviation • square root of variance • roughly the average amount that individual scores deviate from the mean

15. Measures of Variability (cont'd) • Provides a numerical value indicating the amount of variation in a group of scores • standard deviation and the normal curve • z-scores • standardized values transformed from raw scores • mean of z-distribution is always zero; standard deviation always 1

16. Measures of Variability (cont'd) • Provides a numerical value indicating the amount of variation in a group of scores • z-scores • indicates how far above or below a raw score is from its mean in standard deviation units; e.g., a z-score of +1.00 indicates a raw score that is one standard deviation unit above the mean • in a normal distribution, the proportion of scores occurring between any two points can be determined

17. Figure 14.8 Areas under the normal distribution.

18. Examining Relationships Among Variables • Unstandardized difference between means • a comparison of mean differences between levels of a categorical independent variable • Standardized difference between means • effect size • Cohen’s d is a common measure of effect size • mean difference is divided by standard deviation • small, medium, and large effect sizes are indicated by values of at least .2, .5, and .8 respectively

19. Examining Relationships Among Variables (cont'd) • Correlation Coefficient • numerical representation of the strength and direction of relationship between two variables • value ranges from +1.0 to -1.0; absolute value indicates strength of relationship; sign indicates direction • positive correlation indicates that the two variables vary together in the same direction; negative correlation means that they move in opposite directions

20. Examining Relationships Among Variables (cont'd) • Correlation Coefficient • Pearson correlation (r) used with two quantitative variables; only appropriate if data is related in a linear fashion • partial correlation is a technique that involves examining correlation after controlling for one or more variables • a scatterplot can be used to judge the strength and direction of a correlation

21. Figure 14.10 Correlations of different strengths and directions.

22. Regression Analysis • Statistical technique designed to predict dependent variable based on one or more predictor values • simple regression involves the use of one independent or predictor variable • multiple regression involves two or more independent or predictor variables • prediction is made using the regression equation

23. Regression Analysis (cont'd) • Statistical technique designed to predict dependent variable based on one or more predictor values • prediction is made using the regression equation • y-intercept - point where regression line crosses y-axis • regression coefficient - predicted change in the dependent variable (Y) given a one unit change in the independent variable (X)

24. Regression Analysis (cont'd) • Statistical technique designed to predict dependent variable based on one or more predictor values • prediction is made using the regression equation • partial regression coefficient

25. Contingency Tables • Table used to examine relationship between two categorical variables • Cells may contain frequencies or percentages