Introduction to Descriptive Statistics Objectives: Determine the general purpose of correlational statistics in assessment & evaluation “Data have a story to tell. Statistical analysis is detective work in which we apply our intelligence and our tools to discover parts of that story.”-Hamilton (1990)

Correlation Once you know: Once you know: –Middle –Spread –Shape –Relative position of specific cases It is now useful to know relationships between variables. It is now useful to know relationships between variables.

Correlation Direction of Relationships Direction of Relationships Positive or Negative Positive or Negative Magnitude of Relationships Magnitude of Relationships Weak, Moderate, Strong Weak, Moderate, Strong Scatterplots Scatterplots Outliers Outliers

Correlation Quantitative index of association Quantitative index of association Scaling of Pearson r Scaling of Pearson r –1 = perfect negative relationship –1 = perfect negative relationship 0 = no relationship 0 = no relationship +1 = perfect positive relationship +1 = perfect positive relationship Most common measure of association for interval and ratio variables Most common measure of association for interval and ratio variables

Examples Parent educational level and student academic achievement Parent educational level and student academic achievement Parent income or SES and student academic achievement Parent income or SES and student academic achievement Coping strategies and perceived stress Coping strategies and perceived stress

Correlation For positive correlations between two variables: For positive correlations between two variables: High values on x tend to be associated with high values on y High values on x tend to be associated with high values on y Low values on x tend to be associated with low values on y Low values on x tend to be associated with low values on y

r= NC State System Level Data

Correlation For negative correlations between two variables: For negative correlations between two variables: Low values on x tend to be associated with high values on y Low values on x tend to be associated with high values on y High values on x tend to be associated with low values on y High values on x tend to be associated with low values on y

r=-.613

r= NC State System Level Data

r= NC State System Level Data

Interpretation Guidelines Correlation is not causality. Correlation is not causality. Correlation is necessary for causal inference, but not sufficient. Correlation is necessary for causal inference, but not sufficient. Causal inference requires experimental designs. Causal inference requires experimental designs.

Interpreting the Correlation Coefficient Correlation does NOT imply causation!!! Correlation does NOT imply causation!!! Possible Explanations: 1.X causes Y 2.Y causes X 3.A third factor, or multiple extraneous factors, cause both X and Y XY Y X X a Y

Interpretation Guidelines Rum use and number of people entering the priesthood. Rum use and number of people entering the priesthood. Square footage of home and student academic achievement. Square footage of home and student academic achievement. Percent of women in a state who earn high salaries and percent of public officials who are women. Percent of women in a state who earn high salaries and percent of public officials who are women.

Interpretation Guidelines The third variable problem. The third variable problem. –SES and home size. The risk factor vs. causal agent problem. The risk factor vs. causal agent problem. –Length of time smoking and life expectancy. The direction of causality problem. The direction of causality problem. –Productivity and job satisfaction

Interpreting Magnitude

What would you expect? Perceived stress Perceived stress Depression Depression

r=.582

What would you expect? Depression Depression Self-acceptance Self-acceptance

r=-.596

Uses of Correlation In Assessment Inter-rater reliability Inter-rater reliability Split-half reliability Split-half reliability Construct validity Construct validity Concurrent validity Concurrent validity Convergent and Discriminant validity Convergent and Discriminant validity

An Example Using the PRI

Determine whether each statement is TRUE or FALSE: T/F1. A measure of central tendency is a summary score that represents a set of scores. T/F2. The mean is a score that occurs most frequently. T/F3. A distribution of scores may have more than 1 mode (bi- or multi- modal) or no mode (amodal). T/F4. The mode is the point in a distribution where 50% of the scores are above and 50% are below. T/F5. The exact median can always be computed by averaging the two middle scores together. T/F6. The median is calculated by dividing the total sum of the scores by the number of scores. T/F7. The mode is the measure of central tendency that is used the most. T/F8. The mean should be used with categorical data. T/F9. To describe a distribution, a measure of central tendency AND variability should be reported. T/F10. In groups with a narrow spread of scores, the range and SD are larger than in groups where scores spread out. Describing Data: Self Assessment

Fill in the blank with the appropriate descriptive statistic. 11. The _____ is the measure of central tendency that should be used with nominal data. 12. The _____ is not appropriate when extreme scores are present, because it will be misleading. In these cases, the ____ should be used _____ and _____ are most often used for descriptive statistics and not for inferential statistics The _____ is used for both descriptive and inferential statistics The _____ is the distance between the highest and lowest scores The _____ is the average distance of scores from the mean _____ is used most extensively to describe the variability of a distribution.