1. Lesson Fourteen Interpreting Scores

2. Contents

3. Five Questions about Test Scores 1. The general pattern of the set of scores  How do scores run or what do they look like?The general pattern of the set of scores 2. What ’ s the typical level of performance? What does the typical individual look like? 3. How do scores spread out away from the average value? 4. What does an individual score on a test mean? 5. How are two sets of scores related?

4. General Pattern of Scores What do scores look like? –List all scores, but no sense of the scores. –List scores from lowest to highest  “ order ”, out of chaos, but if too many scores, it ’ s difficult to list. –List scores with frequency marked  reduce the length of the distribution, but still have a long table if N is big. –Group the scores into intervals (i.e., grouped frequency distribution)  ideal: 10 ~15 groups –Convert grouped frequency distribution into a graphical representation

5. General Pattern of Scores Frequency distribution: a table that shows how often each score has occurred Grouped frequency distribution: normally 10 ~ 15 groups Graphic representation –Histogram (bar graphs) (Bailey 94) –Frequency polygon (Bailey 95)

6. Frequency Distribution Score f 2 4 5 6 7 9 11 12 14 121211222121211222

7. Grouped Frequency Distribution Scores f 46-49 42-45 38-41 34-37 : 14-17 10-13 6-9 2-5 1152:54441152:5444

8. Graphic Representation Histogram Bar graph Frequency polygon

9. Measures of Central Tendency (1) 2. What ’ s the average level of performance?  measures of central tendency –To identify the typical score (i.e., the “ middle ” of the group) Mean (x bar ) Median (m) Mode (M)

10. Central Tendency (2) Mean –The sum of all scores divided by the total number of the scores –Arithmetic average –The most commonly used –Mean =  X N

11. Central Tendency (3) Median –The middle score that separates the scores into half –There is the same number of scores above and below it (but the list of scores has to be in order, e.g., from low/high to high/low) –The 50 th percentile (p 50 ) –The middle score –e.g., a set of scores: 10, 5, 3  m = 5 –20, 9, 7, 6, 1  m =? – 15, 7, 6, 3  m = ? –Location of the median: (N+1)/2 e.g., N= 15, (15+1)/2 = 8  median = 8 th score

12. Central Tendency (4) Mode: –the score that occurs most frequently –E.g., 18-21= the modal group (in the previous example) –Possible to have bimodal (or even trimodal) groups

13. Measures of Variability (1) 3. How spread out or how close together are the scores? Or how “ widely ” scores spread out around their center? Measures of dispersion (or variability) –Range –Variance –Standard deviation

14. Measures of Variability (2) Range –The difference between the highest and the lowest scores –The highest score – the lowest score (+ 1) –Depending only on two extreme cases, so not very dependable

15. Measures of Variability (3) Variance (s 2 ) A measure of dispersion around the mean (Bailey 92) An indicator of the spread of scores in a distribution S 2 =  (x-x bar) 2 /N - 1

16. Measures of Variability (4) Standard deviation (s) The square root of variance Average of the differences of all score from the mean e.g., two sets of scores –3, 4, 5  mean = 4 –1, 4, 7  mean = 4 –2 nd set is more spread out  larger s s>=0

17. Measures of Variability (5) Under normal distribution, a constant relationship exists between s and the proportion of cases: (Bailey 105) –between mean and either +1s or – 1 s  34.1% cases (between +- 1 s  68.2%) –between mean and either +2 s or – 2 s  47.7% (i.e., between +- 2 s = 95.4%) –between mean and either +3 s or – 3 s  49.9% (i.e., between +- 3 s = 99.8%)

18. Interpretation of Individual Scores 4. What does an individual score on a test mean? What does it say about individual performance? How does this person stand relative to the group? Percentile –The score exceeds a specific fraction of the scores Standard deviation –Provides a common unit of distance from the mean of the group –Individual score can be expressed as a distance above or below the mean in the common unit

19. Measures of Relationship 5. To what extent are scores on two different tests go together? How are two sets of scores on different tests related? What ’ s the relationship between two measurements? Correlation coefficient (r): an index of degree of relationship Correlation only tells you that two things are related, but doesn ’ t mean causality.

20. Correlation -1<= r <= 1 –r: “ + ”  rank people in the same order; exact agreement. “ - ”  rank people in the reverse order; exact disagreement –r: how strong the relationship is (the strength) r= 1  perfect positive relationship –High on one test, must be high on the other r= 0  no relationship –No pattern; random r= -1  strong relationship, but in the opposite direction –High on one test, but low on the other

21. Correlation 3 settings that use correlation coefficients To determine how consistent a measurement procedure is –Info. on stability or reliability To study the relationship between 2 measures –To evaluate one as a predictor of the other Relationships between variables –To understand better how behavior is organized