2. Chapter 6 Foundations of Educational Measurement Part 1 Jeffrey Oescher

3. Descriptive Statistics  Two things you need to know Four types of descriptive statistics How to interpret descriptive statistical results

4. Descriptive Statistics Tell me something about the following scores 33 31 31 31 32 29 30 30 32 30 32 30 28 29 30 29 31 31 28 28 29 30 27 30 29 30 30

5. Descriptive Statistics  The scores vary from 27 – 33.  The typical score appears to be somewhere around 30.  There seems to be about as many high scores as low scores.

6. Descriptive Statistics

7.  Four statistics we’ll study Central tendency Variability Relationship Standard scores

8. Descriptive Statistics  Central Tendency Mean: the arithmetic average Median: the score above and below which one-half of all of the scores in the distribution lie Mode: the most frequently occurring score(s)

9. Descriptive Statistics  Variation Standard deviation: the average deviation of all scores around the mean Variance: the average “squared” deviation of all scores around the mean Range: the difference between the highest and lowest scores in a distribution

10. Descriptive Statistics  Relationships Tell me something about the relationship between the following variables  GRE scores and graduate school performance  Misbehavior in class and achievement  Students’ height and their academic performance

11. Descriptive Statistics  Relationships Pearson correlation coefficient  Magnitude: 0 to 1 Weak 0.0 – 0.3 Moderate 0.4 – 0.7 Strong 0.8 – 1.0  Direction: positive (+) or negative (-)  Notation: r xy Other coefficients

12. Descriptive Statistics  Examples of correlations Weak positive +0.15 Weak negative –0.23 Moderate positive +0.42 Moderate negative –0.51 Strong positive +0.84 Strong negative –0.84

13. Interpreting Descriptive Statistics

14.  How would you describe the following relationships? +0.83 +0.65 +0.21 -0.83 -0.54 -0.03  Which relationship is stronger, +0.83 or - 0.83?  The following link provides the tables discussed above as well as the answers to the questions on the slides. Interpreting Descriptive Statistics

15. Descriptive Statistics  Properties of the normal curve Bell-shaped curve Symmetric around the mean Standard deviation units and proportions under the curve  Approximately 68% of all scores fall between ±1 standard deviation of the mean  Approximately 97% of all scores fall between ±2 standard deviations of the mean

17. Descriptive Statistics  Standard scores How good is an IQ of 130?  The mean of IQ scores is 100 with a standard deviation of 15 A z-score for an IQ of 130 is +2.00 This score is +2.00 standard deviations above the mean This is an excellent score as it is better than about 96% of all other scores How good is a SAT score of 400?  The mean for each section of the SAT is 500 with a standard deviation of 100 A z-score for a SAT score of 400 is –1.00 This score is –1.00 standard deviation below the mean This is not a good score as it is better than only 16% of all other scores

18. Descriptive Statistics  Z-scores A z-score represents the number of standard deviation units a specific score differs from the mean Z = [ x – X ] / s x No matter what the scale of your test or the nature of the scores, you can always convert a raw score to a z-score

19. Descriptive Statistics  Standard scores CEEB scores Normal Curve Equivalents (NCE) Stanines  Figure 7.11 on page 169 is a good reference for the relationships between several standard scores

21. Descriptive Statistics  Standard Scores Which score represents the best effort, 50 or 100?  What are the z-scores for each score if the first test has a mean of 40 and a standard deviation of 10 while the second has a mean of 125 and a standard deviation of 25?  What if the first test has a mean of 35 and a standard deviation of 15 while the second has a mean of 95 with a standard deviation of 5?