1. Welcome to… The Exciting World of Descriptive Statistics in Educational Assessment!

2. Numerical Scales Type 1.Nominal scale-Uses numbers for identification 2.Ordinal scale- Uses numbers for ranking 3.Interval scale- Uses numbers for ranking when units are equidistant 4.Ratio scale-This scale has qualities of equidistant units and absolute zero.

3. Descriptive Statistics- Statistics used to organize and describe data Type Measures of Central Tendency- Statistic methods for observing how data cluster around the mean Normal Distribution-A symmetrical distribution with a single numerical representation for mean, median, and mode Measures of Dispersion-Statistical methods for observing how data spread from the mean.

4. Counting the Data-Frequency Look at the set of data that follows on the next slide. Each time a score occurred, a tally mark was made to count it Which number most likely represents the average score? Which number is the most frequently occurring score?

5. Frequency Distribution Scores 100 99 98 94 90 89 88 82 75 74 68 60 Tally 1 11 1111 1111 11 1111 1111 1 11 1 Frequency 1 2 5 7 10 6 2 1 Average Score? Most Frequent Score?

6. Tally 1 11 1111 1111 11 1111 1111 1 11 1 This frequency count represents data that closely represent a normal distribution.

7. Frequency Polygons Data 100 89 99 89 98 89 94 88 90 75 90 74 90 68 90 60 5432154321 60 68 74 75 88 89 90 94 98 99 100 Scores Frequency

8. Measures of Central Tendency Mean, Median, and Mode Mean- The arithmetic average of a set of scores Median-The middlemost point in a set of data Mode- The most frequently occurring score in a set of data

9. Mean - To find the mean, simply add the scores and divide by the number of scores in the set of data. 98 + 94 + 88 + 75 = 355 Divide by the number of scores: 355/4 = 88.75

10. Median-The Middlemost point in a set of data Data Set 1 100 99 98 97 96 90 88 85 80 79 Data Set 2 100 99 98 97 86 82 78 72 70 68 Median 96 Median

11. Mode-The most frequently occurring score in a set of data. Find the modes for the following sets of data : Data Set 3 99 89 75 Mode: Data set 4 99 88 87 72 70 Mode:

12. Measures of Dispersion Range- Distance between the highest and lowest scores in a set of data. 100 - 65 = 35 RANGE

13. Variance-Describes the total amount that a set of scores varies from the mean. 1. Subtract the mean from each score. When the mean for a set of data is 87, subtract 87 from each score.

14. 100 - 87 = 13 98- 87 = 11 95- 87 = 8 91- 87 = 4 85- 87 = -2 80- 87 = -7 60- 87 = -27 2. Next-Square each difference (multiply each difference by itself) 13 x 13 = 169 11 x 11 = 121 8x 8 = 64 4x 4 = 16 -2 x -2 = 4 -7 x -7 = 49 -27x -27 = + 729 3. Sum these Sum of squares VARIANCE (Continued)

15. 4. Divide the sum of squares by the number of scores. _____divided by_____ = ______ VARIANCE (Final Step) VARIANCE

16. 1. To find the standard deviation, find the square root of the variance. Standard Deviation-Represents the typical amount that a score is expected to vary from the mean in a set of data. √_____ = ______ Std. Deviation

17. 1. find the deviation score (x-M) 2. Divide deviation score by standard deviation Z score- represents the score in terms of standard deviation units (x-M)/SD = Z score

18. Properties of a Normal Distribution or The “Bell Curve” The mean, median, and mode are represented by the same numerical value. Both sides of the curve are symmetrical Anything that occurs naturally and is measurable will be distributed in a normal distribution when sufficient data are collected.

19. How “normal” is the population?

20. Properties of a Skewed Distribution Positively Skewed: More scores fall below the mean Median occurs below the mean Mode occurs below the median Negatively Skewed: More scores fall above the mean Median occurs above the mean Mode occurs above the median Watch out for extreme scores!

21. Z Scores- derived scores that are expressed in standard deviation units A student received a Z score of 2 on a recent statewide exam. What does the 2 mean? This indicates that the student’s score was 2 standard deviations above the mean for that test.