Analysis, Interpretation, and Use of Test Lesson 8.pptx

Assessment 1 lesson

1. Analysis, Interpretation, and Use of Test Data Lesson 8

2. Desired Significant Learning Outcomes Analyze, interpret, and use test data applying: a. Measures of central tendency Measures of variability Measures of position Measures of covariability

3. Measures of Central Tendency

4. 3 Measures of Central Tendency or Measures of Central Location Mean Median Mode

5. MEAN Arithmetic Mean FORMULA: The sum of all the scores The number of scores in the set Mean

6. Scores of 100 College Students in a Final Examination 53 30 21 42 33 41 42 45 32 58 36 51 42 49 64 46 57 35 45 51 57 38 49 54 61 36 53 48 52 49 41 58 42 43 49 51 42 50 62 60 33 43 37 57 35 33 50 42 62 49 75 66 78 52 58 45 53 40 60 33 46 45 79 33 46 43 47 37 33 64 37 36 36 46 41 43 42 47 56 62 50 53 49 39 52 52 50 37 53 40 34 43 43 57 48 43 42 42 65 35

7. The MEAN is the sum of all the scores from 53 down to the last score, which is 35 FORMULA: = 53 +36+57+………… = 35 100

8. Frequency Distribution of Grouped Test Scores Class Interval Midpoint (X) f X 1 f Cumulative Frequency ( cf ) Cumulative Percentage 75-80 70-74 65-69 60-64 55-59 50-54 45-49 40-44 35-39 30-34 25-29 20-24 77 72 67 62 57 52 47 42 37 32 27 22 3 2 8 8 17 18 21 13 9 1 231 134 496 456 884 846 882 481 288 22 100 97 97 95 87 79 62 44 23 10 1 1 100 97 97 95 87 79 62 44 23 10 1 1 TOTAL N 100 Σ X 1 f=4720

9. FORMULA: = 4720 = 47.2 100 1 f

10. MEDIAN Is the value that divides the ranked score into halves, or the middle value of the ranked scores

11. Frequency Distribution of Grouped Test Scores Class Interval Midpoint (X) f X 1 f Cumulative Frequency ( cf ) Cumulative Percentage 75-80 70-74 65-69 60-64 55-59 50-54 45-49 40-44 35-39 30-34 25-29 20-24 77 72 67 62 57 52 47 42 37 32 27 22 3 2 8 8 17 18 21 13 9 1 231 134 496 456 884 846 882 481 288 22 100 97 97 95 87 79 62 44 23 10 1 1 100 97 97 95 87 79 62 44 23 10 1 1 TOTAL N 100 Σ X 1 f=4720

12. Formula Mdn = Lower limit of + size of the (n/2)- cumulative frequency below the medial class median class class interval frequency of the median class

13. Applying the formula: You need a column for cumulative frequency. Determine n/2, which is one-half of number of scores of examinees. Find the class interval of the 50 th score. In this case where there are 100 scores, the 50 th score is in the class interval of 45-49. this class interval of 45-49 becomes the median of the class. Find the exact limits of the median class. In this case, class 44.5-49.5. The lower limit then is 44.5

14. Frequency Distribution of Grouped Test Scores Class Interval Midpoint (X) f X 1 f Cumulative Frequency ( cf ) Cumulative Percentage 75-80 70-74 65-69 60-64 55-59 50-54 45-49 40-44 35-39 30-34 25-29 20-24 77 72 67 62 57 52 47 42 37 32 27 22 3 2 8 8 17 18 21 13 9 1 231 134 496 456 884 846 882 481 288 22 100 97 97 95 87 79 62 44 23 10 1 1 100 97 97 95 87 79 62 44 23 10 1 1 TOTAL N 100 Σ X 1 f=4720

15. Substituting: 5(100/2-44) Median= 44.5 + 18 = 44.5 + 5(6) 18 = 46.17

16. MODE - Is the easiest measure of central tendency to obtain. It is the score or value with the highest frequency in the set of scores. If the scores are arranged in a frequency distribution, the mode is estimated as the midpoint of the class interval which has the highest frequency. This class interval with the highest frequency is also called the modal class. In a graphical representation of the frequency distribution, the mode is the value in the horizontal axis at which the curve is at its highest point. When all the scores in a group have the same frequency, the group of scores has no MODE.

17. Activity: Find the Mean,Mdn and Mo Class Interval Midpoint (X) f X 1 f Cumulative Frequency ( cf ) Cumulative Percentage 80-86 73-79 66-72 59-65 52-58 45-51 38-44 31-37 24-30 17-23 10-16 3-9 8 2 9 7 5 5 18 19 9 8 5 5 TOTAL N 100 Σ X 1 f=

18. Create the table of frequency distribution of the scores in Mathematics of the 100 BSE Students.Calculate the Mean Md, Mo 53 61 66 80 36 42 36 77 52 85 59 69 65 81 39 82 35 68 54 86 62 59 54 77 32 88 48 74 50 35 70 54 52 76 34 77 41 75 60 37 75 52 51 74 33 76 42 64 66 34 62 51 59 59 42 73 52 61 53 34 89 33 56 68 41 71 53 69 42 84 44 58 48 67 36 70 64 68 37 86 45 57 48 62 33 80 65 63 38 87 61 54 44 47 51 69 63 57 40 83

19. How do measures of central tendency determine skewness?

20. Symmetrical Distribution Inthis distribution , the Mean, Median and Mode have the same value. The value of the Median is between the mean and the mode.

21. Positively Skewed When the distribution becomes positively skewed, there are variations in their values. The mode stays at the peak of the curve and its value will be the smallest. The mean will be pulled out from the peak of the distribtion toward the direction of the few high scores..The mean gets the largest value. The Median is between the mode and the Mean.

22. Negatively Skewed The Mode remains at the peak of the curve, but it will have the largest value. The mean will have the smallest value as influenced by the extremely low scores, and the Median still lies between the Mode and the Mean

23. MEASURES OF DISPERSION

24. What do you notice? Different distributions are symmetrical, may have the same values or average Scores in A. range between 0-400 Scores in B range between 125- 275 Scores in C 150-250

25. Measures of variability gives us the estimate to determine how the scores are compresses, which contributes to the “flatness” or “peakedness”of the distribution

26. Indices of Variability a. Range b. Variance and Standard deviation

27. a. RANGE -it is the difference between the highest (X H ) and the lowest Score (X L ) in a distribution. -it is the simplest measure of variability but also considered as the least accurate measure of dispersion because its value is determined by just two scores in a group. It does not take into consideration the spread of all scores. -Its value depend on the highest and lowest scores. -value could be drastically changed by a single value.

28. b. Variance and Standard Deviation the most widely used measure of variability and is considered as the most accurate to represent the deviations of individual scores from the mean values in the distribution.

29. CLASS A CLASS B CLASS C 22 16 12 18 15 12 16 15 12 14 14 12 12 12 12 11 11 12 9 11 12 7 9 12 6 9 12 5 8 12

30. Mean A. ∑ X = 120 B. ∑ X = 120 C. ∑ X = 120 X = 120/10 X=120/10 X=120/10 X= 12 = 12 =12 = 12

31. CLASS A CLASS B CLASS C 22 16 12 18 15 12 16 15 12 14 14 12 12 12 12 11 11 12 9 11 12 7 9 12 6 9 12 5 8 12

32. Recall that ∑(X- ) is the sum of the deviation scores from the mean, which is equal to zero. As such we square each deviation score , then sum up all the squared deviation scores, and divide it by the number of cases. This yields the variance. Getting its square rool is the Standard deviation.

33. population variance no. of scores in the distribution population mean Unbiased estimate

34. CLASS A CLASS B X (X- ) (X- ) 2 X (X- ) (X- ) 2 22 22-12 100 16 16-12 16 18 18-12 36 15 15-12 9 16 16-12 16 15 15-12 9 14 14-12 4 14 14-12 4 12 12-12 12 12-12 11 11-12 1 11 11-12 1 9 9-12 9 11 11-12 1 7 7-12 25 9 9-12 9 6 6-12 36 9 9-12 9 5 5-12 49 8 8-12 16 =12 (X- ) 2 = 276 = 12 (X- ) 2 = 74

35. Computation S A 2 = 276 = 30.67 10-1 S B 2 = 74 = 8.22 10-1

36. RAW SCORE FORMULA

37. Note: Standard deviation is a measure of dispersion, it means that a large SD indicates greater score variability and if SD is small, the scores are closely clustered around the mean.

38. MEASURES OF POSITION. A. Quartile b. Decile c. Percentile

39. QUARTILE Quartlies are the three values that divide a set of scores into four equal parts, with one-fourth of the data values in each part. This means about 25% of the data falls at or below the first quartile (Q1); 50% of the data falls at or below the 2nd quartile (Q2), and 75% falls at or below the 3rd Quartile (Q3). Q2 is also the median. Q1 is the median of the first half of the values, and Q3 the median of the 2nd Half of the values. Thus, the upper quartile represents on average the mark of the top half of the class, while the lower quartile represents that of the bottom half of the class.

40. QUARTILES Quartiles are also used as a measure of the spread of data in the interquartile range (IQR), which is simply the difference between the third and first quartiles (Q 3 -Q 1 ). Half of this gives the semi-interquartile range or quartile deviation (Q). The following example illustrates the abovementioned measures.

41. Example: Given the following scores, find the 1st Quartile, 3rd Quartile, Quartile Deviation. 90, 85, 85, 86, 100, 105, 109, 110, 88, 105, 100, 112 Steps: 1. Arrange the scores in the decreasing order 2. From the bottom, find the points below which 25% of the score value and 75% of the score values fall. 3. Find the average of the two scores in each of these points to determine Q1 and Q3 , respectively. 4. Find Q using the formula. Q= Q3-Q1 2

42. Applying these steps in the above example, we have: 112 110 109 Q 3 = 109+105 =107 105 2 105 100 100 90 88 Q 1 = 88+ 86 =87 86 2 85 85

43. Consequently, applying the formula: Q 3 - Q 1 gives the quartile deviation 2 Q = 107-87 = 10 2

44. DECILE It divides the distribution into 10 equal parts. There are 9 deciles such that 10% of the distribution are equal or less than decile. 1, (D), 20% of the scores are equal or less than decile 2 (D 2 ); and so on. A student whose mark is between the first and second deciles is in decile 2, and one whose mark is above the ninth decile belongs to decile 10. If there are small number of data values, decile is not appropriate to use.

45. PERCENTILE It divides the distribution into one hundred equal parts. In the same manner, for percentiles, there are 99 percentiles such that 1% of the scores are less than the first percentile, 2% of the scores are less than the second percentile, and so on. For example, if you scored 95 in a 100-item test, and your percentile rank is 99th, then this means that 99% of those who took the test performed lower than you.

46. PERCENTILE This also means that you belong to the top 1% of those who took the test. In many cases, percentiles are wrongly interpreted as percentage score. For example, 75% as a percentage scores means you get 75 items correct out of 100 items, which is a mark of grade reflecting performance level. But percentile is a measure of position such that 75th percentile as your mark means that 75% of the students who took the test got lower score than you, or your score is located at the upper 25% of the class who took the same test. Percentiles are commonly used in national assessments or university entrance examinations.