Prof. Ed course MEASURES_OF_VARIATION.pptx

Assessment in Learning 2

1. MEASURES OF VARIATION BY: MARJORIE D. BALBUENA

2. LEARNING OBJECTIVES: Calculate measures of dispersion Provide a sound interpretation of these measures

3. TWO TYPES OF VARIABILITY OR DISPERSION

4. ABSOLUTE Measures of dispersion

5. Ungrouped Data: Subtract the lowest score from the highest score. Range = H – L

6. Ungrouped Data: Find the range of the distribution if the highest score is 100 and the lowest score is 21.

7. Ungrouped Data: Find the range of the distribution if the highest score is 100 and the lowest score is 21. Solution: Range = H – L Range = 100 – 21 Range = 79

8. What is the range?

9. What is the range? Range = H - L Range = 50 – 10 Range = 40

10. Grouped Data: To find the range for a frequency distribution, just get the differences between the upper limit of the highest score and the lower limit of the lowest class interval.

11. Example: Find the range for the frequency distribution Class Interval Frequency 100 - 104 4 105 – 109 6 110 – 114 10 115 – 119 13 120 – 124 8 125 – 129 6 130 - 134 3 N = 50

12. Range = Highest Class Upper Limit – Lowest Class Lower Limit Range = 134.5 – 99.5 Range = 35

13. QUARTILE DEVIATION

14. QUARTILES

15. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR UNGROUPED DATA

16. EXAMPLE:

17. ARRANGED IN ORDER:

18. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR UNGROUPED DATA

19. Assign SERIAL NUMBERS:

20. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR UNGROUPED DATA

21. Q 1

22. Interpretation: 25% of the students have a score less than or equal to 21.5.

23. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR UNGROUPED DATA

24. Q 1 Q 3

25. Interpretation: 7 5% of the students have a score less than or equal to 30.5.

26. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR UNGROUPED DATA

27. ANSWER:

28. Interpretation: If the scores of 8 students in their management statistics quiz have a quartile deviation of 4.5, it means that the middle 50% of the scores (from the 25th percentile to the 75th percentile) deviate from the median by approximately 4.5 points.

29. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR GROUPED DATA Step 1: Find the class interval in which the first quartile ( Q 1 ) falls.  

30. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50

31. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 =  

32. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 1 C lass

33. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR GROUPED DATA Step 2: Find Q 1 Formula: Q 1 = L +   L i   f n

34. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 1 C lass Q 1 = L +   L = 31 – 0.5 = 30.5 =   = 11   f = 13 i = 5

35. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 1 C lass Q 1 = L +   L = 30.5 ; = ; i = 5   Q 1 = 30.5 +   Q 1 = 30.5 +   Q 1 = 30.5 +   Q 1 = 30.5 + 0.58 Q 1 = 31.08

36. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 1 C lass Q 1 = L +   Q 1 = 30.5 +   Q 1 = 30.5 +   Q 1 = 30.5 +   Q 1 = 30.5 + 0.58 Q 1 = 31.08 Interpretation: 25% of the students have a score less than or equal to 31.08.

37. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR GROUPED DATA Step 3: Find the class interval in which the third quartile ( Q 3 ) falls.  

38. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 =  

39. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 3 C lass

40. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR GROUPED DATA Step 4: Find Q 3 Formula: Q 3 = L +   L i   f n

41. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 3 C lass Q 3 = L +   L = 41 – 0.5 = 40.5 =   = 35   f = 10 i = 5

42. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 3 C lass Q 3 = L +   L = 40.5 ; = ; i = 5   Q 3 = 40.5 +   Q 3 = 40.5 +   Q 3 = 40.5 +   Q 3 = 40.5 + 1.25 Q 3 = 41.75

43. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 3 C lass Q 3 = L +   Q 3 = 40.5 +   Q 3 = 40.5 +   Q 3 = 40.5 +   Q 3 = 40.5 + 1.25 Q 3 = 4 1.75 Interpretation: 7 5% of the students have a score less than or equal to 41.75.

44. STEPS IN CALCULATING THE QUARTILE DEVIATION FOR GROUPED DATA Step 5: Calculate for interquartile range and quartile deviation. Formulas: IQR = Q 3 – Q 1 QD =  

45. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 1 = 31.08 ; Q 3 = 4 1.75 IQR = Q 3 – Q 1 IQR = 41.75 – 31.08 IQR = 10.67

46. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 IQR = 10.67 Interpretation: When the IQR is given as 10.67 for the scores of grade 10 students in their math quiz, it means that the middle 50% of the scores (from the 25th percentile to the 75th percentile) fall within a range of 10.67. This range provides insights into the spread of the scores among the students.

47. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Q 1 = 31.08 ; Q 3 = 4 1.75 QD =   QD =   QD = 5.34 QD =  

48. Example: Scores of grade 10 students in their math quiz Scores Frequency (f) < cf 21 – 25 2 2 26 – 30 9 11 31 – 35 13 24 36 – 40 11 35 41 – 45 10 45 46 – 50 5 50 N = 50 Interpretation: When the quartile deviation is given as 5.34 for the scores of grade 10 students in their math quiz, it means that the middle 50% of the scores (from the 25th percentile to the 75th percentile) deviate from the median by approximately 5.34 points. QD = 5.34

49. Assignment: Consider the frequency distribution of scores of the students in Mathematics. Find a) Q 1 ; b) Q 3 ; c) IQR with interpretation ; d) QD with interpretation [20 points] Class Interval Frequency (f) < cf 88 – 96 9 65 80 – 87 10 56 72 – 79 15 46 64 - 71 13 31 56 - 63 9 18 48 - 55 9 9 N = 65