Measures of Central Tendency. For Ungrouped Data

Transcription

1 Measures of Central Tendency For Ungrouped Data

2 Recall: What are the three ways on how we can define central tendency or center of the distribution?

3 We can also say: (1) the point on which a distribution would be balance;

4 We can also say: (2) the value whose average absolute deviation from all other values is minimized; and

5 We can also say: (3) the value whose average squared difference from all the other values is minimized.

6 Recall: What are the measures of central tendency?

7 Do you know? The mean is the point on which the distribution would balance.

8 Do you know? The median is the value that minimizes the sum of absolute deviations.

9 Do you know? The mean is the value that minimizes the sum of squared deviations.

10 The Mean For Ungrouped Data

11 First Type Arithmetic Mean

12 Arithmetic Mean It is also called as simple mean or unweighted mean. It is the sum of a collection of numbers divided by the number of numbers in the collection.

13 Formula Arithmetic Mean: x = Σx n where: x = items/scores n=number of items/scores

14 Example: What is the mean age of a group of children whose ages are 9, 11, 7, 10, 9, 8, 8, 7, 12, 7 and 13?

15 Second Type Weighted Mean

16 Weighted Mean It takes into consideration the proper weights assigned to the observed values according to importance.

17 Formula Weighted Mean: x = Σwx w where: x=items/scores w=weight of each item/score

18 Example: A student took 3 exams in Math. He finished the first exam in 45 minutes and got a grade of 88; 60 minutes on the second exam and got 92; and 90 minutes on the third exam and got 85. What was the student s mean score for the three exams?

19 Third Type Mean for Simple Frequency Distribution

20 Formula Mean for Simple Frequency Distribution: x = Σfx n Where: f=frequency; x=items/scores; and n=number of items

21 Example: x f Σfx n=20 Σfx=

22 Example: x f Σfx n = 20 Σfx = 114

23 Let s Practice Measures of Central Tendency for Ungrouped Data

24 Problem 1: A freshman college student for the following final grades with the number of units for each subject. (a) Find the weighted mean grade and (b) if all the subjects have uniform or equal number of units, what would be the mean grade?

25 Table 1. A freshman college student final grades with the number of units for each subject. Subjects Grades Units Math English Filipino History PE Chemistry Lecture Chemistry Laboratory

26 Problem 2: Gottfried Wilhelm Leibniz answered 20 calculus problems. He spent 1 ½ hours for the first 6 problems; 45 minutes for 3 problems; and 3 hours for 11 problems. What was the average time he spent for each problem?

27 Recall: What are the symbols we use for each of measure of central tendency?

28 Something to think about If you are given a dataset/frequency distribution that is already grouped, what will be the most challenging part? Why?

29 Measures of Central Tendency For Grouped Data

30 The Mean For Grouped Data

31 Long Method/Midpoint Method Formula: x = Σfx n Where: f=frequency; x=class marks; and n=number of samples

32 Example: Class Intervals f (frequency) n=65 x (class mark) fx (frequency x class mark)

33 Example: Class Intervals f (frequency) x (class mark) fx (frequency x class mark) , , n=65 Σfx=5,758

34 Short Method or Assumed Mean Method It is the class mark of the class interval near the center of the distribution or the class mark of the interval with the highest frequency.

35 Formula Assumed Mean Method: x = x 0 + Σfd n w

36 Where: x = assumed mean 0 f = frequency n = simple size d = coded value w = class width

37 Example: Class Intervals f (frequency) x (Class Mark) fx (frequency x class mark) fd n=65

38 Steps involved in the calculation of Mean by the Short Method For Grouped Data

39 Step 1: Tabulate the scores in a frequency distribution.

40 Example: Class Intervals f (frequency) x (Class Mark) fx (frequency x class mark) fd n=65

41 Step 1: Take the midpoint of an interval somewhere near the center of the frequency distribution and, if possible, the interval should contain the largest frequency.

42 Example: Class Intervals f (frequency) x (Class Mark) n=65 fx (frequency x class mark) fd

43 Note: That will be our assumed mean. Therefore, x = 93. 0

44 Step 3: Fill in the column for the deviations from the assumed mean in units of class intervals (d). Starting with 0 for the class interval having the x 0 going up assign positive values and going down assign negative values for d.

45 Example: Class Intervals f (frequency) x (Class Mark) d (coded value) n=65 fd

46 Step 4: The fd column is the product of f and d. The values greater than the x are 0 positive and the values less than x are negative. 0

47 Example: Class Intervals f (frequency) x (Class Mark) d (coded value) n=65 fd

48 Step 5: Get the sum of the fd values.

49 Example: Class Intervals f (frequency) x (Class Mark) d (coded value) n=65 Σfd=-41 fd

50 Step 6: Determine the class width (w).

51 Therefore, The class width is 7.

52 Step 7: Substitute the values obtained to the formula.

53 Therefore, The x =

54 Something to think about What can you say to the mean we obtained using the Midpoint Method and Assumed Mean Method?

55 The Median For Grouped Data

56 Formula: x = LL R + n 2 f f w

57 Where: LL R = lower real limit f = cumulative less than frequency f = frequency n = sample w = class size

58 Example: Class Intervals f (frequency) n=65 f (cumulative less than frequency)

59 Steps Involved in the Calculation of Median For Grouped Data

60 Step 1: Record the cumulative frequencies.

61 Example: Class Intervals f (frequency) f (cumulative less than frequency) n=65

62 Step 2: Determine n 2.

63 Therefore, n 2 =

64 Step 3: Identify the class interval in which the 32.5 th case falls.

65 Example: Class Intervals f (frequency) f (cumulative less than frequency) f = f= n=65

66 Therefore, it is the 4 th class interval with exact limits LL R is 82.5.

67 Step 4: Determine the class width (w).

68 Therefore, the class width is 7.

69 Step 5: Substitute the values obtained to the formula.

70 Answer: The median is 89.2

71 The Mode For Grouped Data

72 Formula: x = LL R + du du + dl w

73 Where: LL R = lower real limit du = difference between the highest frequency and the frequency of the interval below it dl = difference between the highest frequency and the frequency of the interval above it w =class size

74 Example: Class Intervals f n=65

75 Steps Involved in the Calculation of Mode For Grouped Data

76 Step 1: Tabulate the scores in a frequency distribution.

77 Example: Class Intervals f n=65

78 Step 1: Determine the class interval with the highest frequency.

79 Example: Class Intervals f n=65

80 Then, the LL will be the R lower real limit of the class interval with the highest frequency.

81 Thus, LL R = 89. 5

82 Step 2: Determine du. du is the difference between the highest frequency and the frequency of the interval below it.

83 Example: Class Intervals f n=65

84 Thus, du = = 1

85 Step 3: Determine dl. dl is the difference between the highest frequency and the frequency of the interval above it.

86 Example: Class Intervals f n=65

87 Thus, dl = = 2

88 Step 4: Determine the class width.

89 Thus, the class width is 7.

90 Step 5: Substitute the values obtained to the formula.

91 Thus, the mode is 91.8.

92 Kinds of Mode For Grouped Data

93 The True Mode Kinds of Mode

94 Formula: x = 3Median 2Mean

95 From the example: using the true mode formula, the mode is 90.4.

96 Crude Mode Kinds of Mode

97 Crude Mode It is the midpoint of the class interval with the highest frequency.

98 Formula: x = lower limit + upper limit 2

99 From the example: using crude mode method, the mode is 93.