625156784-MEASURES-OF-CENTRAL-TENDENCY-FOR-GROUPED-DATA-ppt (1).pptx

Grouped Data

1. MEASURES OF CENTRAL TENDENCY OF GROUPED DATA

2. MEAN OF A GROUPED DATA When the amount of data values is too large, the data is usually grouped into class intervals. Within a class interval, we count the number of data values (frequency), and determine the midpoint of the interval (class mark).

3. Formula for the Mean of Grouped Data Where M = mean f = frequency x = midpoint

4. EXAMPLE 1 Below are the scores of 30 students in a 60-item test in Math. CLASS INTERVAL FREQUENCY (f) 56-60 2 51-55 1 46-50 41-45 2 36-40 4 31-35 6 26-30 3 21-25 4 16-20 5 11-15 3

5. EXAMPLE 1 Below are the scores of 30 students in a 60-item test in Math. i = 5 Σ f = 30 Σ fx = 910 Mean = = = 30.33   CLASS INTERVAL FREQUENCY (f) CLASS MARK (x) fx 56-60 2 58 116 51-55 1 53 53 46-50 48 41-45 2 43 86 36-40 4 38 152 31-35 6 33 198 26-30 3 28 84 21-25 4 23 92 16-20 5 18 90 11-15 3 13 39

6. EXAMPLE 2 Calculate the mean of the test scores. CLASS INTERVAL FREQUENCY (f) 41-45 1 36-40 8 31-35 8 26-30 14 21-25 7 16-20 2

7. EXAMPLE 2 Calculate the mean of the test scores. i = 5 Σ f = 40 Σ fx = 1200 Mean = = = 30   CLASS INTERVAL FREQUENCY (f) CLASS MARK (x) fx 41-45 1 43 43 36-40 8 38 304 31-35 8 33 264 26-30 14 28 392 21-25 7 23 161 16-20 2 18 36

8. MEDIAN OF A GROUPED DATA The median is the middle value in a set of data arranged in increasing or decreasing order. It usually describes the middle score of such data.

9. Formula for the Median of Grouped Data Mdn = l + ( ) i f be the frequency of the median class i be the class size; the length of class interval cf be the cumulative frequency of the class interval preceding that of the median class l lower boundary of the true limit of the median class  

10. EXAMPLE 1 Given the data below, find the median CLASS INTERVAL FREQUENCY 11-15 3 16-20 5 21-25 3 26-30 3 31-35 6 36-40 4 41-45 2 46-50 1 51-55 1 56-60 2

11. EXAMPLE 1 Given the data below, find the median f = 30 ---- = 15   CLASS INTERVAL FREQUENCY CUMULATIVE FREQUENCY TRUE LIMITS 11-15 3 3 10.5 – 15.5 16-20 5 8 15.5 – 20.5 21-25 3 11 20.5 – 25.5 26-30 3 14 25.5 – 30.5 31-35 6 20 30.5 – 35.5 36-40 4 24 35.5 – 40.5 41-45 2 26 40.5 – 45.5 46-50 1 27 45.5 – 50.5 51-55 1 28 50.5 – 55.5 56-60 2 30 55.5 – 60.5

12. Since cf = 20 is greater than and nearest to 15, then the median class is the class interval 31-35. l = 30.5 the lower boundary of the true limit of the median class. cf = 14 cumulative frequency of the class interval preceding the median class f = 6 frequency of the class interval median class i = 5 class size, i = 35.5 – 30.5 = 5 Mdn = l + ( ) i = 30.5 + ( )(5) = 30.5 + = 31.33  

13. MODE OF A GROUPED DATA For grouped data, we need to have the class interval, along the frequency and true limits. To find the mode class, we look at the class intervals with the largest frequency.

14. MODE OF A GROUPED DATA Suppose there are n data values grouped into class intervals. Let l be the lower boundary of true limit of the mode class i be the class size; length of the interval class f₁ be the frequency of the mode class f₀ be the frequency of the class interval preceding the mode class f₂ be the frequency of the class interval succeeding the mode class Then the median of the grouped data is Mo = l + i ( )  

15. EXAMPLE 1 Given the data below, find the mode CLASS INTERVAL FREQUENCY 11-15 3 16-20 5 21-25 3 26-30 3 31-35 6 36-40 4 41-45 2 46-50 1 51-55 1 56-60 2

16. EXAMPLE 1 Given the data below, find the mode Since the highest frequency is 6, then the mode class is the class interval 31-35. The class size i is equal to 5. Also, l = 30.5 f₁ = 6 f₀ = 3 f ₂ = 4 Mo = l + i ( ) = 30.5 + 5 ( ) = 30.5 + 3 = 33.5   CLASS INTERVAL FREQUENCY TRUE LIMITS 11-15 3 10.5 – 15.5 16-20 5 15.5 – 20.5 21-25 3 20.5 – 25.5 26-30 3 25.5 – 30.5 31-35 6 30.5 – 35.5 36-40 4 35.5 – 40.5 41-45 2 40.5 – 45.5 46-50 1 45.5 – 50.5 51-55 1 50.5 – 55.5 56-60 2 55.5 – 60.5

17. 3