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# 07 central tendency

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### 07 central tendency

1. 1. 14-04-2012 Research Methodology Dr. Nimit Chowdhary, Professor Mean Arithmetic mean Geometric Mean  Median Harmonic Mean Mode Trimmed mean© Dr. Nimit Chowdhary Research Methodology 2 Saturday, April 14, 2012 1
2. 2. 14-04-2012  Measures of central  There are hundred of tendency refer to the colleges in India summary measures  Each college has used to describe the different tuition most "typical" value in  Average college tuition a set of values. is Rs. 2000 Central tendency Mean MedianSaturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 3 Compute mean and median Explain pros and cons of each measure Describe how they change after:  Adding a constant to each value in the data set  Multiplying each value in the data set by a constantSaturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 4 2
3. 3. 14-04-2012 Computing mean is a two step process 1. Find the sum of all values in a group of values 2. Divide the sum by the number of values in the group Sum of all values Mean  Number of valuesSaturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 5POPULATION MEAN SAMPLE MEAN  = Sum the following   = Sum the following X = Sum all X values  x = Sum all x values N = Number of X values  n = Number of x values  = Population mean  x = sample mean  X x x N nSaturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 6 3
4. 4. 14-04-2012BOWLING SCORE Game Score   X / N 1 100 2 170  X  100 170 130 140  540 3 130 N 4 4 140    X / N 540/ 4  135 Mean bowling scoreSaturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 7The median is the midpoint in a set of data List scores from smallest to largest With an odd number of scores, the median is the middle score With an even number of scores, the median is the sum of the middle two scores divided by 2 Median= Sum of middle two scores / 2Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 8 4
5. 5. 14-04-2012BOWLING SCORE Game Score  List the scores in ascending order 1 100 100 130 140 160 170 2 170 3 130  With an odd number of scores, identify the middle 4 140 score 5 160 100 130 140 160 170BOWLING SCORE  List the scores in ascending order Game Score 100 130 140 170 1 100  With an even number of scores, identify the two 2 170 middle scores 3 130 100 130 140 170  The median is the sum of 4 140 the middle scores divided by two 130  140 Median   135 2 5
6. 6. 14-04-2012 MEAN MEDIAN  The mean is better if a  The median is better if a large set of scores does not small set of scores has an have an outlier outlierRs. 60,000/- Rs. 70,000/- Rs. 80,000/- Rs. 90,000/- Rs. 10,000,000/- Mean: = X/ N = (60,000 + 70,000 + 80,000 + 90,000 + 10,000,000) = Rs. 2,060,000 Median: Rs. 80,000/- Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 12 6
7. 7. 14-04-2012Marks obtained by 30 students of class X of acertain school in a mathematics text consistingof 100 marks are as follows: Marks Number of Marks Number of obtained students obtained students 10 1 70 4 20 1 72 1 36 3 80 1 40 4 88 2 50 3 92 3 56 2 95 1 60 4Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 13 Marks obtained (xi) No. of students (fi) f ix i 10 1 10 20 1 20 36 3 108 40 4 160 50 3 150 56 2 112 60 4 240 70 4 280 72 1 72 80 1 80 88 2 176 92 3 276 95 1 95 Total fi=30 fixi = 1779 7
8. 8. 14-04-2012 Mean  fx i i  1779  59.3 f i 30 Class interval No. of students Class mark (xi) f ix i (fi) 10-25 2 17.5 35.0 25-40 3 32.5 97.5 40-55 7 47.5 332.5 55-70 6 62.5 375.0 70-85 6 77.5 465.0 85-100 6 92.5 555.0 fi= 30 fixi =1860.0 Mean  fx i i  1860  62 f i 30Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 16 8
9. 9. 14-04-2012Suppose we have to find the median of thefollowing data which gives the marks, out of50, obtained by 100 students in a test. Marks obtained 20 29 28 33 42 38 43 25 No. of students 6 28 24 15 2 4 1 20Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 17 No. of data points is even, 100 The median is the middle-most value, that is, the value between 50th and 51st points Median will therefore be the average of 50th and 51st pointSaturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 18 9
10. 10. 14-04-2012 Marks obtained No. of students Cumulative frequency 20 6 6 25 20 26 28 24 50 29 28 78 33 15 93 38 4 97 42 2 99 43 1 100Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 19 n/2 th value is 50th, that is 28 (n/2 + 1)th value is 51st , that is 29 So the median is 28  29 Median   28.5 2Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 20 10
11. 11. 14-04-2012 n   2  cf  Median  l   h  f    Where, l = Lower limit of median class n = Number of observations cf = Cumulative frequency of class preceding median class f = Frequency of median class h = Class size Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 21A survey of heights (in cm) of 51 girls of class Xof a school was conducted and the followingdata was obtained Height in cms No. of girls Less than 140 4 Less than 145 11 Less than 150 29 Less than 155 40 Less than 160 46 Less than 165 51Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 22 11
12. 12. 14-04-2012 Create a frequency table Class intervals Frequency Cumm. frequency Below 140 4 4 140-145 7 11 145-150 18 29 150-155 11 40 155-160 6 46 160-165 5 51Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 23  n= 51, therefore n/2 n  =25.5; this means   cf  Median  l   2 h 145-150 is the median  f  class    l= 145  25.5  11  cf= 11 Median  145   5  18    f= 18, and  h= 5 72.5  145  18  149.03Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 24 12
13. 13. 14-04-2012 Two important measures of central tendency- mean and median Grouped and ungrouped data How to calculate mean How to calculate median When to use mean and when to use medianSaturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 25 13