Chapter 4 Measures of Central Tendency and Dispersion

1. Chapter 4Measures of Central Tendency and Dispersion

2. Measures Of Central Tendency And Dispersion • Measures of Central Tendency -- Mean • Arithmetic • Geometric • Harmonic • Weighted Mean • Median, Quartiles, Percentiles, Deciles • Mode • Measures of Variation

3. Measures Of Central Tendency And Dispersion • Range • Mean Deviation • Standard Deviation ( Variance ) • Inter Quartile Range • Coefficient of Variation • Measures of Skewness and Kurtosis • Standardised Variables and Scores

4. Measures of Location or Central Tendency • Measure of Location • Centre of Gravity  • There are three such measures: • Mean • Median, Quartiles, Percentiles and Deciles • Mode

5. Properties of a Measure • It should be easy to understand and calculate • It should be based on all observations • It should not be much affected by a few extreme observations • It should be amenable to mathematical treatment. For example, we should be able to • calculate the combined measure for two sets of observations given the measure for each of the two sets

6. Mean • There are three types of means viz., • Arithmetic Mean • Harmonic Mean • Geometric Mean

7. Sum of Observatio ns = x Number of Observatio ns å xi = n Arithmetic Mean Ungrouped (Raw) Data

8. 2717 2796 3098 3144 3527 3534 3862 4186 4310 4506 4745 4784 4923 5034 5071 5424 5561 6505 6707 6874 Illustration 4.1 Table 4.1 : Equity Holdings of 20 Indian Billionaires ( Rs. in Millions)

9. x Illustration 4.1 For the above data, the A.M. is 2717 + 2796 +…… 4645+….. + 5424 + ….+ 6874 = -------------------------------------------------------------------------- 20 = Rs. 4565.4 Millions

10. å f x i i = x å f i Arithmetic Mean Grouped Data

11. Class Interval ( 1 ) Mid Value of Class Interval ( xi ) ( 3 ) Frequency ( fi ) ( 2 ) fixi Col.(4) = Col.(2) x Col.(3) 2000 – 3000 2 2500 5000 3000 – 4000 5 3500 17500 4000 – 5000 6 4500 27000 5000 – 6000 4 5500 22000 6000 – 7000 3 6500 19500 Sum  fi = 20 fixi = 91000 Illustration 4.2 The calculation is illustrated with the data relating to equity holdings of the group of 20 billionaires given in Table 3.1

12. å f x i = i x å f i Illustration 4.2 values of  fi and  fixi , in formula = 9100 ÷ 20 = 4550

13. å w x i i = x å w i Weighted Arithmetic Mean if the values x1, x2 x3, …. xi, ….xn have weights w1, w2 w3, …. wi, ….,wn then the weighted mean of x is given as

14. Item Monthly Consumption Weight (wi) Rise in Price (Percentage) (pi) wipi Sugar 5 5 20 100 Rice 20 20 10 200 Illustration 4.3

15. å p p w i i å w + 100 200 i 300 25 + 5 20 Illustration 4.3 Therefore, the average price rise could be evaluated as = = = = = 12. Thus the average price rise is 12 % .

16. Geometric Mean The Geometric Mean ( G. M.) of a series of observations with x1, x2, x3, ……..,xn is defined as the nth root of the product of these values . Mathematically G.M. = { ( x1 )( x2 )( x3 )…………….(xn ) } (1/ n ) It may be noted that the G.M. cannot be defined if any value of x is zero as the whole product of various values becomes zero.

17. Illustration 4.5 For the data with values, 2,4, and 8, G.M. = (2 x 4 x 8 ) (1/3) = (64) 1/3 = 4

18. Average Rate of Growth of Production/Business or Increase in Prices If P1 is the production in the first year and Pn is the production in the nth year, then the average rate of growth is given by ( G – 100) % where, G = 100 (Pn / P1 )1/(n-1) or log G = log 100 + { 1/(n–1) } (log Pn – log P1)

19. Example 4.4 The wholesale price index in the year 2000-01 was 145.3. It increased to 195.5 in the year 2005-06. What has been the average rate of increase in the index during the last 5 years. Solution: By using the formula ( 4.8), we have log G = 2 +{ (1/5) ( log 195.5 – log145.3 ) } = 2.02578 Therefore, G = Anti log (2.02578) = 106.11 Thus the average rate of increase = 106.11  100 = 6.11%

20. Combined G.M. of Two Sets of Data  If G1 & G2 are the Geometric means of two sets of data, then the combined Geometric mean, say G, of the combined data is given by : n1 log G1 + n2 log G2 log G = ------------------------------- n1 + n2

21. Combined G.M. of Two Sets of Data • As another example, suppose the average growth rate during the first five years of business is 20 %, and the average growth rate of business during the next five years is 15 %, and we wish to find the average growth rate for the entire period of 10 years. This growth rate can be found by calculating the combined geometric mean of the geometric means 120 and 115, for the two blocks of 5-year periods. Thus, the requisite G.M., say G, can be worked out as follows:

22. Combined G.M. of Two Sets of Data 5 log 120 + 5 log 115 5 x 2.07918 + 5 x 2.06070 log G = ------------------------------- = ---------------------------------- 5 + 5 10 20.6994 = ------------ = 2.06994 10 Therefore, G = antilog 2.06994 = 117.47 Thus the combined average rate of growth for the period of 10 years is 17.47%.

23. Weighted Geometric Mean Just like weighted arithmetic mean, we also have weighted Geometric mean If x1, x2,….,xi,….,xn are n observations with weights w1, w2, …wi,.., wn, then their G.M. is defined as:  wi log xi G.M. = ----------------------  wi

24. Harmonic Mean The harmonic mean (H.M.) is defined as the reciprocal of the arithmetic mean of the reciprocals of the observations.  For example, if x1 and x2 are two observations, then the arithmetic means of their reciprocals viz 1/x1 and 1/ x2 is {(1 / x1)+ (1 / x2)} / 2 = (x2 + x1) / 2 x1 x2 The reciprocal of this arithmetic mean is 2 x1 x2 / (x2 + x1). This is called the harmonic mean.  Thus the harmonic mean of two observations x1 and x2 is 2 x1 x2 ----------------- x1 + x2

25. ´ A . M H . M . Relationship Among A.M. G.M. and H.M. The relationships among the magnitudes of the three types of Means calculated from the same data are as follows:  (i) H.M. ≤ G.M. ≤ A.M.   i.e. the arithmetic mean is greater than or equal to the geometric which is greater than or equal to the harmonic mean. ( ii ) G.M. = i.e. geometric mean is the square root of the product of arithmetic mean and harmonic mean. ( iii) H.M. = ( G.M.) 2 / A .M.

26. Median • whenever there are some extreme values in the data, calculation of A.M. is not desirable. • Further, whenever, exact values of some observations are not available, A.M. cannot be calculated. • In both the situations, another measure of location called Median is used.

27. Median - Ungrouped Data First the data is arranged in ascending/descending order.  In the earlier example relating to equity holdings data of 20 billionaires given in Table 4.1, the data is arranged as per ascending order as follows  2717 2796 3098 3144 3527 3534 3862 4187 4310 4506 4745 4784 4923 5034 5071 5424 5561 6505 6707 6874 Here, the number of observations is 20, and therefore there is no middle observation. However, the two middle most observations are 10th and 11th. The values are 4506 and 4745. Therefore, the median is their average.   4506 + 4745 9251 Median = ----------------- = ----------- 2 2 = 4625.5 Thus, the median equity holdings of the 20 billionaires is Rs.4625.5 Millions.

28. Median - Grouped • The median for the grouped data is also defined as the value corresponding to the ( (n+1)/2 )th observation, and is calculated from the following formula: • ( (n/2) –fc ) • Median = Lm + ----------------- wm • fm •  where, • Lm is the lower limit of 'the median class internal i.e. the interval which contains n/2th observation • fm is the frequency of the median class interval i.e. the class interval which contains the ( (n)/2 )th observation • fc is the cumulative frequency up to the median class- interval • wm is the width of the median class-interval • n is the number of total observations.

29. Class Interval Frequency Cumulative frequency 2000-3000 2 2 3000-4000 5 7 4000-5000 6 13 5000-6000 4 17 6000-70000 3 20 Illustration 4.2

30. Illustration 4.2 Here, n = 20, the median class interval is from 4000 to 5000 as the 10th observation lies in this interval. Further,  Lm = 4000   fm = 6   fc = 7   wm = 1000 Therefore, 20/2 –7 x 1000 Median = 4000 + ------------------------- 6 = 4000 + 3/6 x 1000 = 4000 + 500 = 4500

31. Median • The median divides the data into two parts such that the number of observations less than the median are equal to the number of observations more than it. • This property makes median very useful measure when the data is skewed like income distribution among persons/households, marks obtained in competitive examinations like that for admission to Engineering / Medical Colleges, etc.

32. Graphical Method of Finding the Median • If we draw both the ogives viz. “Less Than “ and “ More Than”, for a data, then the point of intersection of the two ogives is the Median.

33. Quartiles • Median divides the data into two parts such that 50 % of the observations are less than it and 50 % are more than it. Similarly, there are “Quartiles”. There are three Quartiles viz. Q1 , Q2 and Q3. These are referred to as first, second and third quartiles. • The first quartile , Q1, divides the data into two parts such that 25 % ( Quarter ) of the observations are less than it and 75 % more than it. • The second quartile, Q2, is the same as median. The third quartile divides the data into two parts such that 75 % observations are less than it and 25 % are more than it. • All these can be determined, graphically, with the help of the Ogive curve

34. Quartiles

35. Quartiles data Q1 and Q3 are defined as values corresponding to an observation given below : Ungrouped Data Grouped Data (arranged in ascending or descending order) Lower Quartile Q1 {( n + 1 ) / 4 }th ( n / 4 )th  Median Q2 { ( n + 1 ) / 2 }th ( n / 2 )th Upper Quartile Q3 {3 ( n + 1 ) / 4 } th (3 n / 4 )th

36. - ( n / 4 ) f + ´ c Q L w = 1 Q Q f 1 1 Q 1 - ( 3 n / 4 ) f + ´ c Q L w = 3 Q Q f 3 3 Q 3 Quartiles

37. Class Interval Frequency Cumulative frequency 2000-3000 2 2 3000-4000 5 7 4000-5000 6 13 5000-6000 4 17 6000-70000 3 20 Equity Holding Data

38. ( (20/4) – 2) Q1 = 3000 + ---------------  1000 5   ( 5 – 2) = 3000 + --------------------  1000 5   3000 = 3000 + ------------- 5   = 3000 + 600   = 3600 The interpretation of this value of Q1 is that 25 % billionaires have equity holdings less than Rs.

39. (15 – 13) Q3 = -------------  1000 +5000 4   2 = -------  1000 +5000 4 = 5500 The interpretation of this value of Q3 is that 75 % billionaires have equity holdings less than Rs. 5500 Millions.

40. Percentiles (95/100)  n – fc P95 = L P95 + ------------------- x wP95 f P95 where, L P95 is the lower point of the class interval containing 95th percent of total frequency, fc is the cumulative frequency up to the 95th percentile interval, f P95 is the frequency of the 95th percentile interval and wP95 is the width of the 95th percentile interval.

41. Deciles • Just like quartiles divide the data in four parts, the deciles divide the data into ten parts – first deciles ( 10% ) , second ( 20% ) , and so on. In fact, P10 , P20 , ……………….., P90 are the same as deciles. And just as second quartile and median are the same, so the fifth decile i.e.P50 and the median are the same.

42. Mode fm - f0 Mode = Lm + -----------------  wm fm - f0 - f2 • where , • Lm is the lower point of the modal class interval • fm is the frequency of the modal class interval • f0 is the frequency of the interval just before the modal interval • f2 is the frequency of the interval just after the modal interval • wm is the width of the modal class interval

43. Equity Holding Data the modal interval i.e., the class interval with the maximum frequency (6) is 4000 to 5000. Further, Lm = 4000 wm = 1000 fm = 6 f0 = 5 f2 = 4 Therefore

44. Equity Holding Data ( 6 – 5) Mode = 4000 + --------------------  1000 2  6 – 5 – 4 = 4000 +  1000 = 4000 + 333.3 = 4333.3 Thus the modal equity holdings of the billionaires is Rs. 4333.3 Millions.

45. Empirical Relationship among Mean, Median and Mode In a moderately skewed distributions, it is found that the following relationship, generally, holds good : Mean – Mode = 3 (Mean – Median) From the above relationship between, Mean, Median and Mode, if the values of two of these are given, the value of third measure can be found out

46. Equity Holding Data 4333 4500 4565 (mode) (median) (mean)

47. Right Skewed Distribution Mode MedianMean

48. Symmetrical Mode MedianMean

49. Left Skewed Distribution Mean Median Mode

50. Features of a Good Statistical Average • Readily computable, comprehensible and easily understood • It should be based on all the observations • It should be reliable. enough to be taken as true representative of the population • It should not be much affected by the extreme values in the data • It should be amenable to further mathematical treatment. This properly helps in assessing the reliability of conclusions drawn about the population value with the help of sample value • Should not vary much from sample to sample taken from the same population.