VIEWS: 15 PAGES: 16 CATEGORY: Statistics POSTED ON: 1/9/2013 Public Domain
MEASURES OF LOCATION Zubair Latif Main Campus UVAS Lahore MEASURE OF CENTRAL TENDENCY The diagrammatic representation of a set of data can give us some impression about its distribution. Even then there remains a need for a single quantitative measure which could be used to indicate the center of the distribution . An average is a single value which represent a set of data or a distribution as a whole. It is more or less central value round which the observations in the set of data or distribution tend to cluster. Such a central value is also called a measure of central tendency or measure of location. 2 MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES 1. Arithmetic mean 2. Median 3. Mode 4. Geometric mean 5. Harmonic mean 3 MEASURE OF CENTRAL TENDENCY Arithmetic Mean It is obtained by dividing the sum of all the observations by the total number of observations. It is denoted by " X ". For ungrouped data x1 + x2 + x3 + ...xn X X= = n n For grouped data f 1 x1 + f 2 x2 + f 3 x3 ... f k xk fx X= = f 1 + f 2 + f 3 ... f k f 4 Example The following data is the final plant height (cm) of thirty plants of wheat. Construct a frequency distribution 87 91 89 88 89 91 87 92 90 98 95 97 96 100 101 96 98 99 98 100 102 99 101 105 103 107 105 106 107 112 5 MEASURE OF CENTRAL TENDENCY For ungrouped data x1 + x2 + x3 + ...xn X X= = n n 2929 X = = 97.63 30 6 FREQUENCY DISTRIBUTION TABLE Class Class Mid Tally Frequ fx ency Limits Boundaries Points (f) 86---90 85.5---90.5 88 ///// 6 528 91---95 90.5---95.5 93 //// 4 372 96---100 95.5---100.5 98 //////// 10 980 101---105 100.5--105.5 103 ///// 6 618 106---110 105.5– 10.5 108 /// 3 324 111---115 110.5--115.5 113 / 1 113 MEASURE OF CENTRAL TENDENCY For grouped data f 1 x1 + f 2 x2 + f 3 x3 ... f k xk fx X = = f 1 + f 2 + f 3 ... f k f fx 2935 X = 97 .83 f 30 8 Median 9 Median The median is a value that divides a set of data in to two equal parts after arranging the values in ascending or descending order of magnitude. For ungrouped data (i) When "n" is odd then the median is [(n+1)/2]th observation. (ii)When "n" is even then the median is 1/2[(n/2) + (n/2+1)]th observation. 10 Median Example Given below are the marks obtained by 20 students. 53 74 82 42 39 28 20 81 68 58 67 54 93 70 30 61 55 36 37 29. Find Median Solution: First arrange the data in ascending order of magnitude 20 28 29 30 36 37 39 42 53 54 55 58 61 67 68 70 74 81 82 93 Median = 1/2[(n/2) + (n/2+1)]th observation. Median = 1/2[ 10 + 11]th observation = 1/2[ 54 +55] Median = 54.5 11 Median Example Given below are the marks obtained by 9 students. 45 32 37 41 48 36 46 39 36 Find Median Solution: First arrange the data in ascending order of magnitude 32 36 36 37 39 41 45 46 48 Median = [n + 1]/2th observation. Median = [9 + 1]/ 2th observation = 5th observation Median = 39 12 Median For grouped data h n Median = l + ( - c) f 2 l= Lower class boundary of the class containing median h = Class interval of the class containing median* f= Frequency of the class containing median n= Total number of observations c = Cumulative frequency of the class preceding the class containing median. 13 Median Example: Estimate the Median Daily No. of c. f class Income Rs Work boundaries ers( f ) 5-----24 4 4 4.5---24.5 25-----44 6 10 24.5---44.5 45-----64 14 24 44.5---64.5 65-----84 22 46 64.5---84.5 85----104 14 60 84.5---104.5 105---124 5 65 104.5---124.5 125---144 7 72 124.5---144.5 145---164 3 75 144.5---164.5 14 Median Step -1 : Calculate n/2, its helps which class containing the median. n/2 = 75 / 2 = 37.5 see this observation in cumulative frequency column Step- 2: The class containing the median is 64.5 ---- 84.5 Step- 3: l = lower class boundaries containing the median = 64.5 Step- 4: h = class interval of the class containing the median = 20 Step- 5: f = frequency of the class containing median = 22 Step- 6: c = cumulative frequency of the preceding class = 24 15 Median h n Median = l + ( - c) f 2 20 75 Median = 64.5 + ( - 24) 22 2 = 76. 77 16