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EMPIRICAL RELATION BETWEEN THE MEAN, MEDIAN AND THE MODE Tariq Mahmood Bajwa Median in Case of a Frequency Distribution of a Continuous Variable: In case of a frequency distribution, the median is given by the formula : ~ hn X l c f 2 where l= lower class boundary of the median class (i.e. that class for which the cumulative frequency is just in excess of n/2). h= class interval size of the median class f= frequency of the median class n= f (the total number of observations) c= cumulative frequency of the class preceding the median class Example: Going back to the example of the EPA mileage ratings, we have Mileage No. of Class Cumulative Rating Cars Boundaries Frequency Median 30.0 – 32.9 2 29.95 – 32.95 2 class 33.0 – 35.9 4 32.95 – 35.95 6 36.0 – 38.9 14 35.95 – 38.95 20 39.0 – 41.9 38.95 – 41.95 c 8 28 42.0 – 44.9 2 41.95 – 44.95 30 f h= class interval = 3 l n/2 = 30/2 = 15 In this example, n = 30 and n/2 = 15. Thus the third class is the median class. The median lies somewhere between 35.95 and 38.95. Applying the above formula, we obtain X 35.95 15 6 ~ 3 14 35.95 1.93 37.88 ~ 37.9 Interpretation Thisresult implies that half of the cars have mileage less than or up to 37.88 miles per gallon whereas the other half of the cars have mileage greater than 37.88 miles per gallon. Example The following table contains the ages of 50 managers of child- care centers in five cities of a developed country. Ages of a sample of managers of Urban child-care centers 42 26 32 34 57 30 58 37 50 30 53 40 30 47 49 50 40 32 31 40 52 28 23 35 25 30 36 32 26 50 55 30 58 64 52 49 33 43 46 32 61 31 30 40 60 74 37 29 43 54 Having converted this data into a frequency distribution, find the median age. Solution Following the various steps involved in the construction of a frequency distribution, we obtained: Frequency Distribution of Child-Care Managers Age Class Interval Frequency 20 – 29 6 30 – 39 18 40 – 49 11 50 – 59 11 60 – 69 3 70 – 79 1 Total 50 Now, the median is given by, ~ hn X l c f 2 where l= lower class boundary of the median class h= class interval size of the median class f= frequency of the median class n= f (the total number of observations) c= cumulative frequency of the class preceding the median class First of all, we construct the column of class boundary as well as the column of cumulative frequencies. Cumulative Class Frequency Class limits Frequency Boundaries f c.f 20 – 29 19.5 – 29.5 6 6 30 – 39 29.5 – 39.5 18 24 40 – 49 39.5 – 49.5 11 35 50 – 59 49.5 – 59.5 11 46 60 – 69 59.5 – 69.5 3 49 70 – 79 69.5 – 79.5 1 50 Total 50 Now, first of all we have to determine the median class (i.e. that class for which the cumulative frequency is just in excess of n/2). In this example, n = 50 implying that n/2 = 50/2 = 25 Cumulative Class Frequency Class limits Frequency Boundaries f c.f 20 – 29 19.5 – 29.5 6 6 Median 30 – 39 29.5 – 39.5 18 24 class 40 – 49 39.5 – 49.5 11 35 50 – 59 49.5 – 59.5 11 46 60 – 69 59.5 – 69.5 3 49 70 – 79 69.5 – 79.5 1 50 Total 50 Hence, l = 39.5 h = 10 f = 11 and c = 24 Substituting these values in the formula, we obtain: 10 X 39.95 25 24 11 39.95 0.9 40.4 Interpretation Thus, we conclude that the median age is 40.4 years. In other words, 50% of the managers are younger than this age, and 50% are older. Example WAGES OF WORKERS IN A FACTORY Monthly Income No. of (in Rupees) Workers Less than 2000/- 100 2000/- to 2999/- 300 3000/- to 3999/- 500 4000/- to 4999/- 250 5000/- and above 50 Total 1200 In this example, both the first class and the last class are open- ended classes. This is so because of the fact that we do not have exact figures to begin the first class or to end the last class. The advantage of computing the median in the case of an open-ended frequency distribution is that, except in the unlikely event of the median falling within an open-ended group occurring in the beginning of our frequency distribution, there is no need to estimate the upper or lower boundary. EMPIRICAL RELATION BETWEEN THE MEAN, MEDIAN AND THE MODE • This is a concept which is not based on a rigid mathematical formula; rather, it is based on observation. In fact, the word ‘empirical’ implies ‘based on observation’. • This concept relates to the relative positions of the mean, median and the mode in case of a hump- shaped distribution. • In a single-peaked frequency distribution, the values of the mean, median and mode overlap if the frequency distribution is absolutely symmetrical. THE SYMMETRIC CURVE f X Mean = Median = Mode But in the case of a skewed distribution, the mean, median and mode do not all lie on the same point. They are pulled apart from each other, and the empirical relation explains the way in which this happens. Experience tells us that in a unimodal curve of moderate skewness, the median is usually sandwiched between the mean and the mode. The second point is that, in the case of many real-life data-sets, it has been observed that the distance between the mode and the median is approximately double of the distance between the median and the mean, as shown below: f X Median Mode Mean This diagrammatic picture is equivalent to the following algebraic expression: Median - Mode ~ 2 (Mean - Median) ------ (1) The above-mentioned point can also be expressed in the following way: Mean – Mode ~ 3 (Mean – Median) ---- (2) Equation (1) as well as equation (2) yields the approximate relation given below: EMPIRICAL RELATION BETWEEN THE MEAN, MEDIAN AND THE MODE Mode ~ 3 Median – 2 Mean An exactly similar situation holds in case of a moderately negatively skewed distribution. An important point to note is that this empirical relation does not hold in case of a J-shaped or an extremely skewed distribution. Let us try to verify this relation for the data of EPA Mileage Ratings that we have been considering for the past few lectures. Frequency Distribution for EPA Mileage Ratings Class Limit Class Boundaries Frequency 30.0 – 32.9 29.95 – 32.95 2 33.0 – 35.9 32.95 – 35.95 4 36.0 – 38.9 35.95 – 38.95 14 39.0 – 41.9 38.95 – 41.95 8 42.0 – 44.9 41.95 – 44.95 2 Total 30 Number of Cars 0 2 4 6 8 10 12 14 16 Y 29 .9 5 32 .9 5 35 .9 5 38 .9 5 Miles per gallon 41 Histogram .9 5 44 .9 5 X Frequency polygon and Frequency cure Y 16 14 Number of Cars 12 10 8 6 4 2 0 X 5 5 5 5 5 5 5 .4 .4 .4 .4 .4 .4 .4 28 31 34 37 40 43 46 Miles per gallon • As mentioned above, the empirical relation between mean, median and mode holds for moderately skewed distributions and not for extremely skewed ones. • Hence, in this example, since the distribution is only very slightly skewed, • Therefore we can expect the empirical relation between mean, median and the mode to hold reasonable well. Arithmetic Mean: X 37.85 Median: X 37.88 Mode: ˆ X 37.825 Interesting Observation The close proximity of the three measures of central tendency provides a strong indication of the fact that this particular distribution is indeed very slightly skewed. EMPIRICAL RELATION BETWEEN THE MEAN, MEDIAN AND THE MODE Mode 3 Median 2 Mean 3 Median 2 Mean 3(37.88) 2(37.85) 113.64 75.70 37.94 Now, the mode = 37.825 which means that the left- hand side is indeed very close to 37.94 i.e. the right-hand side of the empirical relation. Hence, the empirical relation Mode 3 Median 2 Mean is verified.

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