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THE MODE: The mode is defined as that value which occurs most frequently in a set of data i.e. it indicates the most common result. EXAMPLE: Suppose that the marks of eight students in a particular test are as follows: 2, 7, 9, 5, 8, 9, 10, 9 Obviously, the most common mark is 9. In other words, mode = 9. MODE IN CASE OF RAW DATA PERTAINING TO A CONTINUOUS VARIABLE In case of a set of values (pertaining to a continuous variable) that have not been grouped into a frequency distribution (i.e. in case of raw data pertaining to a continuous variable), the mode is obtained by counting the number of times each value occurs. Let us consider an example. Suppose that the government of a country collected data regarding the percentages of revenues spent on Research & Development by 49 different companies, and obtained the following figures: EXAMPLE Percentage of Revenues Spent on Research and Development Company Percentage Company Percentage 1 13.5 14 9.5 2 8.4 15 8.1 3 10.5 16 13.5 4 9.0 17 9.9 5 9.2 18 6.9 6 9.7 19 7.5 7 6.6 20 11.1 8 10.6 21 8.2 9 10.1 22 8.0 10 7.1 23 7.7 11 8.0 24 7.4 12 7.9 25 6.5 13 6.8 26 9.5 Percentage of Revenues Spent on Research and Development Company Percentage Company Percentage 27 8.2 39 6.5 28 6.9 40 7.5 29 7.2 41 7.1 30 8.2 42 13.2 31 9.6 43 7.7 32 7.2 44 5.9 33 8.8 45 5.2 34 11.3 46 5.6 35 8.5 47 11.7 36 9.4 48 6.0 37 10.5 49 7.8 38 6.9 DOT PLOT The horizontal axis of a dot plot contains a scale for the quantitative variable that we are wanting to represent. The numerical value of each measurement in the data set is located on the horizontal scale by a dot. When data values repeat, the dots are placed above one another, forming a pile at that particular numerical location. R&D 4.5 6 7.5 9 10.5 12 13.5 Dot Plot As is obvious from the above diagram, the value 6.9 occurs 3 times whereas all the other values are occurring either once or twice. Hence the modal value is 6.9. R&D 4.5 6 7.5 9 10.5 12 13.5 ˆ X= 6.9 Also, this dot plot shows that almost all of the R&D percentages are falling between 6% and 12%, most of the percentages are falling between 7% and 9%. We will be interested to note that mode is such a measure that can be computed even in case of nominal and ordinal levels of measurements. For example The marital status of an adult can be classified into one of the following five mutually exclusive categories: Single, married, divorced, separated and widowed. Nominal scale is that where a certain order exists between the groupings. For example: Speaking of human height, an adult can be regarded as tall, medium or short. A company has developed five different bath oils, and, in order to determine consumer-preference, the company conducts a market survey. Number of Respondents favouring various bath-oils 400 Bath oils No. of Respondents 300 200 100 0 I II III IV V Mode The largest number of respondents favaoured bath-oil NO.II, as evidenced by the bar-chart. Thus, we can say that Bath-oil No.II is the mode. THE MODE IN CASE OF A DISCRETE FREQUENCY DISTRIBUTION: In case of a discrete frequency distribution, identification of the mode is immediate; one simply finds that value which has the highest frequency. Example: No. of Passengers No. of Flights An airline found the X f following numbers of 28 1 passengers in fifty flights of a 33 1 forty-seater plane. 34 2 Highest Frequency fm = 13 35 3 occurs against the X value 13. 36 5 37 7 Hence: 38 10 Mode = ˆ X = 39 39 13 40 8 Total 50 THE MODE IN CASE OF THE FREQUENCY DISTRIBUTION OF A CONTINUOUS VARIABLE: In case of grouped data, the modal group is easily recognizable (the one that has the highest frequency). At what point within the modal group does the mode lie? Mode: ˆ f m f1 X l xh f m f1 f m f 2 where l = lower class boundary of the modal class, fm = frequency of the modal class, f1 = frequency of the class preceding the modal class, f2 = frequency of the class following modal class, and h = length of class interval of the modal class EPA MILEAGE RATINGS Class Class Frequency Limit Boundaries 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 EPA MILEAGE RATINGS Class Class No. of Limits Boundaries Cars 30.0 – 32.9 29.95 – 32.95 2 33.0 – 35.9 32.95 – 35.95 4 = f1 36.0 – 38.9 35.95 – 38.95 14 = fm 39.0 – 41.9 38.95 – 41.95 8 = f2 42.0 – 44.9 41.95 – 44.95 2 It is evident that the third class is the modal class. The mode lies somewhere between 35.95 and 38.95. In order to apply the formula for the mode, we note that fm = 14, f1 = 4 and f2 = 8. Hence we obtain: Hence, we obtained: ˆ 14 4 X 35.95 3 14 4 14 8 10 35.95 3 10 6 35.95 1.875 37.825 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 .9 5 44 .9 5 X The frequency polygon of the same distribution was: 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 Frequency curve was as indicated by the dotted line in the following figure: 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 In this example, the mode is 37.825, and if we locate this value on the X-axis, we obtain the following picture: 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 ˆ X = 37.825 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 Convert this data into Frequency Distribution and find the modal age. 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 Mode: ˆ f m f1 X l xh f m f1 f m f 2 where l = lower class boundary of the modal class, fm= frequency of the modal class, f1 = frequency of the class preceding the modal class, f2 = frequency of the class following modal class, and h = length of class interval of the modal class Hence, the mode is given by ˆ 18 6 X 29.5 10 18 6 18 11 12 29.5 10 12 7 120 29.5 19 29.5 6.3 35.8 ˆ X 35.8 Mode: Y 20 No. of Managers 15 10 5 0 X 19.5 29.5 39.5 49.5 59.5 69.5 79.5 Ages of Managers Mode = 35.8 PROPERTIES OF THE MODE •The mode is easily understood and easily ascertained in case of a discrete frequency distribution. •It is not affected by a few very high or low values. The question arises, “When should we use the mode?” The answer to this question is that the mode is a valuable concept in certain situations such as the one described below: EXAMPLE Suppose the manager of a men’s clothing store is asked about the average size of hats sold. He will probably think not of the arithmetic or geometric mean size, or indeed the median size. Instead, he will in all likelihood quote that particular size which is sold most often. This average is of far more use to him as a businessman than the arithmetic mean, geometric mean or the median. The modal size of all clothing is the size which the businessman must stock in the greatest quantity and variety in comparison with other sizes. On the other hand, sometimes a frequency distribution contains two modes in which case it is called a bi-modal distribution as shown below: THE BI-MODAL FREQUENCY DISTRIBUTION f 0 X

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