1 Measures of Central Tendency

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1 CHAPTER 1 Measures of Central Tendency 1.1 INTRODUCTION: STATISTICS It is that branch of science which deals with the collection of data, organising summarising, presenting and analysing data and making valid conclusions and drawing reasonable decisions on the basis of such study. The word statistics refers to some numerical facts relating to any phenomena in social sciences or exact sciences. Facts and figure pertaining to population, production, national income, profits, sales, bank rates, family patterns, dowry system, animal kingdom, plant life, bacteria will all constitute statistics. The word statistics seems to have derived from either the Latin word status or the Italian word statitsta both meaning a political state. The word statistics is presently referred to in two distinct senses. In its first reference as a plural noun, it means an aggregate of collection of numerical or quantitative expressions of facts, i.e., Numerical data or simply data. In its second reference as a singular noun, its means a body of principles and methods used in the collection, presentation, analysis and interpretation of numerical data. Bowley defines statistics as the science of counting in one context. The emphasis made here is only on the collection of data. At another place he says: Statistics may right be called the science of averages. Boddington defines Statistics as the science of estimates and probabilities. According to Lovitt, the science of statistics deals with the collection classification and tabulation of numerical facts. 1. AVERAGES OR MEASURES OF CENTRAL TENDENCY An average is a value which is representative of a set of data. Generally, it is difficult for the human mind to remember huge and unwiedly set of numerical data, to understand their complications and to compare them easily. Statistical methods help us to make the data simple, precise and understandable. In this reference statistical averages plays an important role. In other words, Statistical average is such a simple and precise value, that is used to represent all the values of a statistical series. Some of standard definitions are as follows: 1. According to A.E. Waugh, An average is a single value selected from a group of values to represent them in some way.. According to Croxton and Cowden, An average is a single value within the range of the data that is used to represent all the values in the series.

2 Business Statistics In this definition, two points are significant: (a) Average is a value that lies within the range of data. (b) Its objective is to find out representative value of a statistical series. 3. Simpson and Kafka state, A measure of central tendency is a typical value around which other figures congregates. 1.3 IMPORTANCE OF AVERAGES In view of Prof. Tippet, The average has its limitations, but if they are recognised, there is no single statistical quantity more valuable than average. The average plays an important role in statistical science. In modern practical life averages are required at each step. We usually discuss and refer an average income, average cost, average run rate, average price etc. Dr. Bowley even called Statistics as the science of average. 1.4 REQUISITES FOR A GOOD AVERAGE Following are the requisites of a good average: (a) It should be simple to understand and locate. (b) It should not be affected much by extreme observations. (c) It should be based on all the items or observations. (d) It should be rigidly defined so that the conclusion remains uniform irrespective of enumeration by any person. (e) It should be suitable for mathematical treatment. (f) It should be least affected by fluctuations of sampling. (g) An average should be such that it represents maximum features of a statistical series. (h) An average should be in terms of absolute value, i.e., it should not be expressed in terms of percentage or any other relative measurement. 1.5 TYPES OF AVERAGES There are different types of statistical averages. 1. Averages of position: (a) Median (b) Mode. Mathematical averages (a) Arithmetic mean (b) Geometric mean (c) Harmonic mean (d) Quadratic mean 3. Business averages: (a) Moving average (b) Composite average (c) Progressive average According to syllabus, in this Chapter, we only discuss Averages of position and Mathematical averages in detail (i.e., Mean, Median, Mode, Geometric mean, Harmonic mean) 1.6 AVERAGES OF POSITION Median Median of a distribution is the value of the variable which divides it into two equal parts. According to Connor, The median is that value of the variable which divides the group into two equal parts one

3 Measures of Central Tendency 3 part comprising all values greater and the other all values less than the median. Also according to Secrist Median of a series is the value of that item-actual or estimated when a series is arranged in order of magnitude which divides the distribution in two parts. Median is generally denoted by Calculation of median in individual series In case of ungrouped data, if the number of observations is odd then median is the middle value after the values have been arranged in ascending or descending order of magnitude i.e., if total number of th Ên + 1 items is n (odd) then median value is item. In case of even number of observations there are two middle items and the mean of the value of th th Ên Ên and + 1 items is defined as the median. Example 1: Find out the median of the following items: 5, 7, 9, 1, 10, 8, 7, 15, 1 Items arranged in ascending order of magnitude CHAPTER 1 Serial no. Size of items n the number of items = 9 (odd) M e th Ên + 1 = item th Ê9+ 1 th = item = 5 item = 9. Ans. Example : The following table gives the Economic Advisors index numbers of wholesale prices: Year Index numbers

4 4 Business Statistics Index numbers arranged in ascending order of magnitude Serial no. Size of items n the numer of items = 6 (even) and then Ên th th rd item = 3 item = Ên rd + 1 item = 4 item = Therefore Median = = = = Median = 384. Ans Calculation of median in discrete series In case of discrete frequency (grouped data) distribution median is obtained by considering the cumulative frequencies. Also in discrete series it is checked that values of item (x) are arranged. If not then they are arranged in ascending order with same frequency. The steps for calculating median are: (a) Find  f + 1. Ê f + 1 (b) See the cumulative frequency just greater than. (c) The corresponding value of x is median. Example 3: The following data relate to size of shoes sold at a store during a given week. Find the median. Size of shoes No. of pairs

5 Measures of Central Tendency 5 Size of shoes No. of pairs CHAPTER 1 M Size of shoes No. of pairs c.f. e = ( S f + 1) ( ) = ( 459) = = 9.5 = 9.5 lies in c.f. 94. Its corresponding value is 8.5, so = 8.5. Ans.

6 6 Business Statistics Example 4: Calculate median from the following frequency distribution: Marks No. of students Marks No. of students c.f. ( S f + 1) = ( ) = ( 40) = = 0 = 0 lies in c.f. 3. Its corresponding value is 45 so, = 45. Ans Calculation of median in continuous series In case of continuous frequency distribution, the class corresponding to the cummulative frequency just Sf greater than is called the median class and the value of median is obtained by the formula where, Ê Sf - c Median l = + i f l is the lower limit of the median class. f is the frequency of the median class. i is the width of the class interval. c is the c.f. of the class preceeding the median class. Sf is the total frequency of the data.

7 Measures of Central Tendency 7 Example 5: Calculate the median wages from the following particulars of daily wages of 95 persons: Daily wages in Rs. No. of wage earners CHAPTER 1 Daily wages in Rs. No. of wage earners c.f Ê Sf - c M e = l + i f S f 95 = = 47.5 Ê = Ê = Ê17.5 = Ê175 = = = 7. Ans.

8 8 Business Statistics Example 6: Calculate median from the following data: Age No. of students Age No. of students C.I. c.f Ê Sf - c M e = l + i f S f 100 = = 50 Ê = Ê50-30 = Ê0 5 = Ê100 = = = Ans.

9 Measures of Central Tendency 9 Example 7: Find the median from the following distribution: Size Frequency Below 10 5 Below 0 15 Below 30 3 Below Below Below Below Below Below CHAPTER 1 Size Frequency C.I. f c.f. Below Below Below Below Below Below Below Below Below Ê Sf - c M e = l + i f S f 50 = = 15 Ê = Ê15-95 = Ê30 10 = Ê300 = = = Ans.

10 10 Business Statistics Example 8: Calculate the median from the following data: Class Frequency Above Above Above 0 75 Above 30 5 Above 40 4 Above Above 60 5 Class Frequency C.I. f c.f. Above Above Above Above Above Above Above Ê Sf - c M e = l + i f S f 100 = = 50 Ê = Ê50-48 = Ê 10 = Ê0 = = = Ans.

11 Measures of Central Tendency 11 Example 9: Locate median from the following data: Mid-value Frequency CHAPTER 1 Mid-value Frequency C.I. c.f Ê Sf - c M e = l + i f S f 108 = = 54 Ê = Ê54-36 = Ê18 10 = Ê180 = M = e = 41. Ans.

12 1 Business Statistics Example 10: Find out the missing frequency at = 46 and Σf = 30, from the following data: Class Frequency Given Σf = 30, = 46 Class Frequency c.f D (say) 4 + D D E (say) D + E D + E D + E S f 30 = = 115 Ê Sf - c M e = l+ i f Ê115 -( 4 + D) 46 = Ê115 -( 4 + D) = = 10( 73 -D) 390 = D 10D = = 340 D = 34. Now, 30 = D + E 30 = E 30 = E E = E = 46. Ans. Hence missing frequency for the class interval is 34 and for class interval it is 46.

13 Measures of Central Tendency Merits and demerits of median Merits 1. The value of median is rigidly defined, which is an important property for an ideal average.. The median is very easy to understood and is easy to calculate. 3. Median is not affected by values of extreme items. 4. In certain cases median can be located merely by observation. 5. It can be calculated for distributions with open-end classes. Demerits 1. If the number of items in an individual series is in even number, the value of median cannot be determined exactly. We merely estimate it by taking the mean of two middle terms.. Median is not capable for further algebraic treatment. 3. For calculating median it is necessary to arrange the data in ascending order or descending order, which is not only time consuming but becomes tedious also. 4. Median is a positional average, so its value is not determined by each and every observation. CHAPTER 1 PRACTICE EXERCISE 1 1. Find out median wages of workers from the following data: Name of persons A B C D E F G H I Monthly wages in Rs Compute the median from the following data: [Ans. 0] S. no Marks Find the median from the data given below: [Ans. = 49] Size of items Frequency [Ans. = 7.5] 4. Find out the median from the following data: 35 men get at the rate of Rs per man 40 Rs Rs Rs Rs Rs Rs Rs [Ans. = 8.5]

14 14 Business Statistics 5. The table shows the age distribution of married females according to sample census of 001 in the U.P. state: Age No. of married Age No. of married Age No. of married females females females Calculate the median ages of married females. 6. Find the median from the following table: [Ans. 8.8 years] Size Frequency From the following table, find the median: [Ans. = 31.71] No. of days (absent) No. of students [Ans. = 1.15] 8. The following table gives the marks obtained by 65 students in statistics in certain examination: Examination marks No. of students More than 70% 7 More than 60% 18 More than 50% 40 More than 40% 40 More than 30% 63 More than 0% 65 Calculate the median of the series. 9. Find the median from the data given below: [Ans. = 53.6%] Mid-point Frequency [Ans. = 5.7]

15 Measures of Central Tendency Find out the median from the following frequency distribution: Class Frequency CHAPTER 1 [Ans. = 350] 1.6. Mode Mode is defined to be the size of the variable which occurs more frequently in a set of observations. In other words, mode of the frequency distribution is the value of the variable (x) corresponding to maximum frequency Calculation of mode in individual series In an individual series mode is find by observation and the value occurring maximum number of times is the modal value. If in the given data all the values are distinct then in that case highest value of x is the modal value of given data. Example 11: Find the mode from the following data: (A) 80, 170, 145, 50, 55, 317, 85, 350, 400. (B) 15, 35, 40, 4, 48, 50, 5, 4, 18, 1, 48, 54, 48, 5, 10, 55. (A) For the given data highest frequency is 400 and no one value is repeated. Therefore Mode M o = 400. Ans. (B) Since the frequency 48 occurs the maximum number of times, i.e., 3. Hence the mode for the given data is M o = 48. Ans Calculation of mode in discrete series In the case of discrete series there are two methods to solve the problem. (a) By inspection method: In this method, the size or value having the highest frequency will be identified as mode. Generally, mode is denoted by M o. Example 1: Calculate mode from the following data: Salary (in thousands) Frequency By inspection it is clear that the highest frequency is 18. So, its corresponding value 30 will be mode. Hence, Mode = 30 thousand (b) Grouping method: Grouping method can be used in the following conditions: (i) When the maximum frequency occurs either in very beginning or at the end of the distribution. (ii) The frequencies of the variables increase or decrease in a haphazard way. (iii) When the maximum frequency is repeated or approximately equal concentration is found in two or more neighbouring values. In grouping method, to solve the problem six columns are drawn in addition to the column of values (x) and the frequencies are grouped in the following manner:

16 16 Business Statistics (a) first column: The frequencies given in the problem are shown in the first column. (b) Second column: In this column, frequencies are grouped into two s from the top of the value. (c) Third column: In this column, frequencies are again grouped into two s, but the first frequency is left out i.e., starts from the second frequency. (d) Fourth column: In this column, frequencies are grouped into three s, starting from the first frequency. (e) Fifth column: In this column, frequencies are grouped into three s but this grouping starts from the second frequency. (f) Sixth column: In this column, frequencies are grouped again in three s but this grouping starts from the third frequency. After preparing the grouping table, tallies are marked against the values having highest frequency in first column and the highest total in each of the other column. Finally, the value securing maximum tallies will be modal value. Example 13: Calculate the mode from the following data of the marks obtained by nine students: Marks No. of students Marks No. of students Ans. M o = 14

17 Measures of Central Tendency 17 Example 14: Find the mode in the following given data: Size of shoes Frequency CHAPTER 1 Size Frequency Analysis table } } } } } 55 I = 1 43 III = } } 73 IIII = 5 55 III = I = 1 } } } } } } } } From the Analysis Table, we have size 6 is repeated 5 times therefore its mode. M o = 6. Ans Calculation of mode in continuous series In case of continuous series it should be checked before the calculation of mode that each class interval should be equal or not. If they are not equal, then they should be equalised. After it there are two steps to find the mode. 1. Find the modal class by either observation or by grouping method (in case of irregular distribution).. Then modal value is obtained by the formula f1 - f0 Mo = l+ i f - f - f 1 0