Measures of central tendency median mode

Measures of central tendency median mode

1. DR REKHA CHOUDHARY Department of Economics Jai Narain Vyas University, Jodhpur Rajasthan ECONOMICS BASIC STATISTICS Measures of Centre Tendency: Median, Mode

2. 1.0 Introduction Measures of central tendency are statistical measures which describe the position of a distribution. They are also called statistics of location, and are the complement of statistics of dispersion, which provide information concerning the variance or distribution of observations. In the univariate context, the mean, median and mode are the most commonly used measures of central tendency. Computable values on a distribution that discuss the behavior of the center of a distribution.

3. 1.1 Objectives After going through this unit, you will be able to: Define the term Median, Mode; Explain the status of statistical averages; Describe the types of statistics averages; Describe the calculation of Median and identify the location of Mode by different methods; and Analyse the advantages and disadvantages of Median and Mode.  

4. 1.2 Measures of Central Tendency The value or the figure which represents the whole series is neither the lowest value in the series nor the highest it lies somewhere between these two extremes. 1.The average represents all the measurements made on a group, and gives a concise description of the group as a whole. 2.When two are more groups are measured, the central tendency provides the basis of comparison between them.

5. There are five common type, namely; Arithmetic Mean (AM) Median Mode Geometric Mean (GM) Harmonic Mean (HM) 1.3 Types of Averages Averages Arithmetic Mean (AM) Median Mode Geometric Mean (GM) Harmonic Mean (HM)

6.   1.4 Definition Statistical average Statistical average is such a simple and brief figure which throws light upon the main characteristics of statistical series Simpson and Kafka defined it as “ A measure of central tendency is a typical value around which other figures congregate” Waugh has expressed “An average stand for the whole group of which it forms a part yet represents the whole”. According to Croxton and Cowden, “An average is a single value within the range of the data which is used to represent all the values in the series. Since an average is somewhere with in the range of the data, it is sometimes called a measure of central value”

7. 1.5 Median Median is a central value of the distribution, or the value which divides the distribution in equal parts, each part containing equal number of items. Thus it is the central value of the variable, when the values are arranged in order of magnitude. Connor has defined as “ The median is that value of the variable which divides the group into two equal parts, one part comprising of all values greater, and the other, all values less than median” 1.5.1 Definition

8. Notation M is used for Median Notation N is used for number of items 1.5.2 Calculation of Median 1. Individual Series 1. Arrange the data in ascending or descending order. After arranging and writing serial numbers, the following formula should be used….. M =Size of (N + 1) th item 2 Example : 25, 15, 23, 40, 27, 25, 23, 25, 20 Arranged ascending 15, 20, 23, 23, 25, 25, 25, 47, 40 (Even Numbers) M = Size of (N + 1) th item = Size of (9 + 1) th item = 5 th item Median = 25 2 2 Example: 25, 15, 23, 40, 27, 25, 23, 25, 20, 41 Arranged ascending 15, 20, 23, 23, 25, 25, 25, 47, 40, 41 (Odd Numbers) M =Size of (N + 1) th item = Size of (10 + 1) th item = 5.5 th item = = Size of (5 + 6) th item = 25+25 =50 2 2 2 2 Median = 25

9. 2. Discrete series : i . Arrange the data in ascending or descending order. ii. Calculate the cumulative frequencies. iii. Apply the formula. Example: Example: Size of item 8 10 12 14 16 18 20 Frequency 3 7 12 28 10 9 6 Median =Size of (N +1) th item 2 Median =Size of (75 +1) th item 2 38 th item which lies in 50cf whose value is 14 So Median = 14 Size of item Frequency Cumulative Frequency 8 3 3 10 7 10 12 12 22 14 28 50 16 10 60 18 9 69 20 6 75 Total 75

10. . 3. Continuous series For calculation of median in a continuous frequency distribution… i Calculate the cumulative frequencies. ii. Then ascertain central or median items applying Median = Size of (N ) th item 2 iii. The cumulative frequency in which median item is first-located, the related class-interval iv. Apply the formula. Median = l + i (m – c) f Notation l for lower limit of median class Notation i for class –interval of median class Notation f for frequency of median -class Notation m for median number or N th item 2 Notation c for cumulative frequency of the class just preceding the median class

11.   Example: Continuous series Marks 0-10 10-20 20-30 30-40 40-50 No of students 8 30 40 12 10 Marks No of students (f) Cum. Freq. 0-10 8 8 10-20 30 38 c (l) 20-30 40 f 78 30-40 12 90 40-50 10 100 N =100 Median =Size of (N )th item 2 M =Size of (100)th item = 50 th item 2 Class (20-30) Formula M = l + i ( m-c) f M = 20+ 10 ( 50-38) = 23 40 Median =23

12. 1.6. Advantages and Disadvantages of Median Advantages of Median: •Median can be calculated in all distributions. •Median can be understood even by common people. •Median can be ascertained even with the extreme items. •It can be located graphically •It is most useful dealing with qualitative data Disadvantages of Median : • It is not based on all the values. • It is not capable of further mathematical treatment. • It is affected fluctuation of sampling. • In case of even no. of values it may not the value from the data.

13. 1.7 Mode Mode is the most frequent value or score in the distribution. It is defined as that value of the item in a series. It is denoted by the capital letter Z. highest point of the frequencies distribution curve. Croxton and Cowden : defined it as “the mode of a distribution is the value at the point armed with the item tend to most heavily concentrated. It may be regarded as the most typical of a series of value” The exact value of mode can be obtained by the following formula. Z=L + f1 –f0 x i 2f1 –f0-f2 1.7.1 Definition

14. 1.7.2 Location of Mode 5, 13, 9,7, 1, 9, 2, 9 , 11 Z=L1 + f1 –f0 x i 2f1 –f0-f2 Z = 3M – 2X Individual series By converting individual observation in discrete variable Example: 5, 13, 9,7, 1, 9, 2, 9 and 11 Mode Mode : 9 By converting individual series in continuous series Example : When in a distribution, no value is found to be maximum point of concertation, then the given information should be changed the maximum frequency, which will be the model class. Thereafter, a formula is used to ascertain the modal value By estimating the value of Mode, with the help of Median and Mean

15. (II) Discrete series Weight 48 49 50 51 52 53 No. of Students 4 10 20 11 3 2 Inspection Method Example: The following table gives the weight (in Kgs) of 50 students of a class. Determine the modal weight. Maximum frequency is 20 whose value or size is 50. As such modal weight is 50kg Grouping Method When the distribution of frequencies is not regular and it is difficult to locate maximum frequency, grouping method is adopted. Following are the steps ….. Column ( i ) the frequencies given in the questions are written Column (ii) frequencies are grouped in twos starting from the top Column (iii) frequencies are again grouped in twos, leaving the first frequency. Column (iv) frequencies are group in three’s from the top Column (v) frequencies are again grouped in three’s leaving first frequency Column (vi) frequencies are again grouped in three’s leaving first and second frequencies or grouping will start from the top third frequency In a irregular variable, such a value may be a modal value which is not the highest frequency value but where there is more concentration in its vicinity. The process of grouping will make this point clear

16. ( III) Continuous series Mode for Grouped data Class boundaries Frequency 67.5-87.5 10 87.5-107.5 13 107.5-127.5 15 127.5-147.5 9 147.5-167.5 4 𝒇 0 𝒇 1 𝒇 2  L = Lower Class Boundary of the modal class 𝒇 0 = Preceding frequency of the modal class 𝒇 𝟏 =Highest Frequency 𝒇 𝟐 = Following frequency of the modal class i = Width of class interval 𝑀𝑜𝑑𝑒 Z=L + f1 –f0 x i 2f1 –f0-f2 𝑀𝑜𝑑𝑒 Z=107.5 + 15–13 x 20 2 x15 –13-9 𝑴𝒐𝒅𝒆 = 112.5

17. 1.8 Advantages and Disadvantages of Mode Advantages of Mode : • Mode is readily comprehensible and easily calculated • It is the best representative of data • It is not at all affected by extreme value. • The value of mode can also be determined graphically. • It is usually an actual value of an important part of the series.   Disadvantages of Mode : • It is not based on all observations. • It is not capable of further mathematical manipulation. • Mode is affected to a great extent by sampling fluctuations. • Choice of grouping has great influence on the value of mode .

18. 1.9 Let us Sum up In conclusion, a measure of central tendency is a measure that tells us where the middle of a bunch of data lies. Median is the number present in middle when the numbers in a set of data are arranged in ascending or descending order. If the number of numbers in a data set is even. Then the middle numbers. Mode is the value that occurs most frequently in set of data. You may hear about the median salary for a country or city. When the average income for a country is discussed, the median is most often used because it represents the middle of a group. So, Median and Mode used in daily life also.

19. 1.10 Unit End Questions     1. Calculate median and mode from the following data . Income (in Rs ) No. of persons 50-100 15 50-150 29 50-200 46 50-250 75 50- 300 90 2. Write the advantages and disadvantages of Median and Mode ?

20. 1.11 Suggested Readings Asthana H.S, and Bhushan , B.(2007) Statistics for Social Sciences (with SPSS Applications). Prentice Hall of India B.L.Aggrawal (2009). Basic Statistics . New Age International Publisher, Delhi. Gupta, S.C.(1990) Fundamentals of Statistics . Himalaya Publishing House, Mumbai Elhance , D.N: Fundamental of Statistics Singhal , M.L: Elements of Statistics Nagar, A.L. and Das, R.K.: Basic Statistics Croxton Cowden: Applied General Statistics Nagar, K.N.: Sankhyiki ke mool tatva Gupta, BN : Sankhyiki Balasubramanian  , P., &  Baladhandayutham , A.  (2011).Research methodology in library science. (pp. 164- 170). New Delhi: Deep & Deep Publications.