Measures of Central Tendency: Mean, Median, Mode: By : Girik Pachauri (10/IEC/016) Jaiprakash Nagar (10/IEC/020) Kaushal Kumar (10/IEC/023) Measures of Central Tendency: Mean, Median, Mode 1 Measures of Central Tendency: Mean, Median & Mode
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. Introduction: 2 Measures of Central Tendency: Mean, Median & Mode
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. The average represents all the measurements made on a group, and gives a concise description of the group as a whole. When two are more groups are measured, the central tendency provides the basis of comparison between them. Measures of Central Tendency 3 Measures of Central Tendency: Mean, Median & Mode
1. Arithmetic Mean: Arithmetic mean is a mathematical average and it is the most popular measures of central tendency. It is frequently referred to as ‘mean’ it is obtained by dividing sum of the values of all observations in a series (ƩX) by the number of items (N) constituting the series. Thus, mean of a set of numbers X1, X2, X3,………..Xn denoted by x̅ and is defined as 1. Arithmetic Mean 4 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Arithmetic Mean Calculated Methods : Direct Method : Short cut method : Step deviation Method : 5 Measures of Central Tendency: Mean, Median & Mode
Example : Calculated the Arithmetic Mean DIRC Monthly Users Statistics in the University Library : Month No. of Working Days Total Users Average Users per month Sep-2011 24 11618 484.08 Oct-2011 21 8857 421.76 Nov-2011 23 11459 498.22 Dec-2011 25 8841 353.64 Jan-2012 24 5478 228.25 Feb-2012 23 10811 470.04 Total 140 57064 Example : Calculated the Arithmetic Mean DIRC Monthly Users Statistics in the University Library 6 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: = 407.6 7 Measures of Central Tendency: Mean, Median & Mode
Advantages of Mean: : It is easy to understand & simple calculate. It is based on all the values. It is rigidly defined . It is possible to calculate arithmetic average even if some of the details of the data are lacking. It is a calculated value, and not based on its position in the series. Advantages of Mean: 8 Measures of Central Tendency: Mean, Median & Mode
Disadvantages of Mean:: It is affected by extreme values . It cannot be calculated for open end classes. It cannot be located graphically It gives misleading conclusions. It has upward bias. Disadvantages of Mean: 9 Measures of Central Tendency: Mean, Median & Mode
2.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” 2.Median 10 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Calculation of Median –Discrete series : Arrange the data in ascending or descending order. Calculate the cumulative frequencies. Apply the formula. 11 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Calculation of median – Continuous series For calculation of median in a continuous frequency distribution the following formula will be employed. Algebraically, 12 Measures of Central Tendency: Mean, Median & Mode
Example: Median of a set Grouped Data in a Distribution of Respondents by age : Age Group Frequency of Median class(f) Cumulative frequencies( cf ) 0-20 15 15 20-40 32 47 40-60 54 101 60-80 30 131 80-100 19 150 Total 150 Example: Median of a set Grouped Data in a Distribution of Respondents by age 13 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Median (M)=40+ 40+ = = 40+0.52X20 = 40+10.37 = 50.37 14 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Advantages of Median: Median can be calculated in all distributions. Median can be understood even by common people. Median can be located even if the data are incomplete. It can be located graphically. It is most useful dealing with qualitative data 15 Measures of Central Tendency: Mean, Median & Mode
Disadvantages of Median:: A slight change in the series may bring drastic change in median value. It is not capable of further mathematical treatment. It is affected by fluctuation of sampling. In case of even number of items or continuous series, median is an estimated value other than any value in the series. Disadvantages of Median: 16 Measures of Central Tendency: Mean, Median & Mode
3. 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. It is the highest point of the frequencies distribution curve . 3. Mode 17 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: 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 1 + 18 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Monthly rent (Rs) Number of Libraries (f) 500-1000 5 1000-1500 10 1500-2000 8 2000-2500 16 2500-3000 14 3000 & Above 12 Total 65 Example: Calculate Mode for the distribution of monthly rent Paid by Libraries in Karnataka 19 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Z=2000+ Z =2000+ Z=2400 Z=2000+0.8 ×500=400 20 Measures of Central Tendency: Mean, Median & 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. Advantages of Mode : 21 Measures of Central Tendency: Mean, Median & Mode
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 . Disadvantages of Mode : 22 Measures of Central Tendency: Mean, Median & Mode
Conclusion: A measure of central tendency is a measure that tells us where the middle of a bunch of data lies. Mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average. Conclusion 23 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Median is the number present in the 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 median is the mean of the two middle numbers. Mode is the value that occurs most frequently in a set of data. 24 Measures of Central Tendency: Mean, Median & Mode
References: 1. Balasubramanian , P., & Baladhandayutham , A. (2011). Research methodology in library science . (pp. 164-170). New Delhi: Deep & Deep Publications. 2. Sehgal , R. L. (1998). Statistical techniques for librarians . (pp. 117-130). New Delhi: Ess Ess Publications. 3. Busha,Charles , H., & Harter,Stephen , P. (1980). Research methods in librarianship: techniques and interpretation . (pp. 372-395). New York: Academic Press. 4. Krishnaswami , O. R. (2002). Methodology of research in social sciences . (pp. 361-366). Mumbai: Himalaya Publishing House. 5. Kumar,Arvind . (2002). Research methodology in social science . (pp. 278-289). New Delhi: Sarup & Sons. References 25 Measures of Central Tendency: Mean, Median & Mode
PowerPoint Presentation: Thank You 26 Measures of Central Tendency: Mean, Median & Mode