Measures of central tendency median mode

1. DR REKHACHOUDHARY
Department of Economics
Jai NarainVyas University,Jodhpur
Rajasthan
ECONOMICS
BASIC STATISTICS

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 = 5th 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 = 5.5th 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
38th 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 = 50th 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
(I) 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.