2. Department of Economics
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. Department of Economics
1.1 Objectives
After going through this unit, you will be able to:
Define the term Arithmetic Mean;
Explain combined and weighted Arithmetic Mean;
Describe the relation between Mean, Mode and Median;
Describe the calculation of Mean in in individual, discrete and continuous
series by different methods; and
Explain the advantages and disadvantages of Mean.
4. Department of Economics
1.2 Measures of Central Tendency
It refers to a single central number or value that condenses the mass data &
enables us to give an idea about the whole or entire data. Central tendency is a
statistical measure that determines a single value that accurately describes the
center of the distribution and represents the entire distribution of scores. 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.
Types:
1. Arithmetic Mean
2. Median
3. Mode
4. Geometric Mean
5. Harmonic Mean
5. Department of Economics
1.3 Arithmetic Mean
It is the most commonly used measure of central tendency. It is also called as
‘Average’ .It is defined as additional or summation of all individual
observations divided by the total number of observation.
Horace Secrist, “The arithmetic mean of a series is the figure obtained by
dividing the sum of value of all items by their number”
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 ( X) by the number of
Ʃ items (N) constituting the series. Thus, mean of a set of numbers X1, X2, X3,
………..Xn denoted by and is defined as
1.3.1 Definition
Arithmetic Mean = 𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
6. Department of Economics
1.3.2 Calculation of Mean
1. Individual Series
• Direct Method = ƩX
N
X = Stands for value
̅X͞ = Stands for Mean
ƩX =Stands for summation of values
N = Stands for numbers of items
Example: Roll
No.
1 2 3 4 5 6 7 8 9 10 Total
Mark
s
110 190 160 165 200 190 150 200 165 160 1690
X͞ = ƩX = 1690 = 169 marks
N 10
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2. Discrete series :
Example:
Wages (X) Rs. 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5
No. of persons (f) 35 40 48 100 125 87 43 22
Wages
(X) Rs.
No. of persons
( f)
Total Wages
(f x X)
4.5 35 157.5
5.5 40 220.0
6.5 48 312.0
7.5 100 750.0
8.5 125 1062.5
9.5 87 826.5
10.5 43 451.5
11.5 22 253.0
N= 500 ƩfX= 4033.0
• Direct Method
X = ƩfX
N
= 4033
500
Mean Wage =Rs. 8.07
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2. Discrete series :
1. One of the values should be taken as assumed average (A)
2. dx = (X- A)
3. Deviation multiplying by their frequency and there after summate the product (Ʃfdx)
4. Formula X̅ = A + Ʃfdx
• N
Wages
(X) Rs.
No. of
persons
( f)
Deviatio
ns from
A= 8.5
dx
Total Wages
(f x dx)
4.5 35 -4 -140
5.5 40 -3 -120
6.5 48 -2 -96
7.5 100 -1 -100
8.5 125 0 0
9.5 87 1 87
10.5 43 2 86
11.5 22 3 66
N= 500 Ʃfdx= -217
Short-cut Method
X̅ = A + Ʃfdx
N
= 8.5 + (-217)
500
= 8.066
X̅ Arithmetic mean
͞A Assumed mean
Ʃfdx Product of deviations from assumed
mean and f
N Total no. of items
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3. Continuous series
• Direct method
1. In the continuous series, the arithmetic average is determined in the same manner as in case
of discrete series except that central or mid-value of class groups are assumed as ‘X’ .
Example:
Height (in
ft.) (X)
Frequency
(f)
0-7 26
7-14 31
14-21 35
21-28 42
28-35 82
35-42 71
42-49 54
49-56 19
360
Height (in
ft.)
Mid-
value
(X)
Frequency (f) f x X
0-7 3.5 26 91.0
7-14 10.5 31 325.5
14-21 17.5 35 612.5
21-28 24.5 42 1029.0
28-35 31.5 82 2583.0
35-42 38.5 71 2733.5
42-49 45.5 54 2457.0
49-56 52.5 19 997.5
360 (N) ƩfX
10829.0
X =ƩfX = 10829
N 360
Mean height =
30.08 ft
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• Short cut method
Height (in ft.) Mid-value
(X)
Frequency (f) Deviation
from A =31.5
dx
f x dx
0-7 3.5 26 -28 -728
7-14 10.5 31 -21 -651
14-21 17.5 35 -14 -490
21-28 24.5 42 -7 -294
28-35 31.5 82 0 0
35-42 38.5 71 7 497
42-49 45.5 54 14 756
49-56 52.5 19 21 399
Total 360 (N) Ʃfdx = -511
X̅ =A + Ʃfdx
N
= 31.5 + (-511)
360
= 30.08 ft.
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• Step Deviation method
1.One value, usually the middle value is assumed as an (A)
2. d́ = X – A
i
3. Find out Ʃf d́
4. Formula X̅ =A + Ʃf d́ x i
N
X̅ =A + Ʃf d́ x i
N
= 31.5 + (-73) x 7
360
= 30.08 ft.
Height (in ft.) Mid-value
(X)
Frequency (f) Deviation
from A =31.5
d́ =X – A
i
f x d́
0-7 3.5 26 -4 -104
7-14 10.5 31 -3 -93
14-21 17.5 35 -2 -70
21-28 24.5 42 -1 -42
28-35 31.5 82 0 0
35-42 38.5 71 1 71
42-49 45.5 54 2 108
49-56 52.5 19 3 57
Total 360 (N) Ʃfd́ = -73
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1.4 Combined Arithmetic Mean
If mean of two or more components of a group are given separately along with the
number of items, the combined mean of the whole group can be ascertained
Formula
Combined Mean X̅ = X̅ 1N1 + X̅ 2N2 + X̅ 3N3 …………….+ X̅ nNn
N1 +N2 +N3…………….Nn
X̅ 1 , X̅ 2,X̅ 3 are the arithmetic mean of different components
N1, N2, N3 are the number of items of different components
13. Department of Economics
1.5 Weighted Arithmetic Mean
Example:
X̅ = 3 x 3.75 + 4 x 3.5 + 2 x 4
3 + 4 + 2
Credit
(Weight)
Grade
3.00 3.75
4.00 3.5
2.00 4
Sometimes we associate with the numbers X1, X2, X3,……..XN with certain weighting
factors w1, w2, w3,……..wN depending on the importance of that number
X̅ = 𝑤1 𝑋1 + 𝑤2 𝑋2 + 𝑤3 𝑋3 + …….. + 𝑤𝑛𝑋𝑁 = Ʃ𝑤𝑋 N
𝑤1 + 𝑤2 + 𝑤3 + …….. + 𝑤N Ʃ𝑤
Weighted mean = 3.69
14. Department of Economics
1.6. Advantages and Disadvantages of Mean
Advantages of Mean:
• It is easy to understand & simple calculate.
• It is based on all the values.
• It is rigidly defined .
• It is easy to understand the arithmetic average even if some of the details of
the data are lacking.
• It is not based on the position in the series.
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.
15. Department of Economics
1.7 Relation between Mean , Mode and Median
Mode = 3 Median – 2 Mean
• Symmetrical distribution
• Asymmetrical distribution
• Symmetrical distribution
1. The observations are equally
distributed.
2. The values of mean, median and
mode are always equal. i.e. Mean =
Median = Mode
• Asymmetrical distribution
1. The observations are not equally distributed.
2. Two possibilities are there:
Positively Skewed
Negatively Skewed
16. Department of Economics
1.8 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. The term average is used frequently in everyday
life to express an amount that is typical for a group of people or things. ...
Averages are useful because they: summaries a large amount of data into a
single value; and. indicate that there is some variability around this single value
within the original data . The mean is often used in research, academics and in
sports. When you watch a cricket match and you see the player's batting
average, that number represents the total number of hits divided by the number
of times at bat. In other words, that number is the mean. So, mean is also used
in daily life.
17. Department of Economics
1.9 Unit End Questions
1. Calculate Arithmetic Mean 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 arithmetic mean?
3. Write the formula of Combined arithmetic mean ?
18. Department of Economics
1.10 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.
