Prof. Rajkumar TeotiaInstitute of Advanced Management and Research (IAMR)
Address: 9th Km Stone, NH-58, Delhi-Meerut Road, Duhai,Ghaziabad (U.P) - 201206
Ph:0120-2675904/905 Mob:9999052997 Fax: 0120-2679145e mail: [email protected]
Measures of central tendency
A measure of central tendency is a single value that
attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location.
The mean, median and mode are all valid measures of central tendency but, under different conditions, some measures of central tendency become more appropriate to use than others.
Measures of central tendency
Mean or Arithmetic mean refers to the sum of all variables divided by the number of variables. It is the most popular technique of finding out the average of values. It is also referred to as arithmetic mean or average in day-to-day life.
ARITHMETIC MEAN
For Ungrouped data, arithmetic mean may be calculated by applying any of the following methods:
Direct method
Short-cut method
ARITHMETIC MEAN – FOR UNGROUPED DATA
It is calculated by the following formula:
Where, X = arithmetic mean of the population N = total number of observations in a population. Σ x = Sum of all observation
Direct method:
X = Σ x N
Example: Find out the arithmetic mean of the following observation 10, 15, 30, 7, 42, 79 and 83
Solution: Symbolically, the arithmetic mean is
Σ x = 10 + 15 + 30 + 7 + 42 + 79 + 83 = 266N = 7
X = Σ x = 266 = 38 N 7
X = Σ x N
In the case of short-cut method, the concept of arbitrary mean is followed. The formula for calculation of the arithmetic mean by the short-cut method is given below:
Where, A = arbitrary or assumed mean d = deviation from the arbitrary or assumed mean (d = X - A) N = total number of observations in a population
Short-cut method:
X = A + Σ d N
Example: Data of imports of a certain firms for the year 2010 are mentioned in the following table:
From the above data calculate average value of imports for these
firms using short-cut method.
firms A B C D E F GValue of imports (Rs)
27 32 34 39 56 62 70
Solution: The formula for calculation of the arithmetic mean by the short-cut method is given below:
X = A + Σ d N
Firms Importsx
d = x – AA = 39
ABCDEFG
27323439566270
-12-7-50
172331
N = 7 Σd = 47
By applying the following formula arithmetic mean will be
= 39 + 47 7
= Rs. 45.71
X = A + Σ d N
For grouped data, arithmetic mean may be calculated by applying any of the following methods:
Direct method Short-cut method Step-deviation method
ARITHMETIC MEAN – FOR GROUPED DATA
In the case of direct method, the following formula is used
Where, X = arithmetic mean of the population N = total number of observations in a population. X = mid-point of various classes f = the frequency of each class
DIRECT METHOD – FOR GROUPED DATA
X = Σ f X N
Example: The following table gives the marks of 58 students in Statistics. Calculate the average marks of this group.
Solution:In the case of direct method, the following formula is used
X = Σ fx = 1940 = 33.45 Marks N 58
X = Σ f x N
Marks Mid-pointx
No. of Studentsf
f x
0-10 5 4 20
10-20 15 8 120
20-30 25 11 275
30-40 35 15 525
40-50 45 12 540
50-60 55 6 330
60-70 65 2 130
N = 58 Σ fx = 1940
In the case of short-cut method, the concept of arbitrary mean is followed. The formula for calculation of the arithmetic mean by the short-cut method is given below:
Where,A = arbitrary or assumed meanf = frequency d = deviation from the arbitrary or assumed mean (d = x - A) When the values are extremely large and/or in fractions, the use of
the direct method would be very cumbersome. In such cases, the short-cut method is preferable. This is because the calculation work in the short-cut method is considerably reduced
SHORT-CUT METHOD FOR GROUPED DATA
Example: The following table gives the marks of 58 students in Statistics. Calculate the average marks of this group. By Short-cut Method
Solution:
Here Assumed mean (A) = 35 And (d) = x – A
Marks M id -p o in t
f d = x – A
fd x
0 -1 0 5 4 -3 0 -120 1 0 - 2 0 1 5 8 -2 0 -160 2 0 - 3 0 2 5 1 1 -1 0 -110 3 0 - 4 0 3 5 1 5 0 0 4 0 - 5 0 4 5 1 2 1 0 1 2 0 5 0 - 6 0 5 5 6 2 0 1 2 0 6 0 - 7 0 6 5 2 3 0 60
∑ f d = - 9 0
Following formula will be used for calculating arithmetic mean by step- deviation method
Where, A = arbitrary or assumed mean f = frequency d = deviation from the arbitrary or assumed mean d = X – A i i = class size
STEP-DEVIATION METHOD FOR GROUPED DATA
X = A + Σ f d x i N
Example: The following table gives the marks of 58 students in Statistics. Calculate the average marks of this group By Step-deviation method
Solution: Following formula will be used for calculating arithmetic mean by step- deviation method
X = A + Σ f d x i N
MarksMid-point
x fd = x – A
i Fd0-10 5 4 -3 -12
10-20 15 8 -2 -1620-30 25 11 -1 -1130-40 35 15 0 040-50 45 12 1 1250-60 55 6 2 1260-70 65 2 3 6
∑fd = -9
ARITHMETIC MEAN – FOR DISCRETE SERIES (Where frequencies are given)
In the case of direct method, the following formula is used
Where,
X = arithmetic mean of the population N = total number of observations in a population. x = mid-point of various classes f = the frequency of each class
X = Σ f x N
Short – cut method: In case of Short - cut method the following formula is used
ARITHMETIC MEAN – FOR DISCRETE SERIES (Where frequencies are given)
X = A + ∑ f d
N
X = A + ∑ f d
N
Example- from the following data calculate arithmetic mean by direct method
Solution In the case of direct method, the following formula is used
Marks 5 15 25 35 45 55
No. of students 10 20 30 50 40 30
X = Σ f x N
X = Σ fx = 6300 = 35 N 180
Marks (x) No. of students (f) fx
51525354555
102030504030
50300750
175018001650
N = 180 Σ fx = 6300
Example- from the following data calculate arithmetic mean by Short - cut method
Solution: In case of Short - cut method the following formula is used
Marks 5 15 25 35 45 55
No. of students 10 20 30 50 40 30
X = A + ∑ f d
N
Marks (x) No. of students (f) d = X - A fd
51525354555
102030504030
-35-25-15-55
15
-350-500-450-250200450
N = 180 ∑ f d = - 900
X = 40 - 900 = 35 marks
180
Example- from the following data calculate arithmetic mean by direct method
Solution: - let us first convert less than series into grouped data
ARITHMETIC MEAN – FOR GROUPED DATA (In case of less than series)
Marks less than 10 20 30 40 50 60
No. of students 10 20 30 50 40 30
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of students 10 20 30 50 40 30
In the case of direct method, the following formula is used
X = Σ f x = 6300 = 35 N 180
X = Σ f x N
Marks Mid point (x) No. of students (f) fx0-1010-2020-3030-4040-5050-60
51525354555
102030504030
50300750
175018001650
N = 180 Σ fx = 6300
Example- from the following data calculate arithmetic mean by direct method
Solution: - let us first convert more than series into grouped data
ARITHMETIC MEAN – FOR GROUPED DATA (In case of more than series)
Marks more than 0 10 20 30 40 50 60
No. of students 10 20 30 50 40 30 0
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of students 10 20 30 50 40 30
In the case of direct method, the following formula is used
X = Σ f x N
Marks Mid point (x) No. of students (f) fx
0-1010-2020-3030-4040-5050-60
51525354555
102030504030
50300750175018001650
N = 180 Σ fx = 6300
X = Σ f x = 6300 = 35 N 180
Example- from the following data calculate arithmetic mean by direct method
ARITHMETIC MEAN – FOR GROUPED DATA (In case inclusive series)
Marks 1-10 11-20 21-30 31-40 41-50 51-60
No. of students 10 20 30 50 40 30
Solution: -
Now applying the following formula X = A + Σ fd X i
N
= 45.5 + (-180 x 10) = 35.5 180
1
Marks Mid point (x) No. of students (f) d = x – A i
fd
0.5-10.510.5-20.520.5-30.530.5-40.540.5-50.550.5-60.5
5.515.525.535.545.555.5
102030504030
-4-3-2-101
-40-60-60-500
30
N = 180 Σ fd = - 180
To calculate weighted mean, each variable is multiply by its weight, repeat the multiplication process for all variables and divide this sum by sum of all weights.
OR XW
= Σ WX
Σ W
Weighted Arithmetic mean:-
XW = W1X1
+ W2X2 + W3X3 + ……………… + WnXn
W1 + W2 + W1 + …………………..+ Wn
Where,
XW = weighted arithmetic mean
W = Weight of individual variable Σ W = Total Weight
Example: The performance of a student manager in a business school was
evaluated as follows. Calculate weighted mean
Basis Marks obtained (x) Weights (W)
Class testsPresentationAttendanceClass participationFinal examination
3836152055
101551060
Solution:-
Weighted mean = ΣWX = 4495 = 44.95 ΣW 100
Basis Marks obtained (x) Weights (W) WX
Class testsPresentationAttendanceClass participationFinal examination
3836152055
101551060
38054075200
3300
N = 5 Σx = 164 Σw = 100 ΣWX = 4495
If the arithmetic mean and the number of observation of two or more related groups are known we can calculate the combined arithmetic mean of these groups. The combined mean formula is
Combined mean:
X12….k = N1X1 + N2X2 + N3X3 + ……………………+ NKXK
N1 + N2 + N3 + ………………………+ Nk
Example: the mean marks of 60 students in section A is 40 and the mean marks of 40 students in section B is 45. Find the combined mean of the 100 students in both sections.
Solution: Here, N1 = 60, N2 = 40,
X1 = 40, X2 =45
By using the following formula combined mean will be
X12 = N1X1 + N2X2
N1 + N2
= 60 x 40 + 40 x 45 60 + 40
= 42 marks
Some of the important characteristics of the arithmetic mean are:
The sum of the deviations of the individual items from the arithmetic mean is always zero.
i,e ∑ (X – X) = 0
The sum of the squares of deviations of the individual items from the arithmetic mean is always minimum i,e ∑ (X – X )2 is always minimum.
As the arithmetic mean is based on all the items in a series, a change in the value of any item will lead to a change in the value of the arithmetic mean.
In the case of highly skewed distribution, the arithmetic mean may get distorted on account of a few items with extreme values.
CHARACTERISTICS OF THE ARITHMETIC MEAN
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