Page 145 - DMGT404 RESEARCH_METHODOLOGY

Unit 8: Descriptive Statistics

2. When Data are in the form of an Ungrouped Frequency Distribution Notes
Let there be n values X , X , ..... X out of which X has occurred f times, X has occurred f
1 2 n 1 1 2 2
times, ..... X has occurred fn times. Let N be the total frequency, i.e.,
n
n
N =  i f
i 1
Alternatively, this can be written as follows:
Values X X ..... X Total Frequency
1 2 n
Frequency f f ..... f N
1 2 n
(a) Direct Method: The arithmetic mean of these observations using direct method is
given by



X  X  ... X  X  ... ... X  ...  X  ...  X

 1  2  n
n
1
1
2
X  1 F times 2 F times 3 F times
F  F  ... F
1 2 n
Since X + X + ... + X added F times can also be written F X . Similarly, by writing
1 1 1 1 1 1
other observation in same manner, we have
n
n
 f X i  f X
f X  f X  ...  f X i i i
X  1 1 2 2 n n  i 1  i 1
f  f  ...  f n N ...(1)
1 2 n  f i
i 1
(b) Shortcut Method: As before, we take the deviations of observations from an arbitrary
value A. The deviation of ith observation from A is d = X – A.
i i
Multiplying both sides by f we have f d = f (X – A)
i i i i i
Taking sum over all the observations



 fd = f (X – A) = f X – A f = f X – A × N

i
i
i
i
i
i
i
i
i
Dividing both sides by N we have
 f d  f X  f d

i i  i i – A  X – A or X  A  i i  A d
N N N
3. When Data are in the form of a Grouped Frequency Distribution
In a grouped frequency distribution, there are classes along with their respective
frequencies. Let l be the lower limit and u be the upper limit of i th class. Further, let the
i i
number of classes be n, so that i = 1, 2, ..... n. Also let f be the frequency of i th class. This
i
distribution can written in tabular form, as shown.
!
Caution Here u1 may or may not be equal to l2, i.e., the upper limit of a class may or may
not be equal to the lower limit of its following class.
It may be recalled here that, in a grouped frequency distribution, we only know the
number of observations in a particular class interval and not their individual magnitudes.
Therefore, to calculate mean, we have to make a fundamental assumption that the
observations in a class are uniformly distributed. Under this assumption, the mid-value of

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