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Unit 8: Descriptive Statistics
( + )M = L + U = L + (L + h) (where U = L + h) Notes
1 2 o m 2 m 1 m 2 m 1 m m
= ( + ) L + h
1 2 m 1
Dividing both sides by + , we have
1 2
L + 1 ´ h
M = m + ...(1)
o 1 2
By slight adjustment, the above formula can also be written in terms of the upper limit (U ) of
m
the modal class.
é ù
ê
M = U – h + 1 ´ h = U – 1 - 1 ú ´ h
o m + m ë + 2 û
1 2 1
é ù
= U – ê 2 ´ h ú ...(2)
m +
ë 1 2 û
Replacing by f – f and by f – f , the above equations can be written as
1 m 1 2 m 2
f - f
M = L + m 1 ´ h ...(3)
o m 2 f - f - f
m 1 2
f - f
and M = U – m 2 ´ h ...(4)
2 f - f - f 2
o m
m
1
Notes The above formulae are applicable only to a unimodal frequency distribution.
8.2.6 Relation between Mean, Median and Mode
The relationship between the above measures of central tendency will be interpreted in terms of
a continuous frequency curve.
If the number of observations of a frequency distribution are increased gradually, then
accordingly, we need to have more number of classes, for approximately the same range of
values of the variable, and simultaneously, the width of the corresponding classes would
decrease. Consequently, the histogram of the frequency distribution will get transformed into
a smooth frequency curve, as shown in Figure 8.5.
For a given distribution, the mean is the value of the variable which is the point of balance or
centre of gravity of the distribution. The median is the value such that half of the observations
are below it and remaining half are above it. In terms of the frequency curve, the total area under
the curve is divided into two equal parts by the ordinate at median. Mode of a distribution is a
value around which there is maximum concentration of observations and is given by the point
at which peak of the curve occurs.
For a symmetrical distribution, all the three measures of central tendency are equal i.e.
X = M d = M o , as shown in Figure 8.6.
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