Page 155 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes From the given information we can easily determine the number of observations in group II,
i.e, n = n - n - n = 200 - 50 - 90 = 60.
2 1 3
Further the relation between means is given by

n X 1 n X 2 n X 3
2
1
3
X
n 1 n 2 n 3
n n n X n X 1 n X 3
1 2 3 1 3 200 116 50 113 90 115
X 2 = = 120
n 2 60
To determine s , consider the following relationship between variances:
3
2 2 2 2 2 2 2
n = n ( + d ) + n ( + d ) + n ( + d )
1 1 1 2 2 2 3 3 3
2 2 2 2 2
n n d n d
2 1 1 1 2 2 2 2
or 3 d 3
n 3
Here d = 113 - 116 = - 3, d = 120 - 116 = 4, d = 115 – 116 = –1
1 2 3
2
.
200 7 746 50 36 9 60 49 16
2
3 1 = 64. Thus, 3 = 8.
90
7.8.3 Properties of Standard Deviation
1. Standard deviation of a given set of observations is not greater than any other root mean
1 2 1 2
square deviation, i.e. X i X X i A .
n n
2. Standard Deviation of a given set of observations is not less than mean deviation from
mean, i.e., Standard Deviation Mean Deviation from mean.
3. In an approximately normal distribution, X covers about 68% of the distribution,
X 2 covers about 95% of the distribution and X 3 covers about 99%, i.e., almost
whole of the distribution. This is an Empirical Rule that is based on the observations of
several bell shaped symmetrical distributions. This rule is helpful in determining whether
the deviation of a particular value from its mean is unusual or not. The deviations of more
than 2s are regarded as unusual and warrant some remedial action. Furthermore, all
observations with deviations of more than 3s from their mean are regarded as not
belonging to the given data set.

7.8.4 Merits, Demerits and Uses of Standard Deviation

Merits

1. It is a rigidly defined measure of dispersion.

2. It is based on all the observations.
3. It is capable of being treated mathematically. For example, if standard deviations of a
number of groups are known, their combined standard deviation can be computed.

4. It is not very much affected by the fluctuations of sampling and, therefore, is widely used
in sampling theory and test of significance.

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