Question

If ${Q_3} + {\text{ }}{Q_1} = 108$,\[{Q_3}-{\text{ }}{Q_1} = 74\], then Q.D. is _____________.
A. $33$
B. $3.7$
C. $37$
D. $0.68$

Answer 1

Hint: In the question, we have to find the Q.D. that is the Quartile Deviation. Quartile deviation is the difference between the third and first quartile, then the difference is divided by two. By using this concept, we get the required answer.

Formula used: Quartile divides the data set into four parts. Each part contains an equal number of values. The first quartile represents the $25\% $ of the data. The second quartile represents $50\% $ of the data. The third quartile represents $75\% $ of the data. The second quartile also stands for the median of the dataset. The fourth quartile represents the highest $25\% $ of the data set.
To find quartile deviation of a dataset, we need to use the following formula.
$Q.D. = \dfrac{{{Q_3} - {Q_1}}}{2}$
Where, Q3 is the third quartile, Q1 is the first quartile.

Complete step by step solution:
Here, for the given problem, it is stated that ${Q_3} + {\text{ }}{Q_1} = 108$ and \[{Q_3}-{\text{ }}{Q_1} = 74\]
We have to find the quartile deviation by using the formula,
$Q.D. = \dfrac{{{Q_3} - {Q_1}}}{2}$
Quartile deviation is used to analyse the spread of a dataset or a distribution about a central value that is usually obtained by the measures of central tendency. It is also known as semi interquartile range.
Here, $Q.D. = \dfrac{{{Q_3} - {Q_1}}}{2}$
It is given that \[{Q_3}-{\text{ }}{Q_1} = 74\]
So, $Q.D. = \dfrac{{{{74}_{}}}}{2}$
On dividing the terms and we get
$ \Rightarrow 37$
The Quartile deviation is \[37\].

Hence, Option C is correct.

Note: Quartile deviation is easily understood. It is very easy to calculate. It takes an account of the middle $50\% $ of the dataset. It does not involve much complicated calculations. It is not affected by extreme values falling at any end of the data set. This is the best method to deal with the middle values of a data set. Still it is not a satisfactory measure to conclude about any data set. No concrete conclusions can be drawn on the basis of the results obtained by quartile deviation.