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Question: If \({Q_3} + {\text{ }}{Q_1} = 108\),\[{Q_3}-{\text{ }}{Q_1} = 74\], then Q.D. is _____________. A...

If Q3+ Q1=108{Q_3} + {\text{ }}{Q_1} = 108,Q3− Q1=74{Q_3}-{\text{ }}{Q_1} = 74, then Q.D. is _____________.
A. 3333
B. 3.73.7
C. 3737
D. 0.680.68

Explanation

Solution

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%25\% of the data. The second quartile represents 50%50\% of the data. The third quartile represents 75%75\% of the data. The second quartile also stands for the median of the dataset. The fourth quartile represents the highest 25%25\% of the data set.
To find quartile deviation of a dataset, we need to use the following formula.
Q.D.=Q3−Q12Q.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 Q3+ Q1=108{Q_3} + {\text{ }}{Q_1} = 108 and Q3− Q1=74{Q_3}-{\text{ }}{Q_1} = 74
We have to find the quartile deviation by using the formula,
Q.D.=Q3−Q12Q.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.=Q3−Q12Q.D. = \dfrac{{{Q_3} - {Q_1}}}{2}
It is given that Q3− Q1=74{Q_3}-{\text{ }}{Q_1} = 74
So, Q.D.=742Q.D. = \dfrac{{{{74}_{}}}}{2}
On dividing the terms and we get
⇒37\Rightarrow 37
The Quartile deviation is 3737.

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%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.