Lower quartile =42, upper quartile =56 find the coefficient of quartile
deviation
Answers
Step-by-step explanation:
The Quartile Deviation
Formally, the Quartile Deviation is equal to the half of the Inter-Quartile Range and thus we can write it as –
Q_d = \frac{Q_3 – Q_1}{2}
Q
d
=
2
Q
3
–Q
1
Therefore, we also call it the Semi Inter-Quartile Range.
The Quartile Deviation doesn’t take into account the extreme points of the distribution. Thus, the dispersion or the spread of only the central 50% data is considered.
If the scale of the data is changed, the Qd also changes in the same ratio.
It is the best measure of dispersion for open-ended systems (which have open-ended extreme ranges).
Also, it is less affected by sampling fluctuations in the dataset as compared to the range (another measure of dispersion).
Since it is solely dependent on the central values in the distribution, if in any experiment, these values are abnormal or inaccurate, the result would be affected drastically.
Learn more about Range and Mean Deviation here in detail.
The Coefficient of Quartile Deviation
Based on the quartiles, a relative measure of dispersion, known as the Coefficient of Quartile Deviation, can be defined for any distribution. It is formally defined as –
\text{Coefficient of Quartile Deviation = }\frac{Q_3 – Q_1}{Q_3 + Q_1} \times 100
Coefficient of Quartile Deviation =
Q
3
+Q
1
Q
3
–Q
1
×100
Since it involves a ratio of two quantities of the same dimensions, it is unit-less. Thus, it can act as a suitable parameter for comparing two or more different datasets which may or may not involve quantities with the same dimensions.
So, now let’s go through the solved examples below to get a better idea of how to apply these concepts to various distributions.
Solved Examples on Quartile Deviation
Question 1: The number of vehicles sold by a major Toyota Showroom in a day was recorded for 10 working days. The data is given as –
Day Frequency
1 20
2 15
3 18
4 5
5 10
6 17
7 21
8 19
9 25
10 28
Find the Quartile Deviation and its coefficient for the given discrete distribution case.
Solution: We first need to sort the frequency data given to us before proceeding with the quartiles calculation –
Sorted Data – 5, 10, 15, 17, 18, 19, 20, 21, 25, 28
n(number of data points) = 10
Now, to find the quartiles, we use the logic that the first quartile lies halfway between the lowest value and the median; and the third quartile lies halfway between the median and the largest value.
First Quartile Q1 = \frac{n + 1}{4}
4
n+1
th term.
= \frac{10 + 1}{4}
4
10+1
th term = 2.75th term
= 2nd term + 0.75 × (3rd term – 2nd term)
= 10 + 0.75 × (15 – 10)
= 10 + 3.75
= 13.75
Third Quartile Q3 = \frac{3(n + 1)}{4}
4
3(n+1)
th term.
= \frac{3(10 + 1)}{4}
4
3(10+1)
th term = 8.25th term
= 8th term + 0.25 × (9th term – 8th term)
= 21 + 0.25 × (25 – 21)
= 21 + 1
= 22
Using the values for Q1 and Q3, now we can calculate the Quartile Deviation and its coefficient as follows –
Quartile Deviation = Semi-Inter Quartile Range
= \frac{Q_3 – Q_1}{2}
2
Q
3
–Q
1
= \frac{22 – 13.75}{2}
2
22–13.75
=\frac{8.25}{2}
2
8.25
= 4.125
Coefficient of Quartile Deviation
= \frac{Q_3 – Q_1}{Q_3 + Q_1} \times 100
Q
3
+Q
1
Q
3
–Q
1
×100
= \frac{22 – 13.75}{22 + 13.75} \times 100
22+13.75
22–13.75
×100
= \frac{8.25}{35.75} \times 100
35.75
8.25
×100
≈ 23.08