Quartiles - Calculation of Quartiles

Example 1
1) Calculate \[\mathbf{Q}_{1}\] and \[\mathbf{Q}_{3}\] of the first semester examination marks scored by the students as given: 40, 85, 84, 83, 82, 69, 68, 65, 64, 55, 45.

Solution: Arrange the series in ascending order i.e. 40, 45, 55, 64, 65, 68, 69, 82, 83, 84, 85.
n = Total number of observations.
n = 11
\[\mathbf{Q}_{1}=\mathrm{size~of}\left(\frac{n+1}{4}\right)^{\text{th Observaion}}\]
\[\mathrm{Q}_{1}=\mathrm{size~of}\left(\frac{11+1}{4}\right)^{\text{un Omervanon}}\]
\[\mathrm{Q}_1=\mathrm{size~of}\left(\frac{12}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q_1=size~of~3^{rd~Observation.}}\]
\[\mathrm{Q_1=size~of~3^{rd~Observation.}~is~55}\]
\[\therefore\mathbf{Q}_1=\mathbf{5}\mathbf{5}\]

Third Quartile
\[\mathrm{Q}_3=\mathrm{size~of}3\left(\frac{n+1}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q_3=size~of~3}\left(\frac{11+1}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q}_3=\text{size of 3}\left(\frac{12}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q}_{3}=\mathrm{size~of}\left(3\times3\right)^{\text{th Observation}}\]
\[\mathrm{Q_3=size~of~9^{th~Observation.}~is~83}\]
\[\therefore\mathbf{Q}_{3}=\mathbf{83}\]
 \[\boxed{\quad\mathbf{Ans}:\mathbf{Q}_{1}=55,\mathbf{Q}_{3}=83}\]

Example 2
2) Calculate \[\mathbf{Q}_{3}\] for the given distribution. 20, 28, 31, 18, 19, 17, 32, 33, 22, 21.

Solution: Arrange the data in ascending order. 17, 18, 19, 20, 21, 22, 28, 31, 32, 33. 
n = 10.
\[\mathrm{Q}_3=\mathrm{size~of}3\left(\frac{n+1}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q_3=size~of~3}\left(\frac{10+1}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q}_3=\mathrm{size~of}\left(3\times\frac{11}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q}_3=\text{size of 3}\left(\frac{33}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q_{3}=size~of~8.25^{th~Observation}}\]
\[\mathrm{Q_3=sizeof8^{thobservation}+0.25(9^{thobservation}-8^{thobservation})}\]
Q3 = 31 + 0.25 × 1
\[\therefore\mathbf{Q}_{3}=31.25\]
\[\boxed{\begin{array}{c}\mathbf{Ans}:\mathbf{Q}_3=31.25\end{array}}\]

\[\mathbf{Q}_{1}\] and \[\mathbf{Q}_{3}\] for continuous frequency distribution are calculated by applying the following steps.

1. Arrange the data in ascending or descending order.
2. Write the respective frequencies of the class.
3. Find out the cumulative frequency (cf).
4. Determine the quartile class.

Step - I : First find the value of quartile
\[\mathrm{Q}_1=\mathrm{size~of}\left(\frac{n}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q}_1=\mathrm{size~of}\left(\frac{3n}{4}\right)^{\text{th Observation}}\]
Step - II :
\[\mathbf{Q}_i=l+\left(\frac{\frac{in}{4}-cf}{f}\right)\times h\quad i=1,2,3\]
Where
l = Lower limit of quartile class.
f = Frequency of the quartile class.
cf = Cumulative frequency of the class preceding the quartile class.
n = Total of frequency.
h = Upper limit - lower limit of the quartile class.

1) Find out the \[\mathbf{Q}_{1}\] and \[\mathbf{Q}_{3}\] quartiles for the following data.

Step I
\[\mathrm{Q}_1=\mathrm{size~of}\left(\frac{n}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q}_1=\mathrm{size~of}\left(\frac{50}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q_1=size~of~(12.5)^{th~Observation}}\]
12.5 lies in cf 27, therefore first quartile class is 30-40.
∴ l = 30 f = 20 cf = 7 n = 50 h = 10
Step II
\[\mathbf{Q}_1=l+\left(\frac{\frac{n}{4}-cf}{f}\right)\times h\]
\[\mathbf{Q}_1=30+\left(\frac{\frac{50}{4}-7}{20}\right)\times10\]
\[\mathrm{Q}_1=30+\left(\frac{12.5-7}{20}\right)\times10\]
\[\mathrm{Q}_1=30+\left(\frac{5.5}{20}\right)\times10\]
\[\mathrm{Q}_1=30+\left(\frac{55}{20}\right)\]
Q1 = 30 + 2.75
Q1 = 32.75
∴ Q1 = 32.75
Step I
\[\mathbf{Q}_3=\mathrm{size~of}\left(\frac{3n}{4}\right)^{\text{th Observation}}\]
\[\mathrm{Q_3=size~of\left(\frac{3\times50}{4}\right)^{th~Observation}}\]
\[\mathrm{Q}_3=\mathrm{size~of}\left(\frac{150}{4}\right)^\text{th Observation}\]
\[\mathrm{Q_{3}=size~of~37.5^{th~Observation}}\]
37.5 lies in cf 44 hence third quartile class is 40 - 50
∴ l = 40 f = 17 cf = 27 n = 50 h = 10
Step II
\[\mathbf{Q}_3=l+\left(\frac{\frac{3n}{4}-cf}{f}\right)\times h\]
\[\mathrm{Q}_3=40+\left(\frac{\frac{3\times50}{4}-27}{17}\right)\times10\]
\[\mathrm{Q}_3=40+\left(\frac{37.5-27}{17}\right)\times10\]
\[\mathrm{Q}_3=40+\left(\frac{10.5}{17}\right)\times10\]
\[\mathrm{Q}_3=40+\begin{pmatrix}105 \\17\end{pmatrix}\]
Q₃ = 40 + 6.18
∴ Q₃= 46.18
\[\boxed{\quad\mathbf{Ans}:\mathbf{Q}_1=32.75,\mathbf{Q}_3=46.18}\]