Quartile Deviation the Absolute Measure of Dispersion


Statistics / Saturday, September 15th, 2018

Definition of Quartile Deviation

Quartile Deviation divides the total frequency in to four equal parts. The lower quartiles Q1 refers to the values of variates corresponding to the cumulative frequency N/4, upper quartile Q3 refers the value of variants corresponding to cumulative frequency (¾)N.
\[\text{Interquartile Range}={{Q}_{3}}-{{Q}_{1}}\]
\[\text{Quartile Deviation }\left( \text{QD} \right)=\frac{1}{2}\left( {{Q}_{3}}-{{Q}_{1}} \right)\]
\[\text{Relative measure of dispersion coefficient of QD}=\frac{{{Q}_{3}}-{{Q}_{1}}}{{{Q}_{3}}+{{Q}_{1}}}\]
\[{{Q}_{1}}=LL+\frac{\frac{N}{4}-c{{f}_{p}}}{fc}\times CI\]
\[{{Q}_{2}}=LL+\frac{\frac{N}{2}-c{{f}_{p}}}{fc}\times CI\]
\[{{Q}_{3}}=LL+\frac{\frac{3N}{4}-c{{f}_{p}}}{fc}\times CI\]
\[LL=\text{Lower limit of the quartile class}\]
\[\text{CI=Class width}\]
\[fc=\text{Frequency of quartile class}\]
\[\text{N=Total frequency}\]
\[c{{f}_{p}}=\text{Cumulative frequency of class preceding the quartile class}\]

Note:
1. Q3 – Q1 gives the middle 50% of reading.Q3 and Q1 are also known as upper and lower limit of middle 50% of readings.
2. Quartile range is not capable of further algebraic treatment.

In the following three articles in Statistics we will discuss various types of Absolute Measure of Dispersion in details:
1. Range (R)
2. Mean Deviations (M.D.)
3. Standard Deviations (S.D.)

Merits

1. It is easy to compute and understand.
2. Rigidly defined.
3. Not affected by extreme values.

Demerits

1. Not based on all values.
2. Affected by sampling fluctuations.
3. Not capable of further algebraic treatment.

 Example 01

Find the quartile deviation and coefficient of quartile deviation for the given grouped data also compute middle quartile.

ClassFrequency (f)
1-103
11-2016
21-3026
31-4031
41-5016
51-608
Σ f =N =100

Solution:
\[\frac{N}{4}=\frac{100}{4}=25\]

ClassfCf
1-1033
11-201619
21-302645 ← Q1 class
31-403176 ← Q2 and Q3 class
41-501692
51-608100
Σ f =N =100

\[LL=\text{Lower limit of the quartile class}\]
\[\text{CI=Class width}\]
\[fc=\text{Frequency of quartile class}\]
\[\text{N=Total frequency}\]
\[c{{f}_{p}}=\text{Cumulative frequency of class preceding the quartile class}\]
\[{{Q}_{1}}=LL+\frac{\frac{N}{4}-c{{f}_{p}}}{fc}\times CI\]
\[\Rightarrow {{\text{Q}}_{1}}=20.5+\frac{\frac{100}{4}-19}{26}\times 10\]
\[\Rightarrow {{\text{Q}}_{1}}=20.5+\frac{25-19}{26}\times 10=20.5+\frac{60}{26}=22.81\]
\[{{Q}_{2}}=LL+\frac{\frac{N}{2}-c{{f}_{p}}}{fc}\times CI\]
\[\Rightarrow {{Q}_{2}}=30.5+\frac{\frac{100}{2}-45}{31}\times 10\]
\[\Rightarrow {{Q}_{2}}=30.5+\frac{50-45}{31}\times 10=30.5+\frac{50}{31}=32.11\]
\[{{Q}_{3}}=LL+\frac{\frac{3N}{4}-c{{f}_{p}}}{fc}\times CI\]
\[\Rightarrow {{Q}_{3}}=30.5+\frac{\frac{3\times 100}{4}-45}{31}\times 10\]
\[\Rightarrow {{Q}_{3}}=30.5+\frac{75-45}{31}\times 10=30.5+\frac{300}{31}=40.18\]
\[\therefore \text{Quartile Deviation }\left( \text{QD} \right)=\frac{1}{2}\left( {{Q}_{3}}-{{Q}_{1}} \right)\]
\[\Rightarrow QD=\frac{1}{2}\left( 40.18-22.81 \right)=8.685\]
\[\therefore C\text{oefficient of QD}=\frac{{{Q}_{3}}-{{Q}_{1}}}{{{Q}_{3}}+{{Q}_{1}}}\]
\[\Rightarrow C\text{oefficient of QD=}\frac{40.18-22.81}{40.18+22.81}=\frac{17.37}{62.99}\]
\[\therefore C\text{oefficient of QD=0}\text{.276}\]

 Example 02

Find quartile deviation from the following marks of 12 students and also coefficient of quartile deviation: 25, 30, 37, 43, 48, 54, 61, 67, 72, 80, 84, 89.

Solution:

Sl. No.Marks
125
230
337
443
548
654
761
867
972
1080
1184
1289

Here, N = 12
\[{{\text{Q}}_{1}}={{\left( \frac{N+1}{4} \right)}^{th}}={{3.25}^{th}}item\]
\[\Rightarrow {{\text{Q}}_{1}}={{3}^{rd}}\text{item}+0.25\text{ of item}=37+0.25\left( 43-37 \right)\]
\[\therefore {{\text{Q}}_{1}}=38.5\]
\[{{Q}_{3}}={{\left\{ \frac{3\left( N+1 \right)}{4} \right\}}^{th}}={{\left\{ \frac{39}{4} \right\}}^{th}}={{9.75}^{th}}\text{ item}\]
\[\Rightarrow {{Q}_{3}}={{9}^{th}}\text{ item}+0.75\text{ of item}=72+0.75\left( 80-72 \right)\]
\[\therefore {{Q}_{3}}=78\]
\[\therefore \text{Quartile Deviation }\left( \text{QD} \right)=\frac{1}{2}\left( {{Q}_{3}}-{{Q}_{1}} \right)\]
\[\Rightarrow QD=\frac{1}{2}\left( 78-38.5 \right)=19.75\]
\[\therefore C\text{oefficient of QD}=\frac{{{Q}_{3}}-{{Q}_{1}}}{{{Q}_{3}}+{{Q}_{1}}}\]
\[\therefore C\text{oefficient of QD}=\frac{78-38.5}{78+38.5}=0.339\]

 Example 03

Compute quartile deviation and its coefficient for the data given below:

X585960616263646566
f15203235332220108

Solution:

XfCf
581515
592035
603267 ←Q1 class
6135102
6233135
6322157 ←Q3 class
6420177
6510187
668195
N = 195

\[{{\text{Q}}_{1}}={{\left( \frac{N+1}{4} \right)}^{th}}\text{ size}={{\left( \frac{195+1}{4} \right)}^{th}}\text{ size}\]
\[\Rightarrow {{\text{Q}}_{1}}=\text{48}.\text{7}{{\text{8}}^{th}}\text{ size and corresponding to cf 67}\]
\[\therefore {{\text{Q}}_{1}}=60\]
\[{{Q}_{3}}={{\left\{ \frac{3\left( N+1 \right)}{4} \right\}}^{th}}={{\left\{ \frac{3}{4}\times 196 \right\}}^{th}}={{146.33}^{th}}\text{ size}\]
\[\text{It lies in 157},\text{ cf}.\text{ against cf 157},\]
\[\therefore {{Q}_{3}}=63\]
\[\therefore \text{Quartile Deviation }\left( \text{QD} \right)=\frac{1}{2}\left( {{Q}_{3}}-{{Q}_{1}} \right)\]
\[\Rightarrow QD=\frac{1}{2}\left( 63-60 \right)=1.5\]
\[\therefore C\text{oefficient of QD}=\frac{{{Q}_{3}}-{{Q}_{1}}}{{{Q}_{3}}+{{Q}_{1}}}\]
\[\Rightarrow C\text{oefficient of QD}=\frac{63-60}{63+60}=\frac{3}{123}\]
\[\therefore C\text{oefficient of QD}=0.024\]

 

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