1. Introduction

Quartile Deviation is an important measure of dispersion in statistics.
It measures the spread of the middle 50% of observations in a dataset.
It is also known as the Semi-Interquartile Range.

Measures of dispersion help us understand how widely data values
are distributed. The major measures of dispersion are:

1. Range
2. Quartile Deviation
3. Mean Deviation
4. Standard Deviation
5. Variance

2. Definition of Quartile Deviation

Quartile Deviation is defined as half of the difference between
the third quartile and the first quartile.

Where:

• \(QD\) = Quartile Deviation
• \(Q_1\) = First Quartile
• \(Q_3\) = Third Quartile

3. Understanding Quartiles

Quartiles divide an ordered dataset into four equal parts.
There are three important quartiles.

First Quartile (Q1)

The first quartile represents the 25th percentile.
Approximately 25% of observations lie below this value.

\[
Q_1=P_{25}
\]

Second Quartile (Q2)

The second quartile is the median of the dataset.

\[
Q_2=\text{Median}=P_{50}
\]

Third Quartile (Q3)

The third quartile represents the 75th percentile.
Approximately 75% of observations lie below this value.

\[
Q_3=P_{75}
\]

4. Formula of Quartile Deviation

The interquartile range is calculated as:

\[
IQR=Q_3-Q_1
\]

Therefore, Quartile Deviation is:

\[
\boxed{QD=\frac{Q_3-Q_1}{2}}
\]

\[
\boxed{QD=\frac{IQR}{2}}
\]

5. Coefficient of Quartile Deviation

The coefficient of Quartile Deviation is a relative measure
of dispersion. It is useful for comparing two datasets.

\[
\boxed{CQD=\frac{Q_3-Q_1}{Q_3+Q_1}}
\]

In percentage form:

\[
CQD(\%)=
\frac{Q_3-Q_1}{Q_3+Q_1}\times100
\]

6. Quartile Deviation for Individual Series

For an individual series, arrange all observations in ascending
order before calculating quartiles.

First quartile:

\[
Q_1=\text{Value of }
\left(\frac{N+1}{4}\right)^{th}\text{ item}
\]

Third quartile:

\[
Q_3=\text{Value of }
\left(\frac{3(N+1)}{4}\right)^{th}\text{ item}
\]

Where \(N\) is the total number of observations.

Solved Example 1

Calculate Quartile Deviation for the following observations:

\[
12,15,18,20,22,25,28,30,35
\]

Step 1: Calculate the number of observations.

\[
N=9
\]

Step 2: Calculate the position of Q1.

\[
Q_1=\left(\frac{N+1}{4}\right)^{th}\text{ item}
\]

\[
=\left(\frac{9+1}{4}\right)^{th}
=2.5^{th}\text{ item}
\]

Using linear interpolation between the second and third observations:

\[
Q_1=15+0.5(18-15)
\]

\[
Q_1=16.5
\]

Step 3: Calculate the position of Q3.

\[
Q_3=\left(\frac{3(N+1)}{4}\right)^{th}\text{ item}
\]

\[
=\left(\frac{3(10)}{4}\right)^{th}
=7.5^{th}\text{ item}
\]

\[
Q_3=28+0.5(30-28)
\]

\[
Q_3=29
\]

Step 4: Calculate Quartile Deviation.

\[
QD=\frac{Q_3-Q_1}{2}
\]

\[
QD=\frac{29-16.5}{2}
\]

\[
\boxed{QD=6.25}
\]

7. Quartile Deviation for Discrete Series

In a discrete frequency distribution, each observation has
an associated frequency.

\[
N=\sum f
\]

First quartile position:

\[
Q_1=\text{Size of }\left(\frac{N}{4}\right)^{th}
\text{ item}
\]

Third quartile position:

\[
Q_3=\text{Size of }\left(\frac{3N}{4}\right)^{th}
\text{ item}
\]

Solved Example 2

Calculate Quartile Deviation from the following data.

Step 1: Calculate total frequency.

\[
N=\sum f=40
\]

Step 2: Calculate Q1.

\[
Q_1=\frac{N}{4}=\frac{40}{4}=10^{th}\text{ item}
\]

The 10th item lies in the cumulative frequency 13.

\[
Q_1=20
\]

Step 3: Calculate Q3.

\[
Q_3=\frac{3N}{4}=\frac{3(40)}{4}
=30^{th}\text{ item}
\]

The 30th item lies in the cumulative frequency 35.

\[
Q_3=40
\]

Step 4: Calculate Quartile Deviation.

\[
QD=\frac{40-20}{2}
\]

\[
\boxed{QD=10}
\]

8. Quartile Deviation for Continuous Series

For a continuous frequency distribution, the following
interpolation formulas are used.

First Quartile:

\[
\boxed{
Q_1=L+\left(
\frac{\frac{N}{4}-cf}{f}
\right)h
}
\]

Third Quartile:

\[
\boxed{
Q_3=L+\left(
\frac{\frac{3N}{4}-cf}{f}
\right)h
}
\]

Where:

• \(L\) = Lower class boundary of the quartile class
• \(N\) = Total frequency
• \(cf\) = Cumulative frequency preceding the quartile class
• \(f\) = Frequency of the quartile class
• \(h\) = Class width

Solved Example 3

Calculate Quartile Deviation from the following distribution.

Step 1: Calculate total frequency.

\[
N=50,\qquad h=10
\]

Step 2: Calculate Q1.

\[
\frac{N}{4}=\frac{50}{4}=12.5
\]

The quartile class is 10–20.

\[
L=10,\quad cf=5,\quad f=10,\quad h=10
\]

\[
Q_1=10+\left(\frac{12.5-5}{10}\right)10
\]

\[
\boxed{Q_1=17.5}
\]

Step 3: Calculate Q3.

\[
\frac{3N}{4}=\frac{3(50)}{4}=37.5
\]

The quartile class is 30–40.

\[
L=30,\quad cf=35,\quad f=10,\quad h=10
\]

\[
Q_3=30+\left(\frac{37.5-35}{10}\right)10
\]

\[
\boxed{Q_3=32.5}
\]

Step 4: Calculate Quartile Deviation.

\[
QD=\frac{32.5-17.5}{2}
\]

\[
\boxed{QD=7.5}
\]

9. Properties of Quartile Deviation

1. It is a measure of dispersion.
2. It is based on the first and third quartiles.
3. It measures the spread of the middle 50% of observations.
4. It is less affected by extreme values than the range.
5. It is suitable for skewed distributions.
6. It is not based on all observations.
7. Its value is non-negative for valid ordered numerical data.

10. Advantages of Quartile Deviation

• Simple to understand and calculate.
• Less influenced by extreme observations.
• Useful for skewed distributions.
• Can be used with certain open-ended frequency distributions.
• Useful for comparing the central spread through its coefficient.

11. Disadvantages of Quartile Deviation

• It ignores half of the observations outside the interquartile range.
• It is not based on all data values.
• It has limited algebraic applications.
• It is less efficient than standard deviation for many statistical analyses.

12. Difference Between Quartile Deviation and Standard Deviation

13. Difference Between IQR and Quartile Deviation

Interquartile Range is:

\[
IQR=Q_3-Q_1
\]

Quartile Deviation is:

\[
QD=\frac{Q_3-Q_1}{2}
\]

Therefore:

\[
\boxed{IQR=2QD}
\]

For example, if:

\[
Q_1=15,\qquad Q_3=45
\]

\[
IQR=45-15=30
\]

\[
QD=\frac{30}{2}=15
\]

14. Frequently Asked Questions

Q1. What is Quartile Deviation?

Quartile Deviation is half of the difference between the third
quartile and first quartile.

\[
QD=\frac{Q_3-Q_1}{2}
\]

Q2. What is another name for Quartile Deviation?

It is also known as the Semi-Interquartile Range.

Q3. What is the formula for the coefficient of Quartile Deviation?

\[
CQD=\frac{Q_3-Q_1}{Q_3+Q_1}
\]

Q4. Is Quartile Deviation affected by extreme values?

It is relatively less affected by extreme observations because
it uses quartiles rather than the minimum and maximum values.

Q5. What is the relationship between IQR and Quartile Deviation?

\[
IQR=2QD
\]

Q6. Can Quartile Deviation be zero?

Yes. If both quartiles are equal:

\[
Q_1=Q_3
\]

\[
QD=\frac{Q_3-Q_1}{2}=0
\]

15. Practice Questions

Question 1. Calculate Quartile Deviation if:

\[
Q_1=20,\qquad Q_3=60
\]

Question 2. Calculate the coefficient of Quartile Deviation if:

\[
Q_1=15,\qquad Q_3=45
\]

Question 3. If the Interquartile Range is 36, calculate Quartile Deviation.

Question 4. Find Quartile Deviation for the following observations using the \((N+1)/4\) position convention:

\[
5,\ 10,\ 15,\ 20,\ 25,\ 30,\ 35
\]

Question 5. Explain why Quartile Deviation is less affected by extreme values than the range.

Answers

1. \(QD=20\)
2. \(CQD=0.5\)
3. \(QD=18\)
4. \(QD=15\)
5. Because it uses the first and third quartiles instead of the minimum and maximum values.

16. Conclusion

Quartile Deviation is an important statistical measure used to
understand the dispersion of the middle 50% of observations.
It is calculated using the first and third quartiles.

\[
\boxed{QD=\frac{Q_3-Q_1}{2}}
\]

Understanding its definition, formulas, properties and numerical
applications is important for students of Statistics, Mathematics,
B.Tech, B.Sc. and competitive examinations.

PostNetwork Academy