Quartile Deviation for grouped data
Quartile deviation (QD) is an absolute measure of spread or dispersion based on the quartiles. Quartile deviation is defined as half of the distance between the third quartile $Q_3$ and the first quartile $Q_1$. It is also known as semi-interquartile range. Thus the quartile deviation is given by
$QD = \dfrac{Q_3-Q_1}{2}$
where,
- $Q_1$ is the first quartile
- $Q_3$ is the third quartile
Coefficient of Quartile deviation
The relative measure of spread or dispersion corresponding to quartile deviation is the coefficient of quartile deviation (QD). The coefficient of quartile deviation is used to study and compare the degree of variation for two or more data set having different units of measurements.
The coefficient of quartile deviation is defined as
$$
\begin{aligned}
\text{Coeff. of }QD & = \frac{Q_3 - Q_1}{Q_3+Q_1}
\end{aligned}
$$
The formula for $i^{th}$ quartile for grouped data is
$$ \begin{aligned} Q_i=l + \bigg(\frac{\frac{iN}{4} - F_<}{f}\bigg)\times h; \quad i=1,2,3 \end{aligned} $$
where
- $l :$ the lower limit of the $i^{th}$ quartile class
- $N=\sum f :$ total number of observations
- $f :$ frequency of the $i^{th}$ quartile class
- $F_< :$ cumulative frequency of the class previous to $i^{th}$ quartile class
- $h :$ the class width
Semi-interquartile range is less affected by extreme observations, hence it is a good measure of spread or dispersion for skewed data.
Quartile Deviation Calculator for grouped data
Use this calculator to find the Quartile Deviation for grouped (frequency distribution) data.
| Quartile Deviation Calculator (Grouped Data) | |
|---|---|
| Type of Freq. Dist. | DiscreteContinuous |
| Enter the Classes for X (Separated by comma,) | |
| Enter the frequencies (f) (Separated by comma,) | |
| Results | |
| Number of Obs. (N): | |
| First Quartile : ($Q_1$) | |
| Second Quartile : ($Q_2$) | |
| Third Quartile : ($Q_3$) | |
| Quartile Deviation : $QD$ | |
| Coeff. of QD : | |
How to find quartile deviation for grouped data?
Step 1 - Select type of frequency distribution (Discrete or continuous)
Step 2 - Enter the Range or classes (X) seperated by comma (,)
Step 3 - Enter the Frequencies (f) seperated by comma
Step 4 - Click on "Calculate" for quartile deviation calculation
Step 5 - Gives output as number of observation (N)
Step 6 - Calculate three quartiles $Q_1$, $Q_2$ and $Q_3$
Step 7 - Calculate Quartile deviation and coefficient of quartile deviation
Quartile Deviation for grouped data Example 1
A class teacher has the following data about the number of absences of 35 students of a class. Compute semi-interquartile range for the following frequency distribution.
| No.of days ($x$) | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| No. of Students ($f$) | 1 | 15 | 10 | 5 | 4 |
Solution
| $x_i$ | $f_i$ | $cf$ | |
|---|---|---|---|
| 2 | 1 | 1 | |
| 3 | 15 | 16 | |
| 4 | 10 | 26 | |
| 5 | 5 | 31 | |
| 6 | 4 | 35 | |
| Total | 35 |
Inter-quartile range (IQR)
The inter-quartile range is given by $IQR= Q_3-Q_1$.
The formula for $i^{th}$ quartile is
$Q_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,3$
where $N$ is the total number of observations.
First Quartile $Q_1$
$$ \begin{aligned} Q_{1} &=\bigg(\dfrac{1(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{1(35)}{4}\bigg)^{th}\text{ value}\\ &=\big(8.75\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $8.75$ is $16$. The corresponding value of $X$ is the $1^{st}$ quartile. That is, $Q_1 =3$ days.
Thus, $25$ % of the students had absences less than or equal to $3$ days.
Third Quartile $Q_3$
$$ \begin{aligned} Q_{3} &=\bigg(\dfrac{3(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{3(35)}{4}\bigg)^{th}\text{ value}\\ &=\big(26.25\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $26.25$ is $31$. The corresponding value of $X$ is the $3^{rd}$ quartile. That is, $Q_3 =5$ days.
Thus, $75$ % of the students had absences less than or equal to $5$ days.
Quartile deviation
The quartile deviation ($QD$) is
$$ \begin{aligned} QD & = \frac{Q_3 - Q_1}{2}\\ &= \frac{5 - 3}{2}\\ & = 1. \end{aligned} $$
Coefficient of quartile deviation is
$$ \begin{aligned} \text{Coeff. of }QD & = \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{5 - 3}{5 + 3}\\ & = 0.25. \end{aligned} $$
Quartile Deviation for grouped data Example 2
The following table gives the amount of time (in minutes) spent on the internet each evening by a group of 56 students. Compute quartile deviation for the following frequency distribution.
| Time spent on Internet ($x$) | No. of Students ($f$) |
|---|---|
| 10-12 | 3 |
| 13-15 | 12 |
| 16-18 | 15 |
| 19-21 | 24 |
| 22-24 | 2 |
Solution
| Class Interval | Class Boundries | $f_i$ | $cf$ | |
|---|---|---|---|---|
| 10-12 | 9.5-12.5 | 3 | 3 | |
| 13-15 | 12.5-15.5 | 12 | 15 | |
| 16-18 | 15.5-18.5 | 15 | 30 | |
| 19-21 | 18.5-21.5 | 24 | 54 | |
| 22-24 | 21.5-24.5 | 2 | 56 | |
| Total | 56 |
Quartiles
The formula for $i^{th}$ quartile is
$Q_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,3$
where $N$ is the total number of observations.
First Quartile $Q_1$
$$ \begin{aligned} Q_{1} &=\bigg(\dfrac{1(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{1(56)}{4}\bigg)^{th}\text{ value}\\ &=\big(14\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $14$ is $15$. The corresponding class $12.5-15.5$ is the $1^{st}$ quartile class.
Thus
- $l = 12.5$, the lower limit of the $1^{st}$ quartile class
- $N=56$, total number of observations
- $f =12$, frequency of the $1^{st}$ quartile class
- $F_< = 3$, cumulative frequency of the class previous to $1^{st}$ quartile class
- $h =3$, the class width
The first quartile $Q_1$ can be computed as follows:
$$ \begin{aligned} Q_1 &= l + \bigg(\frac{\frac{1(N)}{4} - F_<}{f}\bigg)\times h\\ &= 12.5 + \bigg(\frac{\frac{1*56}{4} - 3}{12}\bigg)\times 3\\ &= 12.5 + \bigg(\frac{14 - 3}{12}\bigg)\times 3\\ &= 12.5 + \big(0.9167\big)\times 3\\ &= 12.5 + 2.75\\ &= 15.25 \text{ minutes} \end{aligned} $$
Thus, $25$ % of the students spent less than or equal to $15.25$ minutes on the internet.
Third Quartile $Q_3$
$$ \begin{aligned} Q_{3} &=\bigg(\dfrac{3(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{3(56)}{4}\bigg)^{th}\text{ value}\\ &=\big(42\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $42$ is $54$. The corresponding class $18.5-21.5$ is the $3^{rd}$ quartile class.
Thus
- $l = 18.5$, the lower limit of the $3^{rd}$ quartile class
- $N=56$, total number of observations
- $f =24$, frequency of the $3^{rd}$ quartile class
- $F_< = 30$, cumulative frequency of the class previous to $3^{rd}$ quartile class
- $h =3$, the class width
The third quartile $Q_3$ can be computed as follows:
$$ \begin{aligned} Q_3 &= l + \bigg(\frac{\frac{3(N)}{4} - F_<}{f}\bigg)\times h\\ &= 18.5 + \bigg(\frac{\frac{3*56}{4} - 30}{24}\bigg)\times 3\\ &= 18.5 + \bigg(\frac{42 - 30}{24}\bigg)\times 3\\ &= 18.5 + \big(0.5\big)\times 3\\ &= 18.5 + 1.5\\ &= 20 \text{ minutes} \end{aligned} $$
Thus, $75$ % of the students spent less than or equal to $20$ minutes on the internet.
Quartile Deviation
The quartile deviation ($QD$) is
$$ \begin{aligned} QD & = \frac{Q_3 - Q_1}{2}\\ &= \frac{20 - 15.25}{2}\\ & = 2.375\text{ minutes}. \end{aligned} $$
Coefficient of quartile deviation is
$$ \begin{aligned} \text{Coeff. of }QD & = \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{20 - 15.25}{20 + 15.25}\\ & = 0.13475. \end{aligned} $$
Quartile Deviation for grouped data Example 3
The Scores of students in a Math test is given in the table below :
| Class Interval | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
|---|---|---|---|---|---|---|
| Frequency ($f$) | 6 | 8 | 12 | 10 | 5 | 4 |
Find quartile deviation and coefficient of quartile deviation for the given grouped data.
Solution
| Class Interval | Class Boundries | $f_i$ | $cf$ | |
|---|---|---|---|---|
| 10-20 | 10-20 | 6 | 6 | |
| 20-30 | 20-30 | 8 | 14 | |
| 30-40 | 30-40 | 12 | 26 | |
| 40-50 | 40-50 | 10 | 36 | |
| 50-60 | 50-60 | 5 | 41 | |
| 60-70 | 60-70 | 4 | 45 | |
| Total | 45 |
Quartiles
The formula for $i^{th}$ quartile is
$Q_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,3$
where $N$ is the total number of observations.
First Quartile $Q_1$
$$ \begin{aligned} Q_{1} &=\bigg(\dfrac{1(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{1(45)}{4}\bigg)^{th}\text{ value}\\ &=\big(11.25\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $11.25$ is $14$. The corresponding class $20-30$ is the $1^{st}$ quartile class.
Thus
- $l = 20$, the lower limit of the $1^{st}$ quartile class
- $N=45$, total number of observations
- $f =8$, frequency of the $1^{st}$ quartile class
- $F_< = 6$, cumulative frequency of the class previous to $1^{st}$ quartile class
- $h =10$, the class width
The first quartile $Q_1$ can be computed as follows:
$$ \begin{aligned} Q_1 &= l + \bigg(\frac{\frac{1(N)}{4} - F_<}{f}\bigg)\times h\\ &= 20 + \bigg(\frac{\frac{1*45}{4} - 6}{8}\bigg)\times 10\\ &= 20 + \bigg(\frac{11.25 - 6}{8}\bigg)\times 10\\ &= 20 + \big(0.6562\big)\times 10\\ &= 20 + 6.5625\\ &= 26.5625 \text{ Scores} \end{aligned} $$
Thus, $25$ % of the students scores less than or equal to $26.5625$ marks in Math test.
Third Quartile $Q_3$
$$ \begin{aligned} Q_{3} &=\bigg(\dfrac{3(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{3(45)}{4}\bigg)^{th}\text{ value}\\ &=\big(33.75\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $33.75$ is $36$. The corresponding class $40-50$ is the $3^{rd}$ quartile class.
Thus
- $l = 40$, the lower limit of the $3^{rd}$ quartile class
- $N=45$, total number of observations
- $f =10$, frequency of the $3^{rd}$ quartile class
- $F_< = 26$, cumulative frequency of the class previous to $3^{rd}$ quartile class
- $h =10$, the class width
The third quartile $Q_3$ can be computed as follows:
$$ \begin{aligned} Q_3 &= l + \bigg(\frac{\frac{3(N)}{4} - F_<}{f}\bigg)\times h\\ &= 40 + \bigg(\frac{\frac{3*45}{4} - 26}{10}\bigg)\times 10\\ &= 40 + \bigg(\frac{33.75 - 26}{10}\bigg)\times 10\\ &= 40 + \big(0.775\big)\times 10\\ &= 40 + 7.75\\ &= 47.75 \text{ Scores} \end{aligned} $$
Thus, $75$ % of the students scores less than or equal to $47.75$ marks in Math Test.
Quartile Deviation
The quartile deviation ($QD$) is
$$ \begin{aligned} QD & = \frac{Q_3 - Q_1}{2}\\ &= \frac{47.75 - 26.5625}{2}\\ & = 10.59375\text{ Scores}. \end{aligned} $$
Coefficient of quartile deviation is
$$ \begin{aligned} \text{Coeff. of }QD & = \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{47.75 - 26.5625}{47.75 + 26.5625}\\ & = 0.28511. \end{aligned} $$
Quartile Deviation for grouped data Example 4
The following data shows the distribution of maximum loads in short tons supported by certain cables produced by a company:
| Maximum load | No. of Cables |
|---|---|
| 9.25-9.75 | 2 |
| 9.75-10.25 | 5 |
| 10.25-10.75 | 12 |
| 10.75-11.25 | 17 |
| 11.25-11.75 | 14 |
| 11.75-12.25 | 6 |
| 12.25-12.75 | 3 |
| 12.75-13.25 | 1 |
Compute quartile deviation and coefficient of quartile deviation for the above frequency distribution.
Solution
| Class Interval | Class Boundries | $f_i$ | $cf$ | |
|---|---|---|---|---|
| 9.25-9.75 | 9.25-9.75 | 2 | 2 | |
| 9.75-10.25 | 9.75-10.25 | 5 | 7 | |
| 10.25-10.75 | 10.25-10.75 | 12 | 19 | |
| 10.75-11.25 | 10.75-11.25 | 17 | 36 | |
| 11.25-11.75 | 11.25-11.75 | 14 | 50 | |
| 11.75-12.25 | 11.75-12.25 | 6 | 56 | |
| 12.25-12.75 | 12.25-12.75 | 3 | 59 | |
| 12.75-13.25 | 12.75-13.25 | 1 | 60 | |
| Total | 60 |
Quartiles
The formula for $i^{th}$ quartile is
$Q_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,3$
where $N$ is the total number of observations.
First Quartile $Q_1$
$$ \begin{aligned} Q_{1} &=\bigg(\dfrac{1(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{1(60)}{4}\bigg)^{th}\text{ value}\\ &=\big(15\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $15$ is $19$. The corresponding class $10.25-10.75$ is the $1^{st}$ quartile class.
Thus
- $l = 10.25$, the lower limit of the $1^{st}$ quartile class
- $N=60$, total number of observations
- $f =12$, frequency of the $1^{st}$ quartile class
- $F_< = 7$, cumulative frequency of the class previous to $1^{st}$ quartile class
- $h =0.5$, the class width
The first quartile $Q_1$ can be computed as follows:
$$ \begin{aligned} Q_1 &= l + \bigg(\frac{\frac{1(N)}{4} - F_<}{f}\bigg)\times h\\ &= 10.25 + \bigg(\frac{\frac{1*60}{4} - 7}{12}\bigg)\times 0.5\\ &= 10.25 + \bigg(\frac{15 - 7}{12}\bigg)\times 0.5\\ &= 10.25 + \big(0.6667\big)\times 0.5\\ &= 10.25 + 0.3333\\ &= 10.5833 \text{ tons} \end{aligned} $$
Thus, $25$ % of the cables less than or equal to $10.5833$ tons of maximum load.
Third Quartile $Q_3$
$$ \begin{aligned} Q_{3} &=\bigg(\dfrac{3(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{3(60)}{4}\bigg)^{th}\text{ value}\\ &=\big(45\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $45$ is $50$. The corresponding class $11.25-11.75$ is the $3^{rd}$ quartile class.
Thus
- $l = 11.25$, the lower limit of the $3^{rd}$ quartile class
- $N=60$, total number of observations
- $f =14$, frequency of the $3^{rd}$ quartile class
- $F_< = 36$, cumulative frequency of the class previous to $3^{rd}$ quartile class
- $h =0.5$, the class width
The third quartile $Q_3$ can be computed as follows:
$$ \begin{aligned} Q_3 &= l + \bigg(\frac{\frac{3(N)}{4} - F_<}{f}\bigg)\times h\\ &= 11.25 + \bigg(\frac{\frac{3*60}{4} - 36}{14}\bigg)\times 0.5\\ &= 11.25 + \bigg(\frac{45 - 36}{14}\bigg)\times 0.5\\ &= 11.25 + \big(0.6429\big)\times 0.5\\ &= 11.25 + 0.3214\\ &= 11.5714 \text{ tons} \end{aligned} $$
Thus, $75$ % of the cables less than or equal to $11.5714$ tons of maximum load.
Quartile Deviation
The quartile deviation ($QD$) is
$$ \begin{aligned} QD & = \frac{Q_3 - Q_1}{2}\\ &= \frac{11.5714 - 10.5833}{2}\\ & = 0.49405\text{ tons}. \end{aligned} $$
Coefficient of quartile deviation is
$$ \begin{aligned} \text{Coeff. of }QD & = \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{11.5714 - 10.5833}{11.5714 + 10.5833}\\ & = 0.0446. \end{aligned} $$
Quartile Deviation for grouped data Example 5
Following table shows the weight of 100 pumpkin produced from a farm :
| Weight ('00 grams) | Frequency |
|---|---|
| $4 \leq x < 6$ | 4 |
| $6 \leq x < 8$ | 14 |
| $8 \leq x < 10$ | 34 |
| $10 \leq x < 12$ | 28 |
| $12 \leq x < 14$ | 20 |
Calculate quartile deviation and coefficient of quartile deviation.
Solution
| Class Interval | Class Boundries | $f_i$ | $cf$ | |
|---|---|---|---|---|
| 4-6 | 4-6 | 4 | 4 | |
| 6-8 | 6-8 | 14 | 18 | |
| 8-10 | 8-10 | 34 | 52 | |
| 10-12 | 10-12 | 28 | 80 | |
| 12-14 | 12-14 | 20 | 100 | |
| Total | 100 |
Quartiles
The formula for $i^{th}$ quartile is
$Q_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,3$
where $N$ is the total number of observations.
First Quartile $Q_1$
$$ \begin{aligned} Q_{1} &=\bigg(\dfrac{1(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{1(100)}{4}\bigg)^{th}\text{ value}\\ &=\big(25\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $25$ is $52$. The corresponding class $8-10$ is the $1^{st}$ quartile class.
Thus
- $l = 8$, the lower limit of the $1^{st}$ quartile class
- $N=100$, total number of observations
- $f =34$, frequency of the $1^{st}$ quartile class
- $F_< = 18$, cumulative frequency of the class previous to $1^{st}$ quartile class
- $h =2$, the class width
The first quartile $Q_1$ can be computed as follows:
$$ \begin{aligned} Q_1 &= l + \bigg(\frac{\frac{1(N)}{4} - F_<}{f}\bigg)\times h\\ &= 8 + \bigg(\frac{\frac{1*100}{4} - 18}{34}\bigg)\times 2\\ &= 8 + \bigg(\frac{25 - 18}{34}\bigg)\times 2\\ &= 8 + \big(0.2059\big)\times 2\\ &= 8 + 0.4118\\ &= 8.4118 \text{ ('00 grams)} \end{aligned} $$
Thus, $25$ % of the pumpkins weight is less than or equal to $8.4118$ ('00 grams).
Third Quartile $Q_3$
$$ \begin{aligned} Q_{3} &=\bigg(\dfrac{3(N)}{4}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{3(100)}{4}\bigg)^{th}\text{ value}\\ &=\big(75\big)^{th}\text{ value} \end{aligned} $$
The cumulative frequency just greater than or equal to $75$ is $80$. The corresponding class $10-12$ is the $3^{rd}$ quartile class.
Thus
- $l = 10$, the lower limit of the $3^{rd}$ quartile class
- $N=100$, total number of observations
- $f =28$, frequency of the $3^{rd}$ quartile class
- $F_< = 52$, cumulative frequency of the class previous to $3^{rd}$ quartile class
- $h =2$, the class width
The third quartile $Q_3$ can be computed as follows:
$$ \begin{aligned} Q_3 &= l + \bigg(\frac{\frac{3(N)}{4} - F_<}{f}\bigg)\times h\\ &= 10 + \bigg(\frac{\frac{3*100}{4} - 52}{28}\bigg)\times 2\\ &= 10 + \bigg(\frac{75 - 52}{28}\bigg)\times 2\\ &= 10 + \big(0.8214\big)\times 2\\ &= 10 + 1.6429\\ &= 11.6429 \text{ ('00 grams)} \end{aligned} $$
Thus, $75$ % of the pumpkins weight less than or equal to $11.6429$ ('00 grams).
Quartile Deviation
The quartile deviation ($QD$) is
$$ \begin{aligned} QD & = \frac{Q_3 - Q_1}{2}\\ &= \frac{11.6429 - 8.4118}{2}\\ & = 1.61555\text{ ('00 grams)}. \end{aligned} $$
Coefficient of quartile deviation is
$$ \begin{aligned} \text{Coeff. of }QD & = \frac{Q_3 - Q_1}{Q_3+Q_1}\\ &= \frac{11.6429 - 8.4118}{11.6429 + 8.4118}\\ & = 0.16111. \end{aligned} $$
Hope you enjoyed the step by step solution to find Quartile deviation for grouped data.
Do read more about how to find quartiles deviation calculator for ungrouped data.
If you have any doubt or queries feel free to post them in the comment section.
