Question

The weight of coffee (in grams) in $70$ packets is given below.
Determine the modal weight of coffee in packets.

Weight (in grams)	$200-201$	$201-202$	$202-203$	$203-204$	$204-205$	$205-206$
No. of Packets	$12$	$26$	$20$	$9$	$2$	$1$

Answer 1

Hint: In this problem we need to find the modal weight of the coffee in the packets. For this we need to first calculate the modal class of the given data. We know that the modal class of a data is nothing but the class which is having the highest frequency. So, from the given data we will mark the model class. Now we will use the formula $\text{Mode}=L+\left( \dfrac{{{f}_{m}}-{{f}_{1}}}{2{{f}_{m}}-{{f}_{1}}-{{f}_{2}}} \right)h$, where ${{f}_{m}}$ is the frequency of the modal class, ${{f}_{1}}$ is the frequency of the class which is previous to the modal class, ${{f}_{2}}$ is the frequency of the class which is next to modal class, $L$ is the lower boundary of the modal class, $h$ is the height of the model class. After having all the values, we will substitute them in the formula and simplify the equation by using the basic mathematical operations to get the required result.

Complete step-by-step solution:
Given data

Weight (in grams)	$200-201$	$201-202$	$202-203$	$203-204$	$204-205$	$205-206$
No. of Packets	$12$	$26$	$20$	$9$	$2$	$1$

In the above data we can observe that the class $201-202$ has the highest frequency which is $26$. Hence the modal class is $201-202$. So, the values of ${{f}_{m}}$, ${{f}_{1}}$, ${{f}_{2}}$, $L$, $h$ are

Weight (in grams)	No. of Packets
$200-201$	$12$
$201-202$	$26$
$202-203$	$20$
$203-204$	$9$
$204-205$	$2$
$205-206$	$1$

From the above table we can write the values ${{f}_{m}}$, ${{f}_{1}}$, ${{f}_{2}}$, $L$, $h$ as
Frequency of the modal class is ${{f}_{m}}=26$,
Frequency of the class which is previous to the modal class is ${{f}_{1}}=12$,
Frequency of the class which is next to modal class is${{f}_{2}}=20$,
Lower boundary of the modal class is $L=201$,
Height of the model class is $h=1$.
Substituting the all the above values in the formula
$\text{Mode}=L+\left( \dfrac{{{f}_{m}}-{{f}_{1}}}{2{{f}_{m}}-{{f}_{1}}-{{f}_{2}}} \right)h$, then we will get
$\text{Mode}=201+\left( \dfrac{26-12}{2\left( 26 \right)-12-20} \right)\times 1$
Simplifying the above equation by using the basic mathematical operation, then we will have
$\begin{align}
  & \text{Mode}=201+\left( \dfrac{14}{52-32} \right) \\
 & \Rightarrow \text{Mode}=201+\dfrac{14}{20} \\
 & \Rightarrow \text{Mode}=201+0.7 \\
 & \Rightarrow \text{Mode}=201.7 \\
\end{align}$
Hence the model of the given data is $201.7$.

Note: Generally, in a ungrouped data mode is the repeated value or most occurred value. For calculating the mode in an ungrouped data, we don’t have any formulas; we just observe the data and write the most repeated value as mode. But when it comes to grouped data, the above-mentioned formula is used to calculate the mode of the data.