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45. The distribution of marks of 100 students in a mathematics examination is as follows:

[tex]
\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Obtained Marks} & 10-25 & 25-40 & 40-55 & 55-70 & 70-85 & 85-100 \\
\hline
\text{No. of Students} & 5 & 21 & 21 & 8 & 25 & 20 \\
\hline
\end{array}
\]
[/tex]

Find the mode.

Answer :

To find the mode of the distribution of marks, we need to identify the modal class and calculate the mode using the modal class parameters. Here’s how you can do it step-by-step:

1. Identify the Modal Class:
- First, we look at the number of students in each class interval.
- The class interval with the highest frequency of students represents the modal class.
- From the given data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Marks Interval} & \text{Number of Students} \\
\hline
10-25 & 5 \\
25-40 & 21 \\
40-55 & 21 \\
55-70 & 8 \\
70-85 & 25 \\
85-100 & 20 \\
\hline
\end{array}
\][/tex]
- The highest number of students is 25, which occurs in the 70-85 marks interval. So, this is the modal class.

2. Parameters of the Modal Class:
- Lower limit ([tex]\(L\)[/tex]) of the modal class: 70
- Frequency of the modal class ([tex]\(f_0\)[/tex]): 25
- Frequency of the class before the modal class ([tex]\(f_1\)[/tex]): 8
- Frequency of the class after the modal class ([tex]\(f_2\)[/tex]): 20
- Class width ([tex]\(h\)[/tex]): Since each class interval is 15 (e.g., 70-85 = 15), the class width is 15.

3. Calculate the Mode:
- The formula to calculate the mode in a grouped frequency distribution is:

[tex]\[
\text{Mode} = L + \left(\frac{f_0 - f_1}{(f_0 - f_1) + (f_0 - f_2)}\right) \times h
\][/tex]

- Substituting the parameters into the formula:

[tex]\[
\text{Mode} = 70 + \left(\frac{25 - 8}{(25 - 8) + (25 - 20)}\right) \times 15
\][/tex]

- Simplifying the calculation:

[tex]\[
\text{Mode} = 70 + \left(\frac{17}{17 + 5}\right) \times 15
\][/tex]

[tex]\[
\text{Mode} = 70 + \left(\frac{17}{22}\right) \times 15
\][/tex]

[tex]\[
\text{Mode} = 70 + (0.7727 \times 15)
\][/tex]

[tex]\[
\text{Mode} \approx 70 + 11.59
\][/tex]

[tex]\[
\text{Mode} \approx 81.59
\][/tex]

Therefore, the mode of the distribution of marks in the mathematics examination is approximately 81.59.

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