The marks distribution of 30 students in a mathematics examination are given in the table below. Find the mode of this data. $\begin{array}{|l|c|c|}\hline \text{Class interval} & \text{Number of students }(f_i) & \text{Class mark }(x_i) \\ \hline 10\text{-}25 & 2 & 17.5 \\ 25\text{-}40 & 3 & 32.5 \\ 40\text{-}55 & 7 & 47.5 \\ 55\text{-}70 & 6 & 62.5 \\ 70\text{-}85 & 6 & 77.5 \\ 85\text{-}100 & 6 & 92.5 \\ \hline \text{Total} & 30 & \\ \hline \end{array}$

Step 1: Identify the modal class

To find the mode for grouped data, we first need to identify the modal class. The modal class is the class interval with the highest frequency. In the given data, the highest frequency is 7, which corresponds to the class interval 40-55.

Step 2: Identify values for the mode formula

From the modal class 40-55, we identify the following values: $l$ (lower limit of the modal class) = 40, $h$ (class size) = 15 (which is $55 - 40$), $f_1$ (frequency of the modal class) = 7, $f_0$ (frequency of the class preceding the modal class) = 3, and $f_2$ (frequency of the class succeeding the modal class) = 6.

Step 3: Apply the mode formula for grouped data

Now we substitute the identified values into the formula for the mode of grouped data. This formula accounts for the distribution of frequencies around the modal class to give a more precise estimate of the mode.

Step 4: Substitute the values into the formula

We replace $l$, $h$, $f_1$, $f_0$, and $f_2$ with their respective numerical values obtained in the previous step. This is the first step in calculating the mode numerically.

Step 5: Calculate the numerator of the fraction

First, we calculate the difference between the frequency of the modal class and the frequency of the preceding class, which is $f_1 - f_0$. This gives us a value of 4.

Step 6: Calculate the denominator of the fraction

Next, we calculate the denominator of the fraction, which is $2f_1 - f_0 - f_2$. We perform the multiplication and subtractions to get a result of 5.

Step 7: Simplify the fraction

Now we substitute the calculated numerator and denominator back into the fraction. The simplified fraction is $\frac{4}{5}$.

Step 8: Multiply the fraction by the class size

We multiply the simplified fraction by the class size $h$. This gives us $12$, which is the adjustment to be added to the lower limit of the modal class.

Step 9: Add the result to the lower limit

Finally, we add the calculated adjustment to the lower limit of the modal class, $l$. This sum gives us the mode of the given data, which is 52.

More from this chapter

• **Example 6.** Find mean of the following data. | Class | 0-15 | 15-30 | 30-45 | 45-60 | 60-75 | 75-90 | |:---: |:---: |:---: |:---: |:---: |:---: |:---: | | **
• Example 2. The mean of the following data is 14. Find the value of k. |c|c|c|c|c|c| x & 5 & 10 & 15 & 20 & 25 f & 7 & k & 8 & 4 & 5
• Find the mean of the following frequency distribution: class intervals 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 16, 6.
• Find the mean of 4, 8, 15, 16, 23, 42
• The marks obtained by 50 students in a test are tabulated below: | Marks: | 0 - 15 | 15 - 30 | 30 - 45 | 45 - 60 | 60 - 75 | |---|---|---|---|---|---| | Number
• A survey regarding the heights (in cm) of 50 girls of class X of a school was conducted and the following data was obtained: | Height (in cm) | Number of girls
• In the following table, the median age of 200 spectators of a football match is 32. Find the missing frequencies p and q. | Age (in years) | Number of Spectator
• The government rescued 100 people after a train accident. Their ages were recorded in the following table. Find their mean age. | Age (in years) | Number of peo

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