Finding the Mode of Grouped Data

To find the mode of grouped data, we use the formula:

Mode $= L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$

Where:

• $L$ is the lower boundary of the modal class.
• $f_1$ is the frequency of the modal class.
• $f_0$ is the frequency of the class preceding the modal class.
• $f_2$ is the frequency of the class succeeding the modal class.
• $h$ is the class width.

Data Table and Modal Class Identification

The given data is:

The class with the highest frequency (30) is 55 - 64. This is the modal class.

Applying the Mode Formula

From the modal class (55 - 64):

• The lower limit is 55. Since the class intervals appear inclusive, the lower boundary ($L$) is $55 - 0.5 = 54.5$.
• The frequency of the modal class ($f_1$) is 30.
• The frequency of the preceding class (45 - 54) ($f_0$) is 20.
• The frequency of the succeeding class (65 - 74) ($f_2$) is 25.
• The class width ($h$) is $(44 - 35 + 1) = 10$.

Calculation Steps

1. Substitute the values into the mode formula: $ \text{Mode} = 54.5 + \frac{30 - 20}{2 \times 30 - 20 - 25} \times 10 $
2. Simplify the fraction: $ \text{Mode} = 54.5 + \frac{10}{60 - 20 - 25} \times 10 $ $ \text{Mode} = 54.5 + \frac{10}{15} \times 10 $
3. Continue the calculation: $ \text{Mode} = 54.5 + \frac{2}{3} \times 10 $ $ \text{Mode} = 54.5 + \frac{20}{3} $ $ \text{Mode} = 54.5 + 6.666... $
4. Calculate the final value and round to two decimal places: $ \text{Mode} \approx 61.166... $ $ \text{Mode} \approx 61.17 $

The mode of the given data is approximately 61.17.