To find the mode of the given data, we need to analyze the grouped frequency distribution. The data is cumulative, and we must first convert it into a frequency distribution table before finding the modal class and applying the formula for the mode.

Step 1: Convert cumulative frequency to frequency

The given data is cumulative frequency:

To calculate frequency, subtract the cumulative frequency of the previous class from the current cumulative frequency:

Step 2: Identify the modal class

The modal class is the class interval with the highest frequency. From the frequency table, the class interval 20 - 30 has the highest frequency of 9. Therefore, the modal class is 20 - 30.

Step 3: Use the mode formula

The formula to find the mode is:

$\text{Mode} = L + \left( \frac{f_m - f_{1}}{2f_m - f_{1} - f_{2}} \right) \cdot h$

Where:

• $L$ = lower boundary of the modal class
• $f_m$ = frequency of the modal class
• $f_{1}$ = frequency of the class before the modal class
• $f_{2}$ = frequency of the class after the modal class
• $h$ = class width

From the table:

• $L = 20$ (lower boundary of the class 20 - 30)
• $f_m = 9$
• $f_{1} = 5$ (frequency of the class 10 - 20)
• $f_{2} = 3$ (frequency of the class 30 - 40)
• $h = 10$ (class width)

Step 4: Substitute values into the formula

$\text{Mode} = 20 + \left( \frac{9 - 5}{2(9) - 5 - 3} \right) \cdot 10$

Simplify the terms: $\text{Mode} = 20 + \left( \frac{4}{18 - 8} \right) \cdot 10$

$\text{Mode} = 20 + \left( \frac{4}{10} \right) \cdot 10$

$\text{Mode} = 20 + 4 = 24$

Final Answer:

The mode of the given data is 24.

Would you like me to explain further or provide another example?

Here are 5 related questions to consider:

1. How do you calculate cumulative frequencies from raw frequency data?
2. Why is the modal class important for grouped data?
3. What is the difference between mode, mean, and median in statistics?
4. How do you handle datasets where there are multiple modal classes?
5. Can this method be applied to continuous data with unequal class widths?

Tip: Always double-check your class intervals and frequencies when converting between cumulative and raw frequency data to avoid errors.