To find the mode of the given data, let's walk through the process. The mode for a grouped frequency distribution is determined using the modal class and the following formula:

$\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$

Where:

• $L$ = Lower boundary of the modal class
• $f_1$ = Frequency of the modal class
• $f_0$ = Frequency of the class preceding the modal class
• $f_2$ = Frequency of the class succeeding the modal class
• $h$ = Class width

Step 1: Identify the modal class.

The modal class is the class with the highest frequency. From your table:

• $20 - 30$: Frequency = 8
• $30 - 40$: Frequency = 26
• $40 - 50$: Frequency = 30
• $50 - 60$: Frequency = 20
• $60 - 70$: Frequency = 16

The highest frequency is 30, which corresponds to the class 40 - 50.
So, the modal class is $40 - 50$.

Step 2: Write down the necessary values.

• $L = 40$ (Lower boundary of the modal class)
• $f_1 = 30$ (Frequency of the modal class)
• $f_0 = 26$ (Frequency of the class before the modal class)
• $f_2 = 20$ (Frequency of the class after the modal class)
• $h = 10$ (Class width, since each interval spans 10 units)

Step 3: Apply the formula.

$\text{Mode} = 40 + \left( \frac{30 - 26}{2(30) - 26 - 20} \right) \times 10$

Simplify: $\text{Mode} = 40 + \left( \frac{4}{60 - 46} \right) \times 10$ $\text{Mode} = 40 + \left( \frac{4}{14} \right) \times 10$ $\text{Mode} = 40 + \left( 0.2857 \right) \times 10$ $\text{Mode} = 40 + 2.857$

$\text{Mode} \approx 42.86$

Final Answer:

The mode of the given data is approximately 42.86.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do we interpret the mode in real-world data?
2. What if there are two classes with the same highest frequency? How do we handle bimodal distributions?
3. Why do we use the formula for mode in grouped data instead of just the most frequent value?
4. Can the mode be an average of two numbers, like the mean or median?
5. What are the limitations of using the mode for analyzing data?

Tip: In skewed distributions, the mode can give a better sense of the "central tendency" than the mean.