To find the mode of the given distribution, we use the formula for the mode in a grouped frequency distribution:

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

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 before the modal class.
• $f_2$ is the frequency of the class after the modal class.
• $h$ is the class width (difference between the upper and lower boundaries of any class).

Step 1: Identify the modal class

The modal class is the class with the highest frequency. From the dataTo find the mode of the given distribution, we use the formula for the mode in a grouped frequency distribution:

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

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 before the modal class.
• $f_2$ is the frequency of the class after the modal class.
• $h$ is the class width (difference between the upper and lower boundaries of any class).

Step 1: Identify the modal class

The modal class is the class with the highest frequency. From the

• Frequencies: 8, 2, 25, 45, 11, 9.

The highest frequency is 45, so the modal class is 40-50.

Step 2: Identify the parameters

• $L = 40$ (lower boundary of the modal class).
• $f_1 = 45$ (frequency of the modal class).
• $f_0 = 25$ (frequency of the class before the modal class).
• $f_2 = 11$ (frequency of the class after the modal class).
• $h = 10$ (class width).

Step 3: Apply the formula

$\text{Mode} = 40 + \left( \frac{{45 - 25}}{{2(45) - 25 - 11}} \right) \times 10$

$\text{Mode} = 40 + \left( \frac{{20}}{{90 - 36}} \right) \times 10$

$\text{Mode} = 40 + \left( \frac{{20}}{{54}} \right) \times 10$

$\text{Mode} = 40 + 3.70 \approx 43.70$

Final Answer:

The mode of the given distribution is approximately 43.70.

Do you want more details on this solution, or have any questions?

---

Here are 5 relative questions that expand on this information:

1. How do we calculate the median for a grouped frequency distribution?
2. What are the advantages of using the mode compared to the mean and median?
3. How would the mode change if the class intervals were unequal?
4. How can we estimate the mean of this distribution using the midpoint formula?
5. How can we visualize the mode of this distribution using a histogram?

Tip: The mode is particularly useful in identifying the most common or frequent value in a dataset, especially when the data is skewed or non-normal.