Obtain mode of the following distribution.
Classes 10-20,20-30 ,30-40 ,40-50
50-60 ,60-70 
Frequency 8 ,2 ,25 ,45 ,11 ,9

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?

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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.

Obtain mode of the following distribution. Classes 10-20, 20-30, 30-40, 40-50, 50-60, 60-70. Frequency 8, 2, 25, 45, 11, 9.

Learn how to find the mode of a grouped frequency distribution with class intervals 10-70. The example uses a detailed step-by-step calculation of the mode, applying the formula for grouped data. This problem includes frequency analysis and is suitable for students in Grades 9-12.

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