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 a frequency distribution, we use the class with the highest frequency as the **modal class**. Let's go through the steps to determine the mode for the given dataTo find the mode of a frequency distribution, we use the class with the highest frequency as the **modal class**. Let's go through the steps to determine the mode for the given 

### Step 1: Identify the modal class
The class intervals and their corresponding frequencies are:
- \( 10-20 \): Frequency = 8
- \( 20-30 \): Frequency = 2
- \( 30-40 \): Frequency = 25
- \( 40-50 \): Frequency = 45
- \( 50-60 \): Frequency = 11
- \( 60-70 \): Frequency = 9

The modal class is the class with the highest frequency. Here, the highest frequency is **45**, which corresponds to the class interval **40-50**.

### Step 2: Use the mode formula
The formula to calculate the mode for a grouped frequency distribution is:
\[
\text{Mode} = L + \left( \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} \right) \times h
\]
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 dataTo find the mode of a frequency distribution, we use the class with the highest frequency as the **modal class**. Let's go through the steps to determine the mode for the given 

### Step 1: Identify the modal class
The class intervals and their corresponding frequencies are:
- \( 10-20 \): Frequency = 8
- \( 20-30 \): Frequency = 2
- \( 30-40 \): Frequency = 25
- \( 40-50 \): Frequency = 45
- \( 50-60 \): Frequency = 11
- \( 60-70 \): Frequency = 9

The modal class is the class with the highest frequency. Here, the highest frequency is **45**, which corresponds to the class interval **40-50**.

### Step 2: Use the mode formula
The formula to calculate the mode for a grouped frequency distribution is:
\[
\text{Mode} = L + \left( \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} \right) \times h
\]
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 
- \( L = 40 \) (lower boundary of the modal class)
- \( f_m = 45 \) (frequency of the modal class)
- \( f_1 = 25 \) (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 values to the formula
\[
\text{Mode} = 40 + \left( \frac{45 - 25}{(45 - 25) + (45 - 11)} \right) \times 10
\]
\[
= 40 + \left( \frac{20}{20 + 34} \right) \times 10
\]
\[
= 40 + \left( \frac{20}{54} \right) \times 10
\]
\[
= 40 + \left( 0.3704 \right) \times 10
\]
\[
= 40 + 3.704
\]
\[
\approx 43.70
\]

### Conclusion
The mode of the given distribution is approximately **43.70**.

Would you like further details on the calculation, or have any questions? 

Here are some related questions you might find helpful:
1. What is the median of the same distribution?
2. How do you calculate the mean for grouped data?
3. Can we determine skewness from the mode and mean?
4. What are other methods to estimate the mode?
5. How does the mode change if we alter the frequency values?

**Tip:** In frequency distributions, the mode is often a useful measure for understanding the most common value or range of values in a dataset.

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 calculate the mode of a given frequency distribution with class intervals. This detailed solution includes step-by-step calculations using the mode formula for grouped data. Suitable for students in Grades 9-12.

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