Find the mode for the following data.
`{:("Class Interval",f),(" "0-9,2),(" "10-19,4),(" "20-29,7),(" "30-39,5),(" "40-49,3):}`

Options

A. `30`
B. `25.5`
C. `32`
D. `33`

Explanation

To find the mode for the given data, we will follow these steps:

### Step 1: Convert the data into continuous class intervals
The original class intervals are:
- 0-9
- 10-19
- 20-29
- 30-39
- 40-49

To convert these into continuous class intervals, we adjust the lower and upper limits:
- For the lower limit, subtract 0.5 (except for the first interval where we keep it as 0).
- For the upper limit, add 0.5.

Thus, the continuous class intervals will be:
- 0 to 9.5
- 9.5 to 19.5
- 19.5 to 29.5
- 29.5 to 39.5
- 39.5 to 49.5

### Step 2: Identify the frequency of each class interval
The frequencies corresponding to the original class intervals are:
- 0-9: 2
- 10-19: 4
- 20-29: 7
- 30-39: 5
- 40-49: 3

Now, we can write the frequencies for the continuous class intervals:
- 0 to 9.5: 2
- 9.5 to 19.5: 4
- 19.5 to 29.5: 7
- 29.5 to 39.5: 5
- 39.5 to 49.5: 3

### Step 3: Determine the modal class
The modal class is the class interval with the highest frequency. From the frequency distribution:
- The highest frequency is 7, which corresponds to the class interval 19.5 to 29.5.

Thus, the modal class is **19.5 to 29.5**.

### Step 4: Use the mode formula
The formula for calculating the mode is:
\[ \text{Mode} = L_1 + \frac{f_m - f_1}{(2f_m - f_1 - f_2)} \times i \]
Where:
- \(L_1\) = lower limit of the modal class = 19.5
- \(f_m\) = frequency of the modal class = 7
- \(f_1\) = frequency of the class preceding the modal class = 4
- \(f_2\) = frequency of the class succeeding the modal class = 5
- \(i\) = width of the class interval = 29.5 - 19.5 = 10

### Step 5: Substitute the values into the formula
Substituting the values into the mode formula:
\[ \text{Mode} = 19.5 + \frac{7 - 4}{(2 \times 7 - 4 - 5)} \times 10 \]
\[ = 19.5 + \frac{3}{(14 - 9)} \times 10 \]
\[ = 19.5 + \frac{3}{5} \times 10 \]
\[ = 19.5 + 6 \]
\[ = 25.5 \]

### Conclusion
The mode of the given data is **25.5**.

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