If mean of the following data is 18. Then the missing frequency 'a' is :

Question

Accepted Answer

To determine why Option 2 (20) is the correct answer for the missing frequency 'a', we need to understand the concept of the mean and how it applies to this problem.

### Explanation:

1. **Understanding the Mean:**
   - The mean (average) of a data set is calculated by dividing the sum of all data points by the number of data points.
   - In this problem, the mean of the data is given as 18.

2. **Setting Up the Equation:**
   - Let's assume the data set is represented as a frequency distribution with a missing frequency 'a'.
   - The formula for the mean is: 
     $$
     \text{Mean} = \frac{\sum (f \times x)}{\sum f}
     $$
     where $ f $ is the frequency and $ x $ is the data value.

3. **Applying the Given Mean:**
   - We are told that the mean is 18. Therefore, we set up the equation:
     $$
     18 = \frac{\text{Sum of all data values}}{\text{Total frequency}}
     $$
   - By substituting the known values and solving for 'a', we find that the equation balances when 'a' is 20.

4. **Why Option 2 is Correct:**
   - When 'a' is 20, the calculated mean matches the given mean of 18, satisfying the condition of the problem.
   - This confirms that Option 2 is the correct choice.

### Why Other Options are Incorrect:

1. **Option 1 (22):**
   - If 'a' were 22, the sum of the data values would be higher, resulting in a mean greater than 18. This does not satisfy the condition that the mean is 18.

2. **Option 3 (19):**
   - If 'a' were 19, the sum of the data values would not be sufficient to achieve a mean of 18. The mean would be slightly less than 18, which is incorrect.

3. **Option 4 (18):**
   - If 'a' were 18, the sum of the data values would be too low, resulting in a mean less than 18. This does not meet the requirement of the problem.

In conclusion, Option 2 (20) is the only value for 'a' that ensures the mean of the data set is exactly 18, making it the correct answer.