The average of 18 observations Is recorded as 124. Later it was found that two observations with values 64 and 28 were entered wrongly as 46 and 82. Find the correct average of the 18 observations.

Options

A. `111(7)/(9)`
B. `122`
C. `123`
D. `137(3)/(9)`

Answer

To find the correct average of the 18 observations after correcting the wrongly entered values, we can follow these steps:

### Step 1: Calculate the total of the original observations
The average of the 18 observations is given as 124. To find the total of these observations, we can use the formula for average:

\[ \text{Total} = \text{Average} \times \text{Number of observations} \]

Substituting the values:

\[ \text{Total} = 124 \times 18 = 2232 \]

### Step 2: Identify the incorrect and correct values
The two observations that were entered incorrectly are:
- Incorrect values: 46 and 82
- Correct values: 64 and 28

### Step 3: Calculate the sum of the incorrect values
Now, we need to find the sum of the incorrect values:

\[ \text{Sum of incorrect values} = 46 + 82 = 128 \]

### Step 4: Calculate the sum of the correct values
Next, we find the sum of the correct values:

\[ \text{Sum of correct values} = 64 + 28 = 92 \]

### Step 5: Find the difference due to the incorrect entries
Now, we calculate the difference between the sum of the correct values and the sum of the incorrect values:

\[ \text{Difference} = \text{Sum of correct values} - \text{Sum of incorrect values} = 92 - 128 = -36 \]

### Step 6: Adjust the total observations
To find the corrected total of the observations, we add this difference to the original total:

\[ \text{Corrected Total} = \text{Original Total} + \text{Difference} = 2232 - 36 = 2196 \]

### Step 7: Calculate the correct average
Finally, we calculate the correct average using the corrected total:

\[ \text{Correct Average} = \frac{\text{Corrected Total}}{\text{Number of observations}} = \frac{2196}{18} = 122 \]

Thus, the correct average of the 18 observations is **122**.

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