The mean of 20 observation is 12. On checking, it was found that two observation were wrongly copied as 3 and 6. If wrong observation are replaced by correct values 8 and 4, then the correct mean is

Options

A. 15
B. 15.15
C. 15.35
D. 16

Explanation

To find the correct mean after replacing the wrong observations with the correct ones, we can follow these steps:

### Step 1: Calculate the total sum of the original observations.
The mean of 20 observations is given as 12. The formula for mean is:

\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Observations}} \]

From this, we can find the total sum:

\[ \text{Total Sum} = \text{Mean} \times \text{Number of Observations} = 12 \times 20 = 240 \]

### Step 2: Identify the wrong and correct observations.
The wrong observations are 3 and 6, and the correct observations are 8 and 4.

### Step 3: Adjust the total sum by removing the wrong observations and adding the correct ones.
To find the adjusted total sum, we need to subtract the wrong observations and add the correct ones:

\[ \text{Adjusted Total Sum} = \text{Original Total Sum} - \text{Sum of Wrong Observations} + \text{Sum of Correct Observations} \]

Calculating the sum of the wrong observations:

\[ \text{Sum of Wrong Observations} = 3 + 6 = 9 \]

Calculating the sum of the correct observations:

\[ \text{Sum of Correct Observations} = 8 + 4 = 12 \]

Now substituting these values into the adjusted total sum formula:

\[ \text{Adjusted Total Sum} = 240 - 9 + 12 = 240 + 3 = 243 \]

### Step 4: Calculate the new mean using the adjusted total sum.
Now we can find the new mean using the adjusted total sum:

\[ \text{New Mean} = \frac{\text{Adjusted Total Sum}}{\text{Number of Observations}} = \frac{243}{20} \]

Calculating the new mean:

\[ \text{New Mean} = 12.15 \]

### Final Answer:
The correct mean after replacing the wrong observations with the correct ones is **12.15**.

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