The mean of a group of 100 observation was found to be 20. Latter it was found that four observation were incorrect, which were recorded as 21, 21, 18 and 20. What is the mean if the incorrect observation are obtained?

Options

A. 18
B. 20
C. 21
D. 22

Explanation

To solve the problem step by step, we will first calculate the sum of the original observations, then adjust for the incorrect observations, and finally find the new mean.

### Step 1: Calculate the original sum of observations
The mean of the group of 100 observations is given as 20.

To find the total sum of these observations, we use the formula:
\[ \text{Sum} = \text{Mean} \times \text{Number of Observations} \]
Substituting the values:
\[ \text{Sum} = 20 \times 100 = 2000 \]

### Step 2: Identify the incorrect observations
The incorrect observations recorded are:
- 21
- 21
- 18
- 20

### Step 3: Calculate the sum of the incorrect observations
Now, we will calculate the sum of these incorrect observations:
\[ \text{Sum of incorrect observations} = 21 + 21 + 18 + 20 = 80 \]

### Step 4: Calculate the correct sum of observations
To find the correct sum of observations, we need to subtract the incorrect observations from the original sum and then add the correct observations. Since we don't know the correct observations yet, we will assume they are \(x_1, x_2, x_3, x_4\).

However, since we only need to adjust the total sum for the incorrect observations, we can simply replace the incorrect observations with the correct ones. The correct sum can be calculated as:
\[ \text{Correct Sum} = \text{Original Sum} - \text{Sum of Incorrect Observations} + \text{Sum of Correct Observations} \]
But since we don't know the correct observations, we can simply adjust the total sum directly.

### Step 5: Calculate the new mean
Since we are only adjusting for the incorrect observations, we can find the new mean by:
\[ \text{New Mean} = \frac{\text{Correct Sum}}{\text{Number of Observations}} \]
Substituting the values:
\[ \text{New Mean} = \frac{2000 - 80}{100} = \frac{1920}{100} = 19.2 \]

### Final Answer
The new mean after correcting the observations is **19.2**.

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