Mean of 100 observation is 45. It was later found that two observations 19 and 31 were incorrectly recorded as 91 and 13. Then the correct means is

Options

A. `44.0`
B. 44.46
C. `45.00`
D. 45.54

Answer

To find the correct mean after identifying the errors in the recorded observations, we can follow these steps:

### Step 1: Calculate the total sum of the observations
Given that the mean of 100 observations is 45, we can calculate the total sum (S) of the observations using the formula:

\[ S = \text{Mean} \times \text{Number of Observations} \]

Substituting the values:

\[ S = 45 \times 100 = 4500 \]

### Step 2: Identify the incorrect observations
The incorrect observations are 91 and 13. We need to remove these from the total sum.

### Step 3: Add the correct observations
The correct observations that should replace the incorrect ones are 19 and 31. We will add these to the total sum after removing the incorrect observations.

### Step 4: Calculate the corrected sum
Now, we can calculate the corrected sum (S'):

\[ S' = S - (\text{Incorrect Observations}) + (\text{Correct Observations}) \]

Calculating the incorrect observations:

\[ \text{Incorrect Observations} = 91 + 13 = 104 \]

Calculating the correct observations:

\[ \text{Correct Observations} = 19 + 31 = 50 \]

Now substituting these values into the corrected sum formula:

\[ S' = 4500 - 104 + 50 \]

Calculating this gives:

\[ S' = 4500 - 104 + 50 = 4500 - 54 = 4446 \]

### Step 5: Calculate the corrected mean
Finally, we calculate the corrected mean using the corrected sum:

\[ \text{Corrected Mean} = \frac{S'}{\text{Number of Observations}} = \frac{4446}{100} = 44.46 \]

Thus, the corrected mean is **44.46**.

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