The mean of 100 items was found to be 30. If at the time of calculation, two items were wrongly taken as 32 and 12 instead of 23 and 11, the correct mean is

Options

A. (a) 29.90
B. (b) 31.32
C. (c) 32.50
D. (d) 33.45

Answer

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

### Step 1: Calculate the total sum of the items based on the initial mean.
The mean is given by the formula:

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

Given that the mean of 100 items is 30, we can calculate the total sum of the items:

\[ \text{Total Sum} = \text{Mean} \times \text{Number of Items} = 30 \times 100 = 3000 \]

### Step 2: Identify the incorrect and correct values.
The two items that were incorrectly recorded are 32 and 12. They should have been recorded as 23 and 11.

### Step 3: Calculate the difference caused by the incorrect values.
Now, we need to adjust the total sum by removing the incorrect values and adding the correct values:

\[ \text{Adjustment} = (\text{Correct Values}) - (\text{Incorrect Values}) = (23 + 11) - (32 + 12) \]

Calculating the sums:

\[ 23 + 11 = 34 \]
\[ 32 + 12 = 44 \]

Now, substituting back into the adjustment formula:

\[ \text{Adjustment} = 34 - 44 = -10 \]

### Step 4: Update the total sum with the adjustment.
Now we will adjust the total sum:

\[ \text{Correct Total Sum} = \text{Original Total Sum} + \text{Adjustment} = 3000 - 10 = 2990 \]

### Step 5: Calculate the correct mean.
Now that we have the correct total sum, we can calculate the new mean:

\[ \text{Correct Mean} = \frac{\text{Correct Total Sum}}{\text{Number of Items}} = \frac{2990}{100} = 29.9 \]

### Final Answer:
The correct mean is **29.9**.

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