The mean of 10 observations is 16.3. By an error one observations is registered as 32 instead of 23. Then the correct mean is

Options

A. 15.6
B. 15.4
C. 15.7
D. 15.8

Answer

To solve the problem step by step, we will follow the given information and apply the necessary formulas.

### Step 1: Understand the given information
We know that:
- The mean of 10 observations is 16.3.
- One observation was incorrectly recorded as 32 instead of 23.

### Step 2: Calculate the total sum of the observations
The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Given that the mean is 16.3 and the number of observations is 10, we can find the sum of the observations:
\[ \text{Sum of observations} = \text{Mean} \times \text{Number of observations} = 16.3 \times 10 = 163 \]

### Step 3: Determine the error in the recorded observation
The error occurred because one observation was recorded as 32 instead of 23. We need to find the difference caused by this error:
\[ \text{Difference} = 32 - 23 = 9 \]
This means that the sum of the observations was increased by 9 due to the error.

### Step 4: Calculate the correct sum of observations
To find the correct sum of observations, we need to subtract the error from the previously calculated sum:
\[ \text{Correct sum of observations} = 163 - 9 = 154 \]

### Step 5: Calculate the correct mean
Now that we have the correct sum of observations, we can find the correct mean using the same formula:
\[ \text{Correct Mean} = \frac{\text{Correct sum of observations}}{\text{Number of observations}} = \frac{154}{10} = 15.4 \]

### Final Answer
The correct mean of the observations is **15.4**.

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