The mean of certain number of observations is 10. Even though one observation is deleted, mean is not altered. Then the deleted observation is_________

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We know that the mean of a certain number of observations is 10. This means that if we have 'n' observations, the sum of those observations divided by 'n' equals 10.

### Step 2: Calculate the total sum of observations
Let’s denote the number of observations as 'n'. The mean (average) is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Given that the mean is 10, we can express the total sum of observations as:
\[ \text{Sum} = \text{Mean} \times \text{Number of observations} = 10 \times n \]

### Step 3: Consider the deletion of one observation
When one observation is deleted, the mean remains the same (10). Now, the number of observations becomes \( n - 1 \). The sum of the remaining observations can be expressed as:
\[ \text{Sum of remaining observations} = \text{Sum} - \text{Deleted observation} \]
Let’s denote the deleted observation as \( x \). Therefore, the sum of the remaining observations is:
\[ 10n - x \]

### Step 4: Set up the equation for the new mean
The mean of the remaining observations is still 10, so we can write:
\[ \text{Mean of remaining observations} = \frac{10n - x}{n - 1} = 10 \]

### Step 5: Solve for the deleted observation
Now, we can set up the equation:
\[ \frac{10n - x}{n - 1} = 10 \]
Multiplying both sides by \( n - 1 \):
\[ 10n - x = 10(n - 1) \]
Expanding the right side:
\[ 10n - x = 10n - 10 \]
Now, we can simplify this equation:
\[ -x = -10 \]
Thus, we find:
\[ x = 10 \]

### Conclusion
The deleted observation is **10**.

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