The mean of 11 obervation is 50. If the mean of first Six observation is 49 and that of last six observation is 52, then find sixth observation.

Answer

To solve the problem step by step, we will follow the information provided in the question and apply the formulas for mean and sum of observations.

### Step 1: Calculate the sum of all 11 observations
Given that the mean of 11 observations is 50, we can find the sum of these observations using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Thus, we can write:
\[ 50 = \frac{S_{11}}{11} \]
Multiplying both sides by 11 gives:
\[ S_{11} = 50 \times 11 = 550 \]

### Step 2: Calculate the sum of the first 6 observations
We know that the mean of the first 6 observations is 49. Using the same formula:
\[ 49 = \frac{S_{6}}{6} \]
Multiplying both sides by 6 gives:
\[ S_{6} = 49 \times 6 = 294 \]

### Step 3: Calculate the sum of the last 6 observations
The mean of the last 6 observations is given as 52. Again, using the mean formula:
\[ 52 = \frac{S_{last 6}}{6} \]
Multiplying both sides by 6 gives:
\[ S_{last 6} = 52 \times 6 = 312 \]

### Step 4: Relate the sums to find the sixth observation
The sixth observation \(X_6\) is included in both the sum of the first 6 observations and the sum of the last 6 observations. Therefore, we can express the relationship as:
\[ S_{11} = S_{6} + S_{last 6} - X_6 \]
Substituting the values we calculated:
\[ 550 = 294 + 312 - X_6 \]
This simplifies to:
\[ 550 = 606 - X_6 \]

### Step 5: Solve for \(X_6\)
Rearranging the equation gives:
\[ X_6 = 606 - 550 \]
Calculating this gives:
\[ X_6 = 56 \]

### Final Answer
The sixth observation is:
\[ \boxed{56} \]

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