If the mean of the observation a , a + 6 , a + 2 , a + 8 and a + 4 is 11 . Find .
the value of a

Answer

To find the value of \( a \) given that the mean of the observations \( a, a + 6, a + 2, a + 8, \) and \( a + 4 \) is 11, we can follow these steps:

### Step 1: Write down the formula for the mean.
The mean of a set of observations is calculated by taking the sum of all observations and dividing it by the number of observations. In this case, we have 5 observations.

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

### Step 2: Set up the equation using the given mean.
We know that the mean is 11, so we can set up the equation as follows:

\[ 11 = \frac{a + (a + 6) + (a + 2) + (a + 8) + (a + 4)}{5} \]

### Step 3: Simplify the sum of the observations.
Now, let's simplify the sum of the observations in the numerator:

\[ a + (a + 6) + (a + 2) + (a + 8) + (a + 4) = a + a + 6 + a + 2 + a + 8 + a + 4 \]

Combining like terms, we get:

\[ 5a + (6 + 2 + 8 + 4) = 5a + 20 \]

### Step 4: Substitute the simplified sum back into the equation.
Now, we can substitute this back into our mean equation:

\[ 11 = \frac{5a + 20}{5} \]

### Step 5: Multiply both sides by 5 to eliminate the denominator.
To eliminate the fraction, multiply both sides of the equation by 5:

\[ 11 \times 5 = 5a + 20 \]

This simplifies to:

\[ 55 = 5a + 20 \]

### Step 6: Solve for \( a \).
Now, we can solve for \( a \) by isolating it on one side of the equation. First, subtract 20 from both sides:

\[ 55 - 20 = 5a \]

This gives us:

\[ 35 = 5a \]

Now, divide both sides by 5:

\[ a = \frac{35}{5} = 7 \]

### Final Answer:
The value of \( a \) is \( 7 \).

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