Question

If the mean of the data \[ p,6,6,7,8,11,15,16 \] is \(3\) times \(p\), then the mean deviation of the data from its mean is

• A. \(2.25\)
• B. \(3.75\)
• C. \(4.4\)
• D. \(2.5\)

Hint:

Mean deviation about the mean is calculated using: \[ \text{M.D.}=\frac{\sum |x_i-\bar{x}|}{n} \] where \(\bar{x}\) is the arithmetic mean.

Answer 1

The Correct Option is B

Step 1: Set up the mean condition.
The data is $p,6,6,7,8,11,15,16$ with $n=8$, and the mean equals $3p$.
Step 2: Find the sum.
The sum is $p+6+6+7+8+11+15+16=p+69$.
Step 3: Solve for $p$.
Since $\dfrac{p+69}{8}=3p$, we get $p+69=24p$, so $23p=69$ and $p=3$. The mean is $3p=9$.
Step 4: Rewrite the data.
With $p=3$ the data is $3,6,6,7,8,11,15,16$ and mean $\bar x=9$.
Step 5: Find absolute deviations from the mean.
They are $|3-9|=6$, $|6-9|=3$, $|6-9|=3$, $|7-9|=2$, $|8-9|=1$, $|11-9|=2$, $|15-9|=6$, $|16-9|=7$. The sum is $6+3+3+2+1+2+6+7=30$.
Step 6: Compute the mean deviation.
Mean deviation $=\dfrac{30}{8}=3.75$.
\[ \boxed{3.75} \]