Question

The mean deviation about the mean for the following data:
\[ 5,6,7,8,6,9,13,12,15 \] is

• A. \(1.55\)
• B. \(2.88\)
• C. \(3.89\)
• D. \(5\)

Hint:

Mean deviation about the mean is calculated by taking the average of absolute deviations from the arithmetic mean.

Answer 1

The Correct Option is B

Step 1: Find the mean of the data.
Given observations are
\[ 5,6,7,8,6,9,13,12,15 \]
Number of observations,
\[ n=9 \]
Sum of observations:
\[ 5+6+7+8+6+9+13+12+15=81 \]
Therefore, mean is
\[ \bar x=\frac{81}{9} \]
\[ \bar x=9 \]

Step 2: Find the absolute deviations from the mean.
Now compute \(|x_i-\bar x|\):
\[ |5-9|=4 \] \[ |6-9|=3 \] \[ |7-9|=2 \] \[ |8-9|=1 \] \[ |6-9|=3 \] \[ |9-9|=0 \] \[ |13-9|=4 \] \[ |12-9|=3 \] \[ |15-9|=6 \]

Step 3: Find the sum of absolute deviations.
\[ 4+3+2+1+3+0+4+3+6 \]
\[ =26 \]

Step 4: Calculate mean deviation about the mean.
Mean deviation about the mean is given by
\[ \text{MD}=\frac{\sum |x_i-\bar x|}{n} \]
Therefore,
\[ \text{MD}=\frac{26}{9} \]
\[ =2.888\ldots \]
\[ \approx 2.88 \]

Step 5: Final conclusion.
Hence, the mean deviation about the mean is
\[ \boxed{2.88} \]