Question

The mean deviation about the mean for the following data is \[ \begin{array}{c|cccc} x_i & 1 & 2 & 4 & 7\\ \hline f_i & 3 & 2 & 4 & 1 \end{array} \]

• A. $3$
• B. $2$
• C. $1.5$
• D. $1.6$

Hint:

First find the mean, then multiply each absolute deviation by its frequency.

Answer 1

The Correct Option is D

Step 1: Recall the formula.
Mean deviation about the mean is $\dfrac{\sum f_i|x_i-\bar x|}{\sum f_i}.$ First we need the mean $\bar x.$
Step 2: Total frequency.
$\sum f_i=3+2+4+1=10.$
Step 3: Find the mean.
$\sum f_i x_i=1\cdot3+2\cdot2+4\cdot4+7\cdot1=3+4+16+7=30.$ So $\bar x=\dfrac{30}{10}=3.$
Step 4: Find each distance from the mean.
$|1-3|=2,\ |2-3|=1,\ |4-3|=1,\ |7-3|=4.$
Step 5: Multiply by frequencies and add.
$\sum f_i|x_i-\bar x|=3(2)+2(1)+4(1)+1(4)=6+2+4+4=16.$
Step 6: Divide.
Mean deviation $=\dfrac{16}{10}=1.6.$ \[ \boxed{1.6} \]