Question

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

• A. $3a^2–26a+55=0$
• B. $3a^2–32a+84=0$
• C. $3a^2–34a+91=0$
• D. $3a^2–23a+44=0$

Answer 1

The correct option is B

$3a^2–32a+84=0$

We know that, if $x_1,x_2,\cdots ,x_n$ are n observations, then their standard deviation is given by
$\sqrt{\frac{1}{n}\sum x_i^2-{\left(\frac{\sum x_i}{n}\right)}^2}$
We have, $(3.5{)}^2=\frac{(2^2+3^2+a^2+{11}^2)}{4}-{\left(\frac{2+3+a+11}{4}\right)}^2$
$\Rightarrow \frac{49}{4}=\frac{4+9+a^2+121}{4}-{\left(\frac{16+a}{4}\right)}^2\Rightarrow \frac{49}{4}=\frac{134+a^2}{4}-\frac{256+a^2+32a}{16}\Rightarrow \frac{49}{4}=\frac{4a^2+536-256-a^2-32a}{16}\Rightarrow 49\times 4=3a^2-32a+280\Rightarrow 3a^2-32a+84=0$