Question

the table below shows the probability distribution of a random variable x.
x p(x)
-8 0.14
-7 0.44
-6 0.01
-5 0.18
-4 0.09
-3 0.14
what is the standard deviation of x?
round your answer to the nearest hundredth.

Explanation:

Step1: Calculate the expected value $E(X)$

$E(X)=\sum_{i}x_{i}P(x_{i})=(-8)\times0.14+(-7)\times0.44+(-6)\times0.01+(-5)\times0.18+(-4)\times0.09+(-3)\times0.14=-6.2$

Step2: Calculate the variance $Var(X)$

$Var(X)=\sum_{i}(x_{i}-E(X))^{2}P(x_{i})=((-8)-(-6.2))^{2}\times0.14+((-7)-(-6.2))^{2}\times0.44+((-6)-(-6.2))^{2}\times0.01+((-5)-(-6.2))^{2}\times0.18+((-4)-(-6.2))^{2}\times0.09+((-3)-(-6.2))^{2}\times0.14$
$=( - 1.8)^{2}\times0.14+( - 0.8)^{2}\times0.44+(0.2)^{2}\times0.01+(1.2)^{2}\times0.18+(2.2)^{2}\times0.09+(3.2)^{2}\times0.14$
$=3.24\times0.14 + 0.64\times0.44+0.04\times0.01 + 1.44\times0.18+4.84\times0.09+10.24\times0.14$
$=0.4536+0.2816 + 0.0004+0.2592+0.4356+1.4336=2.864$

Step3: Calculate the standard deviation $\sigma$

$\sigma=\sqrt{Var(X)}=\sqrt{2.864}\approx1.69$

Answer:

$1.69$