For the following probability distribution determine standard deviation of the random variable $X$

$X$ 2 3 4

$P (X)$ 0.2 0.5 0.3

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Solution

Apply the basic formula of standard deviation

$E (X) = 2 \times 0.2 + 3 \times 0.5 + 4 \times 0.3$
$= 0.4 + 1.5 + 1.2 = 3.1$

$E (X^{2}) = 2^{2} \times 0.2 + 3^{2} \times 0.5 + 4^{2} \times 0.3$
$= 4 \times 0.2 + 9 \times 0.5 + 16 \times 0.3$
$= 0.8 + 4.5 + 4.8 = 10.1$

So, $V a r (X) = E (X^{2}) - (E (X))^{2}$
$V a r (X) = 10.1 - (3.1)^{2}$
$= 10.1 - 9.61$
$= 0.49$

$S . D = \sqrt{V a r (X)}$
$= \sqrt{0.49}$
$= 0.7$

Hence the required standard deviation is $0.7$

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Standard Deviation about Mean for Frequency Distributions

Standard XII Mathematics

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For the following probability distribution determine standard deviation of the random variable X X 2 3 4 P(X) 0.2 0.5 0.3

Apply the basic formula of standard deviation E(X)=2×0.2+3×0.5+4×0.3 =0.4+1.5+1.2=3.1 E(X2)=22×0.2+32×0.5+42×0.3 =4×0.2+9×0.5+16×0.3 =0.8+4.5+4.8=10.1 So, Var(X)=E(X2)−(E(X))2 Var(X)=10.1−(3.1)2 =10.1−9.61 =0.49 S.D=√Var(X) =√0.49 =0.7 Hence the required standard deviation is 0.7

For the following probability distribution determine standard deviation of the random variable $X$

$X$ 2 3 4

$P (X)$ 0.2 0.5 0.3