Random Variables – Measure of Variability

Measure of Variability

The most useful ways to measure this variability is to find the variance and standard deviation.

Variance

Variance is often used to measure how close the values are to the mean. We can calculate it by measuring the spread of the probability distribution from its expected value and it is shown right below:

$Var(X)=E[(X-\mu)^{2}t]$

$Var(X)=E(X)^{2}-[E(X)]^{2}$

Variance is equal to the expected value of a random variable minus the mean value of the probability distribution or variance is equal to the expected value of the squared probability distribution minus the expected value of the probability distribution squared.

It can be thought of as a measure of the difference a random variable will be from the mean, how far away it will be.

Standard Deviation

The standard deviation is simply the square root of the variance. It is often represented by sigma (σ).

$sd(X)=\sqrt{(Var(X))}$

In general, we can interpret the standard deviation by looking at the probability distribution. For many random variables, around 95% of the distribution lies within two standard deviation away from each side of the mean. It is given by the formula Pr(µ-2σ ≤ X ≤ µ+2σ) =0.95. 3σ means that the value is 3 standard deviation away from the mean and 2σ means that the value is 2 standard deviation and so on.

^{Practice Question}

Question 7b from the 2009 VCAA exam 1, Find the Variance of the distribution below.

Step 1 Find the expected value E(x)

$E(x) = 0 \times 0.1 +1 \times 0.2 +2 \times 0.4 +3 \times 0.2 +4 \times 0.1 \newline =2$

Step 2 Find $E(x^{2})$ by doing the same process but squaring all the x values first.

$E(x) = 0 ^{2}\times 0.1 +1^{2} \times 0.2 +2^{2} \times 0.4 +3^{2} \times 0.2 +4^{2} \times 0.1 \newline =5.2$

Step 3 Plug in the two calculated values in the variance formula above

$Var(X)=E(X)^{2}-[E(X)]^{2} \newline =5.2-2^{2}=1.2$

If we were required to find the standard deviation all we would need to do is calculate the square of the variance.

$sd(X)=\sqrt{(Var(X))} \newline =\sqrt{1.2}$

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