New Zealand
Level 8 - NCEA Level 3

# Measures of Variability of Discrete Random Distribution

Lesson

Just as we can calculate the mean for the probability distribution of a Discrete Random Variable, we can also calculate the variance and standard deviation.

We calculate the variance in the exact same way we calculate the variance of a set of data.

The only difference is including the weightings, or the probabilities, into our calculations.

Calculating the Variance of a DRV

As you can see, there are two versions of the Variance formula. Example 1 demonstrates the first formula, and example 2 demonstrates the second.

##### Example 1

Consider the table.

 $x$x $P$P$($($X=x$X=x$)$) $0$0 $1$1 $2$2 $3$3 $4$4 $0.12$0.12 $0.15$0.15 $0.22$0.22 $0.23$0.23 $0.28$0.28
1. Does the table represent a discrete probability distribution?

Yes

A

No

B

Yes

A

No

B
2. Calculate $E\left(X\right)$E(X).

3. Calculate the variance of $X$X.

4. Hence calculate the standard deviation.

##### Example 2

Let's use the same probability distribution from our previous chapter as an example of how to calculate the variance of a probability distribution.

Consider the probability distribution of a DRV given in the table below.

$x$x $1$1 $2$2 $3$3 $4$4 $5$5
$P(X=x)$P(X=x) $0.2$0.2 $0.15$0.15 $0.4$0.4 $0.1$0.1 $0.15$0.15

Calculate the variance and standard deviation of the distribution.

We already know from last time that the mean, or expected value is $E(X)=2.85$E(X)=2.85

Calculating the variance we get:

$Var\left(X\right)=\left(1^2\times0.2+2^2\times0.15+3^2\times0.4+4^2\times0.1+5^2\times0.15\right)-2.85^2$Var(X)=(12×0.2+22×0.15+32×0.4+42×0.1+52×0.15)2.852

$Var\left(X\right)=1.6275$Var(X)=1.6275

To calculate the standard deviation we simply find the square root of the variance.

$\sigma\left(X\right)=\sqrt{1.6275}$σ(X)=1.6275

$\sigma\left(X\right)=1.2757$σ(X)=1.2757

##### Example 3

Consider the graph drawn below of a discrete probability distribution.

1. Calculate the expected value of this distribution.

2. Would you expect the median to be greater than, less than or equal to the expected value of this distribution?

less than

A

greater than

B

equal to

C

less than

A

greater than

B

equal to

C
3. Calculate the variance of this distribution.

4. Hence, calculate the standard deviation.

5. Would you expect the standard deviation of the distribution drawn below to be greater than, less than or equal to the standard deviation of our original distribution?

less than

A

equal to

B

greater than

C

less than

A

equal to

B

greater than

C

##### Example 4

Consider the probability density function given below:

 $p\left(x\right)$p(x) $=$= $\frac{x}{10}$x10​; $x=1$x=1, $2$2, $3$3, $4$4 $0$0, otherwise
1. Complete the table of values for this probability distribution.

 $x$x $P$P$($($X=x$X=x$)$) $1$1 $2$2 $3$3 $4$4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Hence calculate the mean of the distribution.

3. Calculate the standard deviation of the distribution.

### Outcomes

#### S8-4

Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal

#### 91586

Apply probability distributions in solving problems