# AP Statistics : How to find the standard deviation of the sum of independent random variables

## Example Questions

### Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

A high school calculus exam is administered to a group of students. Upon grading the exam, it was found that the mean score was 95 with a standard deviation of 12. If one student's z score is 1.10, what is the score that she received on her test?

108.2

105.3

107.2

109.2

110.1

108.2

Explanation:

The z-score equation is given as: z = (X - μ) / σ, where X is the value of the element, μ is the mean of the population, and σ is the standard deviation. To solve for the student's test score (X):

X = ( z * σ) + 95 = ( 1.10 * 12) + 95 = 108.2.

### Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

and are independent random variables. If has a mean of  and standard deviation of  while variable has a mean of  and a standard deviation of , what are the mean and standard deviation of ?

Explanation:

First, find that has  and standard deviation .

Then find the mean and standard deviation of .

### Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

Consider the discrete random variable  that takes the following values with the corresponding probabilities:

•  with
•  with
•  with
•  with

Compute the variance of the distribution.

Explanation:

The variance of a discrete random variable is computed as

for all the values of  that the random variable  can take.

First, we compute , which is the expected value. In this case, it is .

So we have

### Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

Clothes 4 Kids uses standard boxes to ship their clothing orders and the mean weight of the clothing packed in the boxes is  pounds. The standard deviation is  pounds. The mean weight of the boxes is  pound with a standard deviation of  pounds. The mean weight of the plastic packaging is  pounds per box, with a  pound standard deviation. What is the standard deviation of the weights of the packed boxes?