AP®StatisticsCollege BoardRevision NotesUnit 4: Probability, Random Variables & Probability DistributionsDiscrete Random VariablesMean & Standard Deviation of a Discrete Random Variable

Mean & Standard Deviation of a Discrete Random Variable (College Board AP® Statistics)

Revision Note

Dan Finlay

Expertise

Maths Lead

Mean of a discrete random variable

What is the mean of a discrete random variable?

  • The mean of a discrete random variable is the weighted average of the possible values based on their probabilities

    • It is also known as the expected value

  • The mean of a random variable X is denoted mu subscript X or E open parentheses X close parentheses

How do I calculate the mean of a discrete random variable?

  • To calculate the mean of a discrete random variable X

    • multiply each possible value of x by its probability

    • add these together

  • The formula is mu subscript X equals sum x subscript i times P open parentheses x subscript i close parentheses

    • This is given in the exam

  • If the distribution is symmetrical then the mean is equal to the median of the values

How do I find the mean of a discrete random variable using a calculator?

  • To find the mean of a discrete random variable X using a calculator

    • type the values for X in a list as if they were data values

    • enter their probabilities as the frequencies

    • find the summary statistics and identify the mean x with bar on top

Exam Tip

If you are asked to find the mean in a free response exam question then you must show your working. You can use your calculator to check your answer.

Worked Example

If a person chooses to donate to a charity, they can pay $1, $5 or $10.

Let X represent the amount, in dollars, that a person pays to the charity when donating. The probability distribution of X is shown in the table.

x

1

5

10

P open parentheses x close parentheses

0.6

0.3

0.1

What is the expected value of the amount that a person pays to the charity when donating?

Answer:

Use the formula mu subscript X equals sum x subscript i times P open parentheses x subscript i close parentheses

Multiply each value of xby its probability then add the probabilities together

table row cell mu subscript X end cell equals cell 1 open parentheses 0.6 close parentheses plus 5 open parentheses 0.3 close parentheses plus 10 open parentheses 0.1 close parentheses end cell row blank equals cell 3.1 end cell end table

The expected value of the amount that a person pays to the charity when donating is $3.10

Standard deviation of a discrete random variable

What is the standard deviation of a discrete random variable?

  • The variance of a discrete random variable is the expected value of the squared differences between the values and the mean

  • The standard deviation of a discrete random variable is the positive square root of its variance

  • The standard deviation of a random variable X is denoted sigma subscript X

How do I calculate the standard deviation of a discrete random variable?

  • To calculate the standard deviation of a discrete random variable X

    • subtract the mean from each value of x

    • square them

    • multiply each one by the probability of that value of x occurring

    • add these together

    • take the positive square root

  • The formula is sigma subscript X equals square root of sum open parentheses x subscript i minus mu subscript X close parentheses squared times P open parentheses x subscript i close parentheses end root

    • This is given in the exam

How do I find the standard deviation of a discrete random variable using a calculator?

  • To find the standard deviation of a discrete random variable X using a calculator

    • type the values for X in a list as if they were data values

    • enter their probabilities as the frequencies

    • find the summary statistics and identify the population standard deviation sigma

      • the sample standard deviation, s, should not exist in this case

      • n equals 1 which means n minus 1 equals 0 and numbers cannot be divided by zero

Exam Tip

If you are asked to find the standard deviation in a free response exam question then you must show your working. You can use your calculator to check your answer.

Worked Example

The probability distribution of the discrete random variable X is shown by the following graph.

Bar graph showing probability distribution P(x) with x-values from 1 to 6. The probability for 1 and 6 is 0.1, the probability for 2 and 5 is 0.15 and the probability for 3 and 4 is 0.25

(a) Explain why the expected value of X is 3.5.

Answer:

The distribution is symmetrical therefore the mean is equal to the median which is 3.5

(b) Calculate the standard deviation of X.

Answer:

Use the formula sigma subscript X equals square root of sum open parentheses x subscript i minus mu subscript X close parentheses squared times P open parentheses x subscript i close parentheses end root

Read the probabilities from the graph

Subtract 3.5 from each value of x, square it and multiply it by its probability

Add together the probabilities and take the positive square root of the result

table row cell sigma subscript X end cell equals cell square root of open parentheses 1 minus 3.5 close parentheses squared open parentheses 0.1 close parentheses plus open parentheses 2 minus 3.5 close parentheses squared open parentheses 0.15 close parentheses plus open parentheses 3 minus 3.5 close parentheses squared open parentheses 0.25 close parentheses plus open parentheses 4 minus 3.5 close parentheses squared open parentheses 0.25 close parentheses plus open parentheses 5 minus 3.5 close parentheses squared open parentheses 0.15 close parentheses plus open parentheses 6 minus 3.5 close parentheses squared open parentheses 0.1 close parentheses end root end cell row blank equals cell square root of 2.05 end root end cell row blank equals cell 1.431... end cell end table

The standard deviation of X is 1.43

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    Dan Finlay

    Author: Dan Finlay

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.