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Expected Value in Statistics: Definition and Calculating it

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What is Expected Value?
Find an Expected Value by Hand
Find an Expected Value in Excel
What is an Expected Value used for in Real Life?

What is Expected Value?


Expected value is exactly what you might think it means intuitively: the return you can expect for some kind of action, like how many questions you might get right if you guess on a multiple choice test.

For example, if you take a 20 question multiple-choice test with A,B,C,D as the answers, and you guess all “A”, then you can expect to get 25% right (5 out of 20). The math behind this kind of expected value is:
The probability (P) of getting a question right if you guess: .25
The number of questions on the test (n)*: 20
P x n = .25 x 20 = 5

*You might see this as X instead.

This type of expected value is called an expected value for a binomial random variable. It’s binomial because there are only two possible outcomes: you get the answer right, or you get the answer wrong.

Formula

The expected value formula for binomial random variables is written as E(X)=n*P (or P*n, X*P or P*X).

The expected value formula changes a little if you have a series of trials (for example, a series of coin tosses). When you have a series of trials, you take your basic formula (n*P) for each trial and then add them together. Mathematically, the expected value formula for a series of binomial trials is:
E(X) = x1P1 + x2P2 + x3P3 + . . . + xnPn.
The “…+ xnPn” just means to keep on adding your results together — even if you have 1,000 results you would still keep on adding.

If you make a chart, the math behind finding an expected value becomes clearer.
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Calculate an Expected value in statistics by hand

This article explains how to figure out the expected value for a single item (like purchasing a single raffle ticket) and what to do if you have multiple items. If you have a discrete random variable, read this other article instead: Expected value for a discrete random variable.

Sample question: You buy one $10 raffle ticket for a new car valued at $15,000. Two thousand tickets are sold. What is the expected value of your gain?

Step 1: Make a probability chart (see: How to construct a probability distribution). Put Gain(X) and Probability P(X) heading the rows and Win/Lose heading the columns.
expected value

Step 2: Figure out how much you could gain and lose. In our example, if we won, we’d be up $15,000 (less the $10 cost of the raffle ticket). If you lose, you’d be down $10.  Fill in the data (I’m using Excel here, so the negative amounts are showing in red).

expected value 2

Step 3: In the bottom row, put your odds of winning or losing. Seeing as 2,000 tickets were sold, you have a 1/2000 chance of winning. And you also have a 1,999/2,000 probability chance of losing.

expected value 3

Step 4: Multiply the gains (X) in the top row by the Probabilities (P) in the bottom row.
$14,990 * 1/2000 = $7.495,
(-$10)*(1,999/2,000)= -$9.995

Step 5:Add the two values together:
$7.495 + -$9.995 = -$2.5.

That’s it!

Note on multiple items: for example, what if you purchase a $10 ticket, 200 tickets are sold, and as well as a car, you have runner up prizes of a CD player and luggage set?

Perform the steps exactly as above. Make a probability chart except you’ll have more items:

expected value 4

Then multiply/add the probababilities as in step 4: 14,990*(1/200) + 100 * (1/200) + 200 * (1/200) +  -$10 * (197/200).

You’ll note now that because you have 3 prizes, you have 3 chances of winning, so your chance of losing decreases to 197/200.

Note on the formula: The actual formula for expected gain is E(X)=∑X*P(X) (this is also one of the AP Statistics formulas).  What this is saying (in English) is “The expected value is the sum of all the gains multiplied by their individual probabilities.”

Use our online expected value calculator to calculate simple probabilities!

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Find an Expected Value in Excel


Find an expected value in Excel
Step 1: Type your values into two columns in Excel (“x” in one column and “f(x)” in the next.
Step 2: Click an empty cell.
Step 3: Type =SUMPRODUCT(A2:A6,B2:B6) into the cell where A2:A6 is the actual location of your x variables and f(x) is the actual location of your f(x) variables.
Step 4: Press Enter.

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That’s it!

What is Expected Value in Statistics used for in Real Life?

Expected values for binomial random variables are probably the simplest type of expected values. In real life, you’re likely to encounter more complex expected values that have more than two possibilities. For example, you might buy a scratch off lottery ticket with prizes of $1000, $10 and $1.

Related articles:
Online expected value calculator

If you’re looking for more information on formula variations (this gets a bit more technical!), see this article at Wolfram.

Expected Value in Statistics: Definition and Calculating it was last modified: August 13th, 2015 by Stephanie
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8 thoughts on “Expected Value in Statistics: Definition and Calculating it

  1. Lisa Barcomb

    The way that this seems to be is that you need to know how to set up your tables with the information given to you. And this is where I am seeing were I am having problems, what goes where and why? I am having problems with that formula E(X)=Ex*P(X)I really don’t understand it. I guess if I go back to where this started and re-read it the section maybe I will get the jest of it. I see how they put the tables together thats not hard its just trying to figure out where the information goes.

  2. Stephanie

    Lisa,
    If you follow the steps in this how-to, you can skip using the formula. But you are right–it’s a matter of figuring out where to put the information, which is sometimes a challenge. I’m sure with practice you’ll pick it up,
    Stephanie

  3. Mary Johnson

    I agree with Lisa. I am having a hard time understanding where the information goes. This explanation does help a little, I guess I just need to do it more often.

  4. Donna Allen

    I too agree, sometimes the biggest challenge is to know where to plug in the numbers in the equation. The more problems I practice, the more it seems to click, though. Your explanations on here are clear cut and easy to follow.

  5. Catherine Flanagan

    This blog really helped me figure out probability charts. I agree with the other post that it was hard to figure out at first, but after practicing over and over it finally came to me. I also like that it shows the possibility of winning multiple prizes. I don’t recall the book having an example like this one.

  6. angie widdows

    the examples are so helpful when you make tables. Without making the tables, it gets confusing. The more examples the better.

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