How to Find Expected Value on a TI-84

You need the expected value of a discrete probability distribution for your AP Statistics homework and you have a TI-84 Plus CE in front of you. The calculator does it faster than any formula you could write by hand: press STAT, EDIT, enter your outcomes in L1 and their probabilities in L2, then run 1-Var Stats on the two lists together. The TI-84 outputs the expected value (mean) as x̄, which is the exact sum of each outcome times its probability.

The expected value is the long-run average outcome, not the most likely one. If you roll a fair die, the expected value is 3.5, a number that never appears; the mode is every face equally because all have probability 1/6. Mistaking the two is the most common error AP students make on exam questions about discrete random variables.

Calculator Documentation

The TI-84 Plus CE guidebook (Texas Instruments, 2019) documents 1-Var Stats with a frequency list on page 54 of the statistics section. The same section covers the alternative method using list arithmetic, sum(L1*L2), which gives you Σx before you divide by the total probability. Both methods require that the probabilities in L2 sum to 1; if they do not, your expected value is wrong.

Press STAT, then 1:Edit. If L1 or L2 contain old data, press CLEAR on each column header before entering new numbers.

In L1 enter each possible outcome of the random variable. For the sum of two dice, L1 holds 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. In L2 enter the corresponding probabilities as decimals. For two dice, the chance of a 2 is 1/36 = 0.0278, a 3 is 2/36 = 0.0556, and so on. The probabilities in L2 must sum to exactly 1. A common mistake is entering frequencies instead of probabilities, entering 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 instead of dividing by 36. That gives you the sum of the dice values, not the expected value.

If a problem gives you a probability table with a fraction like 3/10, enter it as 0.3 in L2. The TI-84 does not accept fractions directly in the list; convert to a decimal first.

Press STAT, scroll right to CALC, then select 1:1-Var Stats. On the home screen, type L1, L2 so the command reads 1-Var Stats L1, L2. Press ENTER.

The output scrolls. The value labelled x̄ is the expected value of the probability distribution. For a standard discrete random variable where L2 holds the probabilities directly, x̄ equals Σ (outcome × probability). The same screen shows σx, which is the population standard deviation of the random variable. If you need the variance, square σx, do not use Sx, which is the sample standard deviation and is meaningless for a probability distribution.

The TI-84 Plus CE guidebook confirms that 1-Var Stats L1, L2 computes the mean of the data in L1 weighted by the frequencies in L2. When those frequencies are probabilities, the mean is the expected value and σx is the standard deviation of the random variable. The outputs Σx and Σx² are intermediate sums you can ignore unless you are verifying by hand.

Keystrokes Summary

• STAT → 1:Edit → enter outcomes in L1, probabilities in L2
• STAT → CALC → 1:1-Var Stats → L1, L2 → ENTER
• Read x̄ for expected value, σx for standard deviation

The sum(L1*L2) method is a direct translation of the expected value formula. Press 2ND MATH to open the MATH menu, scroll right to NUM, select 5:sum(. On the home screen type L1*L2) so the command reads sum(L1*L2). Press ENTER.

If your outcomes in L1 are 2 through 12 and your probabilities in L2 sum to 1, sum(L1*L2) returns the same number as x̄ from 1-Var Stats. The sum( command lives in the MATH NUM submenu and returns the sum of the elements in a list; L1*L2 multiplies the two lists element-by-element, giving you a new list where each entry is outcome × probability.

This method is useful when you only need the expected value and not the standard deviation or quartiles. It is also the method to use if you need to compute the variance of a random variable by hand, because you can also compute sum(L1^2 * L2) and subtract the square of the expected value. On a TI-84, 1-Var Stats already gives you σx², so you only need sum(L1*L2) when you are checking work or when the problem asks for the formula.

Entering frequencies instead of probabilities. If you put 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 in L2 for two dice, 1-Var Stats computes the mean of the data in L1 weighted by those frequencies, which gives you 7 exactly. That is the expected value of a single die (3.5) times 2, but only by coincidence. The correct expected value is also 7, but the method is wrong. If you use frequencies that do not sum to 1, you must divide by the total frequency. 1-Var Stats L1, L2 with frequencies gives you the mean of the data, not the expected value of the probability distribution. The distinction matters when the frequencies are not a multiple of the probabilities.

List length mismatch. If L1 has 11 values and L2 has 10, the TI-84 returns an ERR:DIM MISMATCH. Count the entries before running the command. Every outcome in L1 must have a probability in L2.

Using Sx instead of σx. The output screen shows both Sx and σx. For a probability distribution, you always use σx, the population standard deviation. Sx is for sample data. AP Statistics exams deduct points for using Sx when the problem describes a random variable.

Forgetting that 1-Var Stats expects probabilities in decimal form. Fractions like 1/2 entered as the text string "1/2" cause an error. Convert to 0.5 first.