How to Find Variance on a TI-84

You have a test in an hour and need the variance of a data set. Your TI-84 can give it to you in about 20 seconds, but only if you know exactly which keys to press and which output to ignore. The single biggest mistake on this calculator is grabbing the standard deviation, Sx or σx, and reporting it as the variance. Variance is the square of that number. Here is the exact sequence to get variance on ti 84, from entering data to squaring the right standard deviation.

Press STAT, then choose 1:Edit. If L1 already has numbers, clear them: arrow up to highlight L1, press CLEAR, then ENTER. Type each data value, pressing ENTER after each one. For the set {5, 8, 12, 15, 20}, you enter five numbers in L1.

What If You Have Two Data Sets?

Use L1 for the first set and L2 for the second. The 1-Var Stats command works on one list at a time; you run it once per list.

Press STAT, arrow right to CALC, then choose 1:1-Var Stats. Press ENTER again. The calculator outputs a screen with these values, in order: n, x̄, Σx, Σx², Sx, σx, minX, Q1, Med, Q3, maxX. Source: TI-84 Plus CE Guidebook (Texas Instruments), 1-Var Stats outputs. Your variance is not on that screen yet.

The 1-Var Stats output gives you two standard deviations: Sx and σx. They are the square roots of the two variances. The one you need depends on whether your data is a sample or the entire group.

Sx: Sample Standard Deviation

Sx uses n-1 in the denominator. This is Bessel's correction, and it makes the sample variance an unbiased estimator of the group variance. If your data is a subset of a larger group, which it nearly always is in an exam problem, you use Sx. The formula is √(Σ(x - x̄)²/(n-1)).

σx: Population Standard Deviation

σx uses n in the denominator. You use this only when your data set is the entire group. In practice, this is rare. The formula is √(Σ(x - x̄)²/n).

How to Tell Which One the Problem Expects

If the problem says "sample" or gives you a subset of a larger group, use Sx. If it says "population" or "all", use σx. If it says nothing and gives you a small data set, assume it is a sample and use Sx. The most common exam mistake is using σx for a sample, which gives a variance that is too low.

After running 1-Var Stats, you do not re-enter data. Use the shortcut: press VARS, then 5:Statistics (or arrow to it). Choose 3:σx or 4:Sx depending on which standard deviation you need. Press ENTER. The calculator pastes that value onto the home screen. Press x², then ENTER. The result is the variance.

For the data set {5, 8, 12, 15, 20}: Sx ≈ 5.787. Square it: 5.787² ≈ 33.49. That is the sample variance. σx ≈ 5.177. Square it: 5.177² ≈ 26.80. That is the population variance. The difference is about 6.7 squared units, which is large enough to cost you points on an exam.

This is the single most common confusion. The 1-Var Stats output does not show variance; it shows standard deviation. You must square it yourself. The TI-84 Plus CE Guidebook lists Sx and σx as distinct outputs because they are different formulas, but neither is variance. Variance is always the square of the standard deviation you chose.

If you report Sx as the variance, your answer is off by a factor of √(n-1) or √(n). For a sample of size 20, Sx is about 4.36 times the actual variance. That is a catastrophic score.

If your data set is large and you have a frequency column, enter the data values in L1 and the frequencies in L2. Press STAT, CALC, 1:1-Var Stats. This time, press 2nd L1, then a comma, then 2nd L2, then ENTER. The calculator uses the frequencies. Everything else is the same: square Sx or σx for the variance.

Failure case: If you enter frequencies in L2 but do not tell the calculator, it treats every value in L1 as a single observation, ignoring the frequencies. The result is wrong.

Grouped Data and Frequency Tables

For grouped data (e.g., class intervals), enter the class midpoint in L1 and the frequency in L2. The variance you get is an approximation. This method is covered in depth on the variance of grouped data page; here, just know that the same 1-Var Stats procedure applies.

If you get a variance that looks too large, you squared σx instead of Sx, or vice versa. If the variance is negative, you typed a negative number into the square function, variance is always non-negative. If the variance is zero, all your data values are identical. If you cannot find the 1-Var Stats output, you may have pressed 2nd STAT (which is the DISTR menu) instead of STAT. Press STAT once, then arrow right to CALC.

If the calculator shows "ERR: DIM MISMATCH", your L1 and L2 have different numbers of entries. Count them. If it shows "ERR: ARGUMENT", you typed the list names wrong. Clear them and try again.

The most practical thing to do next: open the TI-84 Plus CE Guidebook to page 56-58 and practice running 1-Var Stats on the example data set {1, 2, 3, 4, 5} until you can get Sx = 1.581 and σx = 1.414 without looking at the screen.

In an exam problem, the wording tells you. If it asks for the variance of the sample, use Sx. If it asks for the variance of the population, use σx. If it asks for the variance of a data set that is clearly all the data (e.g., all students in a class), use σx. If it asks for an estimate of the group variance from a sample, use Sx. If the problem uses the letters s² or s²n-1, that is sample variance. If it uses σ² or s²n, that is population variance. If you are still unsure, write down both and explain which one you used, partial credit is better than a wrong answer.

Failure case: A problem asks for "the variance of the heights of all students in the school." That is a group. Use σx. If you use Sx, your variance is too high by a factor of n/(n-1). For a large n, that is close, but wrong.

Symbol	Name	Denominator	When to Use	Square to Get Variance
Sx	Sample standard deviation	n-1	Data is a sample	Sx² = sample variance (s²)
σx	Population standard deviation	n	Data is the entire group	σx² = population variance (σ²)
n	Sample size	—	—	—
x̄	Sample mean	—	—	—
Σx	Sum of data values	—	—	—
Σx²	Sum of squared data values	—	—	—