How to find standard deviation on TI-84: Sx vs sigma-x

Q: Does the AP exam want Sx or sigma-x?

Almost always Sx. AP Statistics problems describe samples, and the course defines the sample standard deviation with n - 1 in the denominator. Use σ_x only when the data set is the complete population you are describing.

Q: Why is Sx bigger than sigma-x?

Both start from the same sum of squared deviations, but Sx divides it by n - 1 while σ_x divides by n. Dividing by a smaller number gives a larger result. For the data set 4, 7, 8, 10, 16 the sum is 80, so Sx = sqrt(80/4) = 4.4721 and σ_x = sqrt(80/5) = 4. This holds whenever the values are not all identical; if every value in the list is the same, the sum of squared deviations is 0 and both standard deviations are 0.

By Jude Wallis · Updated July 31, 2026

Enter your data with STAT then EDIT, then run STAT, CALC, 1-Var Stats. The screen reports Sx, the sample standard deviation that divides by n - 1, and sigma-x, the population version that divides by n. Sample data uses Sx.

AP Statistics: Unit 1 (topics 1.7 Summary Statistics for One Quantitative Variable). Standard deviation is Unit 1 topic 1.7 in the Fall 2026 AP Statistics course, which defines the sample standard deviation with n-1 in the denominator and expects a graphing calculator with statistical capabilities on both sections of the exam.

Type the data in with STAT then EDIT

Press STAT, choose EDIT, and the list editor opens with columns named L1 through L6. Type your values down L1, pressing ENTER after each one. Any list works, but L1 is the default the statistics menus expect.

Clear the old data first, or the calculator will summarize last week's numbers without complaint. Arrow up until the name L1 at the top of the column is highlighted, press CLEAR, then press the down arrow. The column empties while the list itself stays in place.

Check the row count when you finish. If you meant to type 12 values, the last row number should read 12, and a stray extra row is the most common reason a standard deviation comes out wrong.

Run 1-Var Stats

Press STAT, arrow right once to the CALC tab, and choose 1-Var Stats. On current operating systems a setup screen appears: set List to L1, leave FreqList blank, then highlight Calculate and press ENTER. On older operating systems the command drops onto the home screen instead, where you type the list name after it and press ENTER.

The results screen is longer than the display. Only the first few lines fit at once, so hold the down arrow to scroll through the rest before you copy anything.

The lines arrive in this order: $\bar{x}$ (x-bar, the sample mean), $\Sigma x$ (the sum of the values), $\Sigma x^2$ (the sum of the squared values), $\text{Sx}$ (the sample standard deviation), $\sigma_x$ (sigma-x, the population standard deviation), $n$ (the sample size), and then minX, Q1, Med, Q3, and maxX.

Sx and sigma-x are two different numbers

The calculator prints both standard deviations one line apart and never says which one you wanted. That is what makes the wrong choice a silent error: both values are positive and both are in the right units, so a wrong pick looks exactly as reasonable as a right one.

The difference is the denominator. $\text{Sx}$ is the sample standard deviation, and it divides the sum of squared deviations by $n - 1$ :

\text{Sx} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}

$\sigma_x$ is the population standard deviation, and it divides by $n$ , measuring deviations from the population mean $\mu$ (mu):

\sigma_x = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}

Because $n - 1$ is smaller than $n$ , $\text{Sx}$ comes out larger than $\sigma_x$ for the same list whenever the values are not all identical. The one exception is a list in which every value is the same: the sum of squared deviations is 0, so both come out 0. The two are related by $\text{Sx} = \sigma_x \sqrt{\frac{n}{n - 1}}$ , so at $n = 5$ they differ by about 12 percent, while at $n = 500$ they differ by about 0.1 percent. Small data sets are where the wrong line does real damage.

Which line you should report

Use $\text{Sx}$ for sample data, which covers nearly everything in an AP Statistics course. A random sample of 40 households and the 25 athletes enrolled in a study are both samples, so both use $\text{Sx}$ .

Use $\sigma_x$ only when your list holds every member of the group you are describing. The heights of all nine players on a team, when that team is the entire population of interest, is a genuine case for it.

The AP course defines the sample standard deviation $s$ with $n - 1$ in the denominator and calls its square the sample variance. Every inference formula for means then runs on that $s$ . Type $\sigma_x$ into a t-test by mistake and the standard error comes out too small, so the test statistic is too large and the p-value too small, which can push you into rejecting a null hypothesis you had no evidence against.

What else 1-Var Stats hands you

The same screen answers several other questions, which saves entering the data a second time.

$n$ is the sample size. Compare it against the number of values you meant to type.
$\bar{x}$ is the sample mean, the balance point of the data.
$\Sigma x$ and $\Sigma x^2$ are the raw sums, handy when a formula asks for them directly.
minX, Q1, Med, Q3, and maxX are the five-number summary, everything a boxplot needs, from which you also get the IQR as $Q_3 - Q_1$ . The screen does not print the IQR itself, so subtract the two quartiles yourself.

One warning about the quartiles. The TI-84 splits the data at the median and excludes the median itself when the sample size is odd, and this site follows that same median-excluded rule. Some textbooks include the median in both halves and report different quartiles, so a mismatch with an answer key is usually this rule rather than a mistake. The guide on how to make a boxplot works through the five-number summary in detail.

Frequency-weighted standard deviation with a second list

Data often arrives as a frequency table instead of a list of individual values. Put the distinct values in L1 and their counts in L2, then run 1-Var Stats with List: L1 and FreqList: L2. Type the list name L2 by pressing 2nd then the 2 key.

The result is exactly what you would get by typing every value out the long way. A value of 4 with a frequency of 7 contributes seven 4s to the mean, to $n$ , and to the sum of squared deviations, so $n$ on the output is the total count and not the number of rows you typed.

Keep the frequencies as whole counts, and make sure both lists have the same length. Frequencies entered as percentages will not give the standard deviation of the underlying data, and lists of unequal length produce a dimension mismatch error.

Check the calculator against the formula

Running the hand calculation once tells you what the machine is doing: find the mean, subtract it from each value, square the deviations, add them, divide, take the square root. The standard deviation by hand guide walks through every step on a small data set.

To check a list quickly, enter the same values in the standard deviation calculator and confirm both numbers match your screen. To watch the spread react as points move, try the descriptive statistics sandbox.

$\text{Sx}$ is also not the standard error, which describes how much a sample mean varies from sample to sample and equals $\text{Sx}/\sqrt{n}$ . The guide on standard error vs standard deviation keeps the two apart. Once you have $\bar{x}$ , $\text{Sx}$ , and $n$ , you can type them straight into a t-test on the same calculator.

How this fits the Fall 2026 AP course

Standard deviation is Unit 1 topic 1.7, Summary Statistics for One Quantitative Variable, in the AP Statistics course that takes effect in Fall 2026, with the first exam in May 2027. The course describes it as a typical deviation of the data values from their mean and gives the sample formula with $n - 1$ in the denominator.

Unit 1, Exploring One-Variable Data and Collecting Data, carries 20 to 30 percent of the multiple-choice section, the largest share of any unit. A graphing calculator with statistical capabilities is expected on both sections of the exam, and formulas and tables are provided.

$\text{Sx}$ then returns throughout Unit 4, Inference for Quantitative Data: Means. Every t-interval and t-test for a mean is built on the sample standard deviation, so reading the right line becomes a habit that pays off long past Unit 1.

Sx and sigma-x for the data set 4, 7, 8, 10, 16

Enter 4, 7, 8, 10, 16 into L1 and run 1-Var Stats. Confirm the sample standard deviation and the population standard deviation by hand, and read the five-number summary off the same screen.

Find the mean: $\bar{x} = \frac{4 + 7 + 8 + 10 + 16}{5} = \frac{45}{5} = 9$ .
Find each deviation from the mean: $4 - 9 = -5$ , $7 - 9 = -2$ , $8 - 9 = -1$ , $10 - 9 = 1$ , and $16 - 9 = 7$ . As a check they add to $-5 - 2 - 1 + 1 + 7 = 0$ .
Square the deviations and add them: $(-5)^2 + (-2)^2 + (-1)^2 + 1^2 + 7^2 = 25 + 4 + 1 + 1 + 49 = 80$ .
Divide by $n - 1 = 4$ for the sample: $\frac{80}{4} = 20$ , so $\text{Sx} = \sqrt{20} = 4.4721$ .
Divide by $n = 5$ for the population: $\frac{80}{5} = 16$ , so $\sigma_x = \sqrt{16} = 4$ .
Read the rest of the screen and confirm it: $n = 5$ , $\bar{x} = 9$ , $\Sigma x = 45$ , and $\Sigma x^2 = 16 + 49 + 64 + 100 + 256 = 485$ . The five-number summary is minX $= 4$ , Q1 $= \frac{4 + 7}{2} = 5.5$ , Med $= 8$ , Q3 $= \frac{10 + 16}{2} = 13$ , maxX $= 16$ , using the median-excluded quartile rule.

$\text{Sx} = \sqrt{20} \approx 4.4721$ and $\sigma_x = \sqrt{16} = 4$ . The only difference is dividing 80 by 4 instead of by 5, and $\text{Sx}$ is the value to report unless these five numbers are an entire population.

Standard deviation from a frequency table

Ten students take a 10-point quiz. Three score 1 point, four score 4 points, and three score 7 points. Find the sample standard deviation using two lists, then confirm it by hand.

Enter the distinct scores 1, 4, 7 in L1 and the matching frequencies 3, 4, 3 in L2. Run 1-Var Stats with List: L1 and FreqList: L2.
Read $n$ from the output: $3 + 4 + 3 = 10$ students, not the 3 rows you typed.
Find the mean, weighting each score by its frequency: $\bar{x} = \frac{(1)(3) + (4)(4) + (7)(3)}{10} = \frac{3 + 16 + 21}{10} = \frac{40}{10} = 4$ .
Weight each squared deviation by its frequency: $(1 - 4)^2(3) = 9(3) = 27$ , $(4 - 4)^2(4) = 0(4) = 0$ , and $(7 - 4)^2(3) = 9(3) = 27$ . The total is $27 + 0 + 27 = 54$ .
Divide by $n - 1 = 9$ for the sample: $\frac{54}{9} = 6$ , so $\text{Sx} = \sqrt{6} = 2.4495$ .
For comparison, dividing by $n = 10$ gives $\frac{54}{10} = 5.4$ and $\sigma_x = \sqrt{5.4} = 2.3238$ , the smaller of the two here.

$\text{Sx} = \sqrt{6} \approx 2.4495$ points with $n = 10$ , and $\sigma_x = \sqrt{5.4} \approx 2.3238$ points. Report $\text{Sx}$ , since these 10 students are a sample of the students who could take the quiz.

Frequently asked questions

Does the AP exam want Sx or sigma-x?

Almost always $\text{Sx}$ . AP Statistics problems describe samples, and the course defines the sample standard deviation with $n - 1$ in the denominator. Use $\sigma_x$ only when the data set is the complete population you are describing.

Why is Sx bigger than sigma-x?

Both start from the same sum of squared deviations, but $\text{Sx}$ divides it by $n - 1$ while $\sigma_x$ divides by $n$ . Dividing by a smaller number gives a larger result. For the data set 4, 7, 8, 10, 16 the sum is 80, so $\text{Sx} = \sqrt{80/4} = 4.4721$ and $\sigma_x = \sqrt{80/5} = 4$ . This holds whenever the values are not all identical; if every value in the list is the same, the sum of squared deviations is 0 and both standard deviations are 0.

How do I clear old data out of a list?

Press STAT, choose EDIT, then arrow up until the list name at the top of the column is highlighted. Press CLEAR, then press the down arrow. The values disappear and the empty list stays available.

My Q1 does not match the textbook answer. Did I do something wrong?

Probably not. The TI-84 excludes the median from both halves when the sample size is odd, and this site uses that same rule. A textbook that includes the median in both halves reports different quartiles for the same data, which also changes the IQR.

Can 1-Var Stats give the standard deviation of a probability distribution?

Some teachers show the shortcut of putting the values in L1 and their probabilities in the frequency list. Read $\sigma_x$ in that case, since a probability distribution describes a population. Support for non-whole-number frequencies varies by operating system, so computing $\mu = \sum x_i p_i$ and $\sigma = \sqrt{\sum (x_i - \mu)^2 p_i}$ directly is more reliable.