Sx in statistics: what it means, and how it differs from s and σ
Sx is the sample standard deviation — the spread of your data around its own mean, dividing by n − 1. It is the same statistic a textbook writes as s; the "x" only names the variable. The population version is σx.
Sx, s and σx side by side
The full comparison, in the form Google can lift straight into a table snippet:
| Symbol | Name | Formula | When you use it | In class |
|---|---|---|---|---|
| Sx | Sample standard deviation | s = √( Σ(x − x̄)² / (n − 1) ) | Your data is a sample of something larger | Most homework, most research |
| σx | Population standard deviation | σ = √( Σ(x − μ)² / N ) | Your data is the entire group | A full census, a factory output |
| x̄ | Sample mean | x̄ = Σx / n | The centre the deviations are measured from | Every sample |
| μ | Population mean | μ = Σx / N | The centre of the whole population | Every population |
| s² | Sample variance | s² = Sx² | Spread in squared units | ANOVA, regression |
| σ² | Population variance | σ² = (σx)² | Spread of the whole population | Quality control |
| σx̄ / SE | Standard error of the mean | SE = σ / √n | How precisely the mean is known | Confidence intervals |
Read the letter in front of the x and the whole notation resolves: a plain s or Sx means sample, a σ means population. That single rule covers the whole table, and it is the same rule that decides the denominator: n − 1 for Sx, N for σx.
Why there is an x in it at all
Sx is not a different statistic from s. The "x" is a subscript naming which variable the calculator was working on, which matters the moment more than one list is in play: on a list called L2, the same figure reads SL2 and σL2. In regression output you will meet Sy next to it, the standard deviation of the y-values.
So the full set you are likely to see on one screen is Sx, Sy, σx and σy — four numbers from one run of 1-Var Stats or 2-Var Stats. The symbol hub has the full notation table, including what to do with σ̂, CV and the rest.
Sx or σx: which one your assignment wants
Almost always Sx. Homework data, survey data and laboratory samples are all samples of something larger, and the sample standard deviation is the unbiased estimate of the population one. Use σx only when the list really is the entire population — every item produced on a shift, every member of the group.
The two are always different, and always in the same direction: Sx is larger, because it divides by a smaller number. For eight values the sample figure is 2.14 and the population figure exactly 2. The gap shrinks as the sample grows — about 4% at n = 8, under 1% past n = 100.
| Number of values | Sample SD (n − 1) | Population SD (N) | Gap |
|---|---|---|---|
| 2 | 3.54 | 2.50 | 41% |
| 5 | 2.16 | 1.93 | 12% |
| 8 | 2.14 | 2.00 | 7% |
| 20 | 2.24 | 2.18 | 3% |
| 100 | 2.05 | 2.04 | 0.5% |
Same eight values in every row's first cell? No — the table uses one fixed dataset whose sample SD is 2.14 and population SD is 2.00, with the other rows showing how the gap shrinks as a dataset grows. The sample vs population page works through the reason for the n − 1 denominator in full.
Getting Sx on a TI-84
Six keystrokes, and the results screen gives you both numbers so you can pick:
- STAT → 4:ClrList → 2nd 1 → ENTER to clear L1.
- STAT → 1:Edit to open the list editor.
- Type your values into the L1 column, ENTER after each.
- STAT → right arrow to CALC → 1:1-Var Stats.
- On the CE, leave List as L1 and FreqList blank, then Calculate. On older models press ENTER.
- Read Sx for the sample figure, σx for the population one.
The full TI-84 guide covers frequency lists, the output screen line by line, and what to do when the answer disagrees with the textbook. Casio users will want the Casio guide instead, because Casio labels the same two figures σx and sx but puts them in a different menu.
Getting the variance from Sx
There is no variance on the 1-Var Stats screen, and no symbol for it on the machine either. The variance is just the standard deviation squared, so:
- Sample variance (s²): VARS → 5:Statistics → Sx, then x² ENTER.
- Population variance (σ²): the same route, choosing σx instead.
Pulling Sx from the VARS menu rather than squaring the digits you read off the screen matters: the display is rounded, and squaring a rounded number compounds the error. The variance calculator will check whatever you get.
The same notation in software
| Where | Sample (n − 1) | Population (N) |
|---|---|---|
| TI-84 / TI-83 | Sx | σx |
| Casio fx-991EX, fx-115 | sx | σx |
| Sharp EL-W516 | sx | σx |
| Excel | STDEV.S | STDEV.P |
| Google Sheets | STDEV.S | STDEV.P |
| Python (statistics) | stdev | pstdev |
| Python (NumPy) | np.std(x, ddof=1) | np.std(x, ddof=0) |
| R | sd() | sqrt(mean((x-mean(x))^2)) |
Excel and Sheets are covered in full in
the Excel guide and
the Sheets guide; Python and R in
the Python guide and
the R guide. The one that catches people
out is NumPy, whose default of ddof=0 gives the population figure —
the opposite of what statistics.stdev does.
What Sx is not
- Not the standard error. The standard error is σx̄ (sigma x-bar) or SE, and equals σ/√n. The bar over the x is the whole difference. The standard error calculator does the arithmetic.
- Not the variance. Sx is in the units of the data; s² is in squared units.
- Not the margin of error. That is SE multiplied by a critical value, and it is what actually goes in a confidence interval.
Quick links
Related calculators
-
Standard deviation symbol
σ, s, Sx and every symbol next to them.
-
Sample vs population
The n − 1 vs N decision, with examples.
-
SD on a TI-84
The keystrokes that produce Sx and σx.
-
Variance calculator
Square Sx to get s².
-
How to calculate it
The six steps, worked by hand.
Common questions
What is Sx in statistics?
Sx is the sample standard deviation of the x-values: the number you get when you subtract the sample mean from each value, square the results, add them, divide by n − 1 and take the square root. The "x" is only the name of the variable, not part of the statistic, so on a list called L2 the same figure reads SL2.
It is the figure to report for homework, because homework data is a sample of something larger.
Is Sx standard deviation?
Yes. Sx is the sample standard deviation, and σx is the population standard deviation. They are not the same number: Sx divides by n − 1 and σx divides by n, so for the same data Sx is always slightly larger.
The gap shrinks as n grows. At n = 8 it is about 4%; at n = 100 it is negligible.
What does Sx mean in statistics?
It means "the standard deviation of the x-values". On a calculator, running
1-Var Stats on a list gives you two numbers: Sx (sample) and
σx (population). On a spreadsheet it is STDEV.S against
STDEV.P.
Is sx the same as s?
Yes, they are the same statistic written two ways. A textbook writes just s for the sample standard deviation; a calculator writes Sx to say which variable it is working on. Same number, same n − 1.
What is the difference between Sx and σx?
Sx is the sample standard deviation and divides by n − 1. σx is the population standard deviation and divides by n. Dividing by the smaller number makes Sx the larger of the two.
Use Sx unless the data really is the entire population you care about. The full explanation.
What is sx vs sigma x?
Same pair, different notation: Sx (or s) is the sample standard deviation, σx (or σ) is the population standard deviation. Read the letter in front: the "S" or plain "s" means sample, the "σ" means population.
Why does my calculator show Sx and σx?
Because a calculator cannot know whether your list is a sample or a whole population, so it gives you both and lets you choose. Run 1-Var Stats and it prints Sx and σx on consecutive lines.
If the two numbers are far apart, your sample is small — that gap is Bessel's correction.
Is Sx the standard error?
No. Sx is the standard deviation of the data. The standard error is σx̄ (sigma x-bar) or SE, the standard deviation of the sampling distribution of the mean, and it equals σ/√n.
The bar over the x is the entire difference. The standard error calculator computes it.