Sample Standard Deviation Calculator
Paste your sample to get the sample standard deviation s, using n − 1, along with the sample variance, mean and standard error.
- n − 1 formula
- Standard error
- Step-by-step
- Paste from Excel
Your data
8 values.
Results
Sample standard deviation (s)
19.3962
Sample variance (s²)
376.2143
Sample mean (x̄)
267.25
Standard error (SE)
6.8576
Center
- Count (n)
- 8
- Sum (Σx)
- 2,138
- Mean
- 267.25
- Median
- 265.5
- Mode
- No mode
Spread
- Sample standard deviation (s)
- 19.3962
- Population standard deviation (σ)
- 18.1435
- Sample variance (s²)
- 376.2143
- Population variance (σ²)
- 329.1875
- Sum of squares (SS)
- 2,633.5
- Mean absolute deviation
- 15
- Standard error of the mean
- 6.8576
- Coefficient of variation (s ÷ mean)
- 7.26%
- Minimum
- 240
- Maximum
- 299
- Range
- 59
Quartiles and outliers
- Q1
- 253
- Q3
- 281
- Interquartile range (IQR)
- 28
- Five number summary
- 240, 253, 265.5, 281, 299
- Outlier fences
- 211 and 323
- Outliers (1.5 × IQR)
- None
Empirical rule (using s)
- Within 1 SD (68% if normal)
- 5 of 8 (62.5%)
- Within 2 SD (95% if normal)
- 8 of 8 (100%)
- Within 3 SD (99.7% if normal)
- 8 of 8 (100%)
Step-by-step working
Count the values
n = 8
Find the mean
x̄ = Σx ÷ n
Σx = 251 + 287 + 263 + 240 + 299 + 275 + 268 + 255 = 2,138
x̄ = 2,138 ÷ 8 = 267.25
Subtract the mean from each value and square the result
(x − x̄)²
(251 − 267.25)² = (−16.25)² = 264.0625
(287 − 267.25)² = (19.75)² = 390.0625
(263 − 267.25)² = (−4.25)² = 18.0625
…and the same for every value. The table lists them all.
Add up the squared deviations (sum of squares)
SS = Σ(x − x̄)²
SS = 264.0625 + 390.0625 + 18.0625 + 742.5625 + 1,008.0625 + 60.0625 + 0.5625 + 150.0625 = 2,633.5
Divide by n − 1 to get the sample variance
s² = SS ÷ (n − 1)
s² = 2,633.5 ÷ 7 = 376.2143
Take the square root to get the standard deviation
s = √s²
s = √376.2143 = 19.3962
| # | x | x − x̄ | (x − x̄)² | z |
|---|---|---|---|---|
| 1 | 251 | −16.25 | 264.0625 | −0.8378 |
| 2 | 287 | 19.75 | 390.0625 | 1.0182 |
| 3 | 263 | −4.25 | 18.0625 | −0.2191 |
| 4 | 240 | −27.25 | 742.5625 | −1.4049 |
| 5 | 299 | 31.75 | 1,008.0625 | 1.6369 |
| 6 | 275 | 7.75 | 60.0625 | 0.3996 |
| 7 | 268 | 0.75 | 0.5625 | 0.0387 |
| 8 | 255 | −12.25 | 150.0625 | −0.6316 |
| Σ | 2,138 | 0 | 2,633.5 |
Rows with |z| above 3 are highlighted. The Σ row of x shows Σx.
What is the sample standard deviation?
The sample standard deviation s estimates how spread out a whole population is, using only a sample from it. It works like the ordinary standard deviation, with one change: the sum of squared deviations is divided by n − 1 instead of n. This is Bessel’s correction. A sample’s values are, on average, closer to their own mean than to the unknown population mean, so dividing by n would make the spread look smaller than it really is. Use the sample standard deviation for survey responses, lab measurements, a batch of products or any data that stands in for a larger group. It is what Excel’s STDEV.S, the TI-84’s Sx and most statistics software report by default.
How to find the sample standard deviation
- Enter the sample values, separated by commas, spaces or new lines, or paste a column from a spreadsheet.
- Leave the data type on Sample so the working divides by n − 1.
- Read s, the sample variance s², the sample mean x̄ and the standard error of the mean (s ÷ √n).
- Check the step-by-step working, then copy the results or each value’s z-score.
Sample standard deviation formulas
- Sample standard deviation
s = √( Σ(x − x̄)² ÷ (n − 1) )
x̄ is the sample mean and n the sample size.
- Sample variance
s² = Σ(x − x̄)² ÷ (n − 1)
The square of s.
- Sample mean
x̄ = Σx ÷ n
The centre the deviations are measured from.
- Standard error of the mean
SE = s ÷ √n
How precisely the sample mean estimates the population mean.
Sample or population: which one should you use?
Why use this sample standard deviation calculator
- Working shown with your numbersEvery answer comes with the formula and each line of arithmetic filled in with your data, plus a table of deviations, so you can check homework or copy the method into an exam answer.
- Standard error includedThe standard error of the mean (s ÷ √n) and the coefficient of variation are shown next to s, ready for a lab report or a confidence interval.
- Paste straight from a spreadsheetCopy a column from Excel, Google Sheets or Numbers and paste it. Commas, spaces, tabs and new lines all work, headers like “Score” are skipped, and multi-column pastes let you pick one column.
- Frequency tables and grouped dataType each value once with how often it appears, or enter class intervals such as 10-19 and the calculator uses their midpoints. No need to repeat a value 30 times.
- See the spread, not just the numberA histogram with the mean and a one standard deviation band, plus a box plot, shows how the data is spread and where any outliers sit.
- Private, free and instantCalculations run in your browser as you type. Nothing is sent to a server, there is no sign-up, no limit on how many numbers you enter and no “show steps” paywall.
When you need the sample standard deviation
- Surveys and research. Describe survey responses or experimental groups before running a t-test or building a confidence interval.
- Science labs. Summarise repeated measurements, report mean ± standard deviation, and find the standard error for error bars.
- Statistics homework. Check answers for AP Statistics, intro stats and lab reports, and copy the step-by-step working.
- Quality control. Track how much part sizes, fill weights or process times vary, and spot values outside the usual range.
- Clinical and health data. Describe blood pressure, weight or test results from a group of patients as mean ± SD.
- A/B tests and experiments. Measure the spread of each group before comparing means.
The same statistics in Excel, Google Sheets and a TI-84
To check an answer or redo it in a spreadsheet, use these functions (they work the same in Google Sheets). On a TI-83 or TI-84, enter the data in L1, press STAT, CALC, 1-Var Stats; for a frequency table put the counts in L2 and set FreqList to L2.
| Statistic | Excel and Google Sheets | TI-84 (1-Var Stats) |
|---|---|---|
| Sample standard deviation (s) | =STDEV.S(A2:A20) | Sx |
| Population standard deviation (σ) | =STDEV.P(A2:A20) | σx |
| Sample variance (s²) | =VAR.S(A2:A20) | Sx² |
| Population variance (σ²) | =VAR.P(A2:A20) | σx² |
| Mean | =AVERAGE(A2:A20) | x̄ |
| Median | =MEDIAN(A2:A20) | Med |
| Mode | =MODE.MULT(A2:A20) | Not shown |
| Q1 and Q3 | =QUARTILE.INC(A2:A20, 1) and 3, or QUARTILE.EXC | Q1, Q3 (textbook method) |
| Sum of squares (SS) | =DEVSQ(A2:A20) | Not shown (Σx² is) |
| Mean absolute deviation | =AVEDEV(A2:A20) | Not shown |
| Standard error of the mean | =STDEV.S(A2:A20)/SQRT(COUNT(A2:A20)) | Sx ÷ √n |
| Z-score of a value | =STANDARDIZE(x, mean, sd) | (x − x̄) ÷ Sx |
| Percentile from a z-score | =NORM.S.DIST(z, TRUE) | normalcdf(−1E99, z) |
What this calculator does that most others leave out
We compared the top standard deviation, variance, z-score and quartile calculators in October 2026. Most return one number for one mode, show the formula without your numbers, or cap the number of values. This one was built to cover the gaps.
- Both versions at once. Most calculators make you pick sample or population first. Here s, σ, s² and σ² are always on screen.
- Working with your own numbers. Not just the formula: the mean, every squared deviation, the sum of squares and the final square root are written out with your data.
- Frequency and grouped data. Enter value and count pairs or class intervals instead of typing repeated numbers.
- Quartiles that match your class. Choose the textbook / TI-84 method or Excel’s QUARTILE.INC or QUARTILE.EXC.
- Z-scores and outliers. Every value gets a z-score, and values beyond 1.5 × IQR or 3 standard deviations are flagged.
- No limits, fully private. Paste thousands of values or open a CSV file. Everything is processed on your device and never sent anywhere.
More math and statistics tools
Only need the mean, median and mode, or a weighted average for grades? Use the average calculator. For percentages and percent change, try the percentage calculator, and to type formulas like σ = √(Σ(x − μ)² ÷ N) for a report, use the online math editor. Browse all student tools and calculators.
Worked example: reaction times from 8 people
Data (milliseconds): 251, 287, 263, 240, 299, 275, 268, 255. 1. Sample mean: x̄ = 2138 ÷ 8 = 267.25. 2. Sum of squared deviations: SS = 2633.5. 3. Sample variance: s² = 2633.5 ÷ (8 − 1) = 376.2143. 4. Sample standard deviation: s = √376.2143 = 19.3962 ms. 5. Standard error of the mean: 19.3962 ÷ √8 = 6.8576 ms. Dividing by 8 instead would give 18.1435, the population value, which understates the spread of all people’s reaction times.
What to expect from the result
You need at least two values; with one value n − 1 is zero and s is undefined. The standard error assumes the values are independent. For very small samples, s itself is a rough estimate, so report the sample size with it.
Sample Standard Deviation Calculator questions
Why does the sample formula divide by n − 1?
A sample’s values sit closer to their own mean than to the true population mean, so dividing the sum of squares by n would underestimate the spread. Dividing by n − 1 (Bessel’s correction) fixes that bias. The difference is large for small samples and tiny for big ones.
What symbol is used for sample standard deviation?
A lowercase s, and s² for the sample variance. The sample mean is written x̄ (x-bar). Population values use Greek letters: σ for standard deviation and μ for the mean. On a TI-84 the sample SD is labelled Sx and the population SD σx.
What is the standard error of the mean?
It is s ÷ √n, the expected spread of the sample mean if you repeated the sampling many times. It shrinks as the sample gets bigger and is used for confidence intervals and error bars.
Is sample standard deviation always bigger than population standard deviation?
For the same data, yes, slightly, because it divides by a smaller number. The ratio is √(n ÷ (n − 1)), so the gap is about 5% with 10 values and under 1% with 60.
Which Excel function gives the sample standard deviation?
=STDEV.S(range), or the older =STDEV(range). Both divide by n − 1. =STDEV.P gives the population version.