Standard Deviation Calculator
Get sample or population SD with a row-by-row deviation table, the definition formula and the computational Σx² − n·x̄² formula. Copy the APA result or switch to population variance without re-pasting the data.
- No upload
- Worked steps
- Free, no sign-up
100% Local in Browser — Your Data Never Leaves This Tab
Comma, space, tab or newline separated. Paste an Excel column directly; blank lines and labels are ignored.
Try a sample
Paste a dataset to begin
The worked steps, table, box plot and summary update as you type. Nothing is sent anywhere.
How it works
- 1
Paste and sort
Enter a column of observations. The page sorts them, computes the mean, and builds a three-column table: the value, its deviation from the mean, and the squared deviation. The middle column sums to zero, which is a useful sanity check.
- 2
Pick the denominator
Sample mode divides the sum of squares by n − 1 (Bessel’s correction); population mode divides by N. The toggle changes the variance, the SD and the APA copy, while the table and the raw sum of squares stay the same.
- 3
Check both formulas
Method 1 substitutes into s = √(Σ(xᵢ − x̄)² / (n − 1)). Method 2 uses s = √((Σxᵢ² − n·x̄²) / (n − 1)). Both must land on the same number; seeing them agree is the point of showing both.
More tools
- Median CalculatorPaste a dataset and get the median with the sorted list, the exact middle position or pair, and a box plot. Odd and even counts are handled separately, and the numbers never leave your browser.
- Mean CalculatorAdd a list of numbers and see the mean with the sum, the count and every substitution in Σx / n. The dataset stays in your browser and carries over to the median, SD, IQR and z-score pages.
- Average CalculatorFind the average of any list with the sum, count and substitution shown line by line. Also called an average finder; the page reports the arithmetic mean and compares it with the median so a skew does not fool you.
- Interquartile Range (IQR) CalculatorCompare Q1, Q2, Q3 and IQR under the TI-83/84/AP exclusive rule, Tukey’s inclusive hinges and Excel QUARTILE.INC interpolation. See the 1.5 × IQR fences, potential outliers and a live box plot.
- Z-Score CalculatorTurn x, μ and σ into a z-score and a left-tail, right-tail or two-tailed probability, or standardize every value in your dataset at once. The standard normal curve fills the same area the table reports.
What it won't do
- Sample vs population is not a cosmetic toggle. Use n − 1 when the data is a sample from a larger population; use N when the list is the complete population. The difference matters most for small n.
- The table uses the arithmetic mean. A frequency table must be expanded into repeated observations first; this page does not silently weight values.
- The normal approximation in the z-score page is separate from the SD calculation here. SD describes spread; a z-score standardizes one value using a chosen μ and σ.
- All calculations are local. A very large paste can be slow in the browser, but no row is sent to a server and no account is required.
Questions
Should I use n or n − 1 for standard deviation?
Use n − 1 for a sample and N for a population. Dividing by n − 1 is Bessel’s correction: it makes the sample variance an unbiased estimate of the population variance. The page labels the denominator in every formula so the choice is visible.
What is the difference between the two formulas?
The definition formula squares each deviation from the mean: Σ(xᵢ − x̄)². The computational formula uses Σxᵢ² − n·x̄², which is easier to evaluate by hand. They are algebraically equivalent; the page calculates both and shows the same variance.
Why does my TI-84 give a slightly different answer?
The usual cause is the sample/population setting: TI-84 Sx uses n − 1 and σx uses N. Rounding the mean before squaring can also move the last digit. Keep full precision until the final report.
What does the deviation table show?
xᵢ is each sorted observation, xᵢ − x̄ is its signed distance from the mean, and (xᵢ − x̄)² is the squared distance used in the sum. The bottom row is Σ(xᵢ − x̄)², the numerator of the variance.
How do I copy the APA result?
Press “Copy APA Format (M = …, SD = …)”. It copies the mean and the sample SD, because APA-style reports normally describe a sample. If you are reporting a population, use the population SD shown on the page and adapt the notation.
Is standard deviation the same as variance?
No. Variance is the average squared deviation; standard deviation is its square root. SD is in the same units as the data, which is why it is easier to interpret, while variance is the quantity that adds nicely in later formulas.