Standard Deviation Calculator - Sample and Population σ

A standard deviation calculator measures how far a list of numbers typically sits from its mean. The statistic is not “the average difference.” It is the square root of an average of squared differences, which penalizes outliers more than a plain mean absolute deviation and keeps the result in the same units as the data. If exam scores are points, σ is points. If masses are grams, s is grams.

The fork that trips homework is sample versus population. Population standard deviation σ uses every member of the set you actually care about and divides the sum of squared deviations by n. Sample standard deviation s treats the list as a draw from a larger unseen set and divides by n − 1 (Bessel's correction) so the sample variance is unbiased for the population variance under ordinary sampling assumptions. This page makes you pick the fork before you copy a number, because silently defaulting to n − 1 when a problem said “population” is how a correct Σ(x − mean)² still loses the mark.

Computation stays in the browser. Paste commas, spaces, or a column of new lines. The default list 2, 4, 4, 4, 5, 5, 7, 9 is the well-known eight-point example whose mean is 5, whose sum of squared deviations is 32, whose population SD is exactly 2, and whose sample SD is √(32/7) ≈ 2.138. Use it as a checksum: if another site disagrees on this list, one of the two tools is using the other denominator.

Using this sample and population SD tool is a short loop:

1. Paste numbers separated by commas, spaces, semicolons, or new lines. A spreadsheet column usually pastes as one value per line.
2. Toggle Sample (n−1, s) or Population (n, σ). Sample needs at least two values. Population accepts a single value (variance and SD are then 0).
3. Read count, mean, variance, standard deviation, min, max, and range. Σ(x − mean)² is shown so you can check the numerator by hand.
4. Copy the summary or Clear the box. The last list and toggle are stored in this browser for up to 30 days, not on a server.

Limits: at most 10,000 numeric tokens. Scientific notation such as 1e-3 is accepted. Grouping commas like 1,234 split into 1 and 234 because a comma is a delimiter. Letters, units glued to a number (5cm), and empty lists are rejected. This is a descriptive calculator: it does not compute a confidence interval, a z-score of a new observation, or a p-value.

For the default eight numbers the sum is 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40, so the mean μ = 40 / 8 = 5. Deviations from 5 are −3, −1, −1, −1, 0, 0, 2, 4. Squares are 9, 1, 1, 1, 0, 0, 4, 16. Those squares sum to 32. Population variance is 32 / 8 = 4. Population standard deviation is √4 = 2. If this calculator shows anything else on that list in population mode, something is wrong - not a rounding disagreement.

Why square at all? Absolute deviations −3 and +4 would partially cancel if you added them raw; squaring makes every miss positive and weights a miss of 4 four times a miss of 2 in the variance (16 versus 4), not twice. Taking the square root after averaging returns to the original unit so you can say “typical distance from the mean is 2,” not “typical squared distance is 4.”

Population mode is the right button when the list is the whole set: every tree in a 12-tree orchard, every invoice issued this Tuesday, every student in a closed class of eight. It is also the right button when a worksheet explicitly says population or σ. Pair the mean with the average calculator if you want median and mode as well; this page is built around variance and SD, not a full descriptive dump.

Same eight numbers, same Σ(x − mean)² = 32, but now the denominator is 7. Sample variance = 32 / 7 ≈ 4.571429. Sample standard deviation s = √(32/7) ≈ 2.138090. That is not a different data set. It is a different claim: you are estimating the spread of a larger population from eight observations, and the sample mean already used those eight numbers, which systematically shrinks the apparent scatter if you still divide by n.

Bessel's correction is the name for dividing by n − 1 rather than n when computing sample variance. Degrees of freedom language: one degree of freedom was spent estimating the mean, so n − 1 remain for spread. If n = 1, n − 1 = 0 and sample variance is undefined; this calculator refuses that case instead of printing Infinity or silently switching to population. A single number has no sample spread around a mean that was forced to equal that number.

Unbiasedness applies to the variance s², not automatically to s itself. The square root is a nonlinear function, so E[s] is still a slightly biased estimator of σ in small samples. Intro stats courses still report s as “the” sample standard deviation, and so does this tool. If you need a bias-corrected σ̂ with extra constants, you are past what a list-paste widget should silently apply.

Variance is the mean of squared deviations (with n or n − 1 in the denominator). Standard deviation is its square root. They rank data sets the same way: larger variance always means larger SD because square root is increasing on [0, ∞). What changes is interpretability. Variance of 4 (points²) is awkward in a parent newsletter. SD of 2 points is a sentence.

Algebra likes variance. Variances of independent random variables add; standard deviations do not. If you later build a standard error of a mean, you will take s / √n, which starts from sample SD, not from a hand-wavy “average difference.” If you later want a p-value for a mean, you still need a test statistic and a reference distribution - the p-value calculator - not only s. This page stops at the descriptive layer: count, mean, variance, SD, min, max, range.

Range is max − min. For the default list, min 2, max 9, range 7. Range uses two points and ignores everything in the middle, so it is a poor substitute for SD, but it is a superb typo detector (a 90 that should have been 9). Min and max are printed so you can see which tail moved.

Use sample (s, n − 1) when the numbers are a sample and the goal is to estimate the spread of a larger population: eight scores drawn from a national exam, 40 reaction times from a lab that will run hundreds more, a random subset of invoices. Inferential formulas you meet next (t intervals, many software defaults) assume s, not σ.

Use population (σ, n) when the list is the entire population of interest, or when the problem statement says population. Every student in a class of eight is a population for “this class.” Those same eight scores are a sample for “all students who took the exam nationwide.” The arithmetic does not know your research question; the toggle is the research question.

Do not mix denominators mid-writeup. Quoting σ = 2 in one sentence and s ≈ 2.138 in the next for the same eight numbers without naming the formula looks like two data sets. If a rubric says “show the formula,” copy Σ(x − mean)² from the result panel and write the denominator you toggled. If you need a percent-style error against a known reference value rather than spread around a mean, that is a different statistic - the percent error calculator, not SD.

1. Count n. Confirm no extra blank that became a zero.
2. Sum the values. Mean = sum / n.
3. For each x, compute (x − mean), then square it. Add those squares. That total is Σ(x − mean)².
4. Population: divide by n. Sample: divide by n − 1. That quotient is the variance.
5. Square root of the variance is the standard deviation.

A common slip is averaging the deviations before squaring (they can cancel toward zero) or taking the square root of Σ(x − mean)² without dividing. Another is using n − 1 on a population problem because a graphing calculator defaulted to sample. Walk the default list until 2 and ≈2.138 feel inevitable, then paste your real data.

If you need to draw a comparison group or a simulated sample of the same size, the random number generator can produce integers or decimals in a range. That is sampling practice, not a substitute for measuring the spread of the list you already have.

It does not compute a weighted standard deviation, a grouped (frequency table) SD, interquartile range, or median absolute deviation. It does not drop outliers unless you delete them from the paste. It does not implement Bessel correction variants other than n versus n − 1. It does not replace a statistics package for huge data files: the cap is 10,000 points, and IEEE-754 doubles will lose integers past about 15–16 significant digits.

Parsing is delimiter-based. A US thousands separator is a split, not grouping. Locale decimals that use a comma as the decimal mark will split as well. Convert to a plain period decimal before pasting. The tool will not infer units or convert centimeters to meters; inconsistent units make a numerically valid SD that is scientifically meaningless.

Excel and Google Sheets split the same fork this page names in words. STDEV.S (or older STDEV) is sample, denominator n − 1. STDEV.P is population, denominator n. VAR.S and VAR.P are the matching variances. If a sheet shows 2.138 on 2, 4, 4, 4, 5, 5, 7, 9 you are in sample mode; if it shows 2.000 you are in population mode. Pasting the same eight cells into this textarea should match the function you actually called, not the function you assumed.

Python's statistics module defaults to sample (stdev / variance) and offers pstdev / pvariance for population. NumPy's np.std historically defaulted to population (ddof=0) which surprises people who just used Excel. The lesson is not “memorize every library,” it is “name the denominator.” This calculator puts the name in the toggle so a copied result is labeled Sample (n−1, s) or Population (n, σ) instead of a bare 2.14 that could be either.

A two-pass algorithm (mean first, then sum of squares) is what this page uses. For ordinary homework lists it is exact enough. Pathological lists with huge offsets (values near 10¹⁶ plus a tiny scatter) can lose the scatter in double precision; that is an IEEE-754 limit, not a different SD formula. If your data live at that scale, center them or use a statistics package with compensated summation. Ten thousand points is the paste cap so a browser tab stays responsive; it is not a claim that 10,001 observations are statistically forbidden.