Standard Deviation Calculator

Divide by n−1 or by N, and your answer changes

Every other statistic on this page is unambiguous. The standard deviation is not: there are two of them, they differ by a single character in the denominator, and nothing in your data tells you which one is correct. That decision is yours, it is the first thing this calculator asks you for, and it is the most common reason two people analysing the same numbers report different spreads.

What each symbol means

• xᵢ — each individual value in the dataset.
• μ (mu) — the population mean; x̄ (x-bar) — the sample mean.
• N — the number of values in a population; n — the number in a sample.
• Σ (sigma) — “sum of”, add up the terms that follow.
• (xᵢ − mean)² — the squared deviation of one value from the mean.

Watch the gap open on a real dataset. Take 10, 12, 8, 15, 11. The mean is (10 + 12 + 8 + 15 + 11) ÷ 5 = 11.2. The squared deviations are (10 − 11.2)² = 1.44, (12 − 11.2)² = 0.64, (8 − 11.2)² = 10.24, (15 − 11.2)² = 14.44 and (11 − 11.2)² = 0.04, which add to 26.80. Everything up to this point is shared. Only the last step forks: 26.80 ÷ 5 = 5.36 gives a population SD of √5.36 = 2.3152, while 26.80 ÷ 4 = 6.70 gives a sample SD of √6.70 = 2.5884. Same five numbers, and the reported spread differs by (2.5884 − 2.3152) ÷ 2.3152 = 11.8% — yet both figures are correct answers to different questions.

The sample figure is always the larger of the two, and deliberately so. This is Bessel’s correction. The sample mean is not handed to you from outside — it is computed from the very values whose deviations you are about to measure, so those values sit closer to their own mean than they do to the unknown population mean. Dividing by n would therefore understate the true spread every single time. NIST’s handbook frames the same point as a degree of freedom: one is spent estimating the mean from the sample, leaving N − 1 to estimate the scale, which is exactly the divisor the sample formula uses.

The correction shrinks as data accumulates, because the ratio n ÷ (n − 1) approaches 1. With five values the divisor changes by a fifth and the answer moves visibly; with five hundred it changes by a fifth of a percent and the two formulas are practically indistinguishable. That is why the sample-versus-population argument is fierce on small classroom datasets and irrelevant on large ones. If your five numbers are the only five that exist — five machines on a line, five sitting members of a committee — take the population figure. If they stand in for something larger you cannot measure in full, take the sample figure. The calculator will show you both, but it does not make the choice for you, because the choice is a claim about where your data came from and only you know that.

Standard error is not a smaller standard deviation

Standard deviation and standard error are printed a few centimetres apart, share a symbol, and measure completely different things. The standard deviation describes the data: how far a typical value sits from the mean. The standard error describes an estimate: how far the sample mean is likely to sit from the population mean you were actually trying to learn about. One is a fact about the numbers in front of you; the other is a statement about your uncertainty.

The arithmetic makes the distinction concrete: SE = s ÷ √n. For the same five values, 2.5884 ÷ √5 = 2.5884 ÷ 2.2361 = 1.1576. The spread of the data is 2.5884 and it is not going to change if you collect more of the same kind of data — a wider population is a wider population. The precision of the mean is 1.1576 and it improves the moment you add observations, because n grows underneath the square root. Quadruple the sample and the standard error halves while the standard deviation stands still. Any figure that gets smaller as you collect more data is describing your confidence, not your data.

The confidence interval reported alongside it inherits both facts. It is built from the standard error, so it narrows with sample size, and it uses a Student-t critical value rather than a normal one, so the multiplier itself widens as the sample shrinks — a small sample is penalised twice, once through a larger standard error and once through a larger multiplier. It also assumes your values are an independent random sample of the population you are generalising to. Five readings taken from the same batch, the same hour, or the same respondent are not independent, and no width of interval repairs that; the interval will look reassuringly tight and mean nothing.

Two limitations follow directly. This tool computes descriptive statistics only — it will not run a hypothesis test, produce a p-value, fit a regression, or measure correlation, so a confidence interval here is a description of precision and not a decision procedure. And it cannot tell whether the interval is honest, because it cannot see how your data was collected.

One mistyped digit can multiply the spread sevenfold

Because deviations are squared before they are averaged, distance from the mean is punished quadratically and a single wrong value can dominate the whole statistic. Keep the same dataset, 10, 12, 8, 15, 11, and transpose the digits of one entry so that 15 is typed as 51 — the most ordinary data-entry slip there is.

The mean moves from 11.2 to 92 ÷ 5 = 18.4. The squared deviations become 70.56, 40.96, 108.16, 1062.76 and 54.76, and their sum jumps from 26.80 to 1337.20, of which one term alone supplies 1062.76 ÷ 1337.20, or about 79%. The sample variance is 1337.20 ÷ 4 = 334.30 and the sample standard deviation is √334.30 = 18.2839. That is 18.2839 ÷ 2.5884 = 7.06 times the original spread, from one keystroke.

Now look at what did not move. Sorted, the clean data reads 8, 10, 11, 12, 15 and the corrupted data reads 8, 10, 11, 12, 51. The median is 11 in both. Under the type-7 convention this page uses, Q1 is 10 and Q3 is 12 in both, so the IQR is 2 in both. Rank-based measures barely notice the error that multiplied the standard deviation sevenfold. That asymmetry is the whole argument for reporting the median and the IQR alongside the mean and the SD when data is skewed, heavy-tailed, or of uncertain provenance.

It is also why the two outlier methods on this page disagree so instructively. The IQR rule places the upper fence at Q3 + 1.5 × IQR = 12 + 3 = 15, so 51 is far beyond it and gets flagged immediately; in the clean dataset the largest value is exactly 15, sitting on the fence rather than past it, and nothing is flagged. The z-score rule fares much worse: the z-score of 51 is (51 − 18.4) ÷ 18.2839 = 1.78, comfortably inside the default threshold of 3, so the z method waves it through. The outlier inflated the very standard deviation the z-score divides by and thereby hid itself. On small datasets the IQR fences are the more trustworthy of the two, which is why the calculator offers both and defaults to running them together.

A flag is a prompt to look, never a licence to delete — the NIST handbook is explicit that outliers must be reviewed rather than removed, because the same extreme value may be a typo, a rare but genuine observation, or the most important thing in the dataset. This calculator will not decide which. It also does not judge whether the standard deviation is the right measure of spread for your data at all; on skewed or heavy-tailed distributions the median, IQR, or MAD often describe it far more faithfully, and all three are computed for you above.

A bigger standard deviation does not mean more variable data

Standard deviation carries the units of your data, which makes it useless for comparison the moment two datasets differ in scale or in unit. A spread of 2.5884 is enormous around a mean of 11.2 and negligible around a mean of a million, and a spread measured in grams cannot be ranked against one measured in seconds at all. Comparing the raw figures answers a question nobody asked.

The coefficient of variation strips the units out by dividing: CV = SD ÷ mean × 100, also called the relative standard deviation. For the working dataset that is 2.5884 ÷ 11.2 × 100 = 23.11%, a pure number that can be set beside the CV of any other dataset regardless of what either one measures. Its limitations are sharp, though. It is undefined when the mean is 0, and it becomes wildly unstable when the mean merely approaches 0, since a shrinking denominator sends it to arbitrarily large values. It also needs a genuine ratio scale with a meaningful zero: applied to temperatures in Celsius or to test scores centred on an arbitrary baseline, a CV is arithmetic without meaning.

Compare mode goes further and treats two datasets as one problem. It reports each group’s own statistics, then the pooled standard deviation — √(((n₁−1)s₁² + (n₂−1)s₂²) ÷ (n₁+n₂−2)) — which merges the two spreads into a single estimate weighted by each group’s size, so a group of 200 counts for far more than a group of 5. Cohen’s d then expresses the gap between the two means in units of that pooled spread, which is what lets you say a difference is large or small rather than merely present.

What none of this does is tell you whether the difference is real. An effect size is a description of the data you have, not evidence about the population it came from; that requires a hypothesis test, and this page does not perform one. Two groups can differ substantially in a sample by chance alone, especially at the sample sizes where the sample-versus-population divisor still matters.

The 68–95–99.7 rule belongs to the bell curve, not to your data

The empirical rule is a property of the normal distribution: about 68% of a normal population lies within one standard deviation of its mean, about 95% within two, about 99.7% within three. It is not a property of standard deviation, and it is not a property of your dataset. That is why this calculator prints the theoretical percentage and the percentage your values actually achieve side by side rather than only the textbook figure.

Run it on the working dataset and the gap is immediate. Using the sample SD, one standard deviation around the mean spans 11.2 − 2.5884 = 8.6116 to 11.2 + 2.5884 = 13.7884. That window contains 10, 11 and 12 but excludes 8 and 15, so actual coverage is 3 of 5 = 60%, against a theoretical 68%. Two standard deviations span 6.0232 to 16.3768, which captures all five values: 100% coverage against a theoretical 95%. Neither number indicates an error. With five observations, coverage can only land on a multiple of 20%, so exact agreement with 68% is arithmetically impossible.

On larger datasets a persistent gap means something different: your data is not shaped like a bell. Standard deviation is a single number summarising an entire distribution, and radically different shapes can share one. A symmetric dataset, a strongly right-skewed one, and a bimodal one with two separated clusters can all report the same SD while behaving nothing alike — the skewed one will overrun the upper window and underfill the lower, and the bimodal one may leave the region around its own mean almost empty. The histogram and box plot above exist precisely so the shape is visible rather than inferred from one number.

Read the rule as a sanity check, never as a guarantee, and never invert it into a claim about probability for an individual value. This tool does not test your data for normality and never asserts that it is normal; it shows you the comparison and leaves the reading to you.

Grouped data buys convenience with a midpoint you never observed

Frequency mode and grouped-data mode look like the same feature and are not. In frequency mode you enter each distinct value with the number of times it occurs, and the result is exact — mathematically identical to typing that value out that many times, just far less tedious. Nothing is approximated, and the standard deviation you get is the standard deviation of the underlying data.

Grouped-data mode is different in kind. You no longer have values, only class intervals and their counts, so the calculator substitutes the midpoint of each interval for every observation inside it — the same construction a statistics textbook uses. That substitution is an assumption, and what it costs you depends entirely on how the values actually sit inside their classes. When observations really do sit symmetrically within them the estimate is close. When they pile up against one edge, the midpoint systematically misplaces them, and the error grows with the width of the class. An open-ended top class has no midpoint at all, so it has to be closed by an assumption before anything can be computed. Every figure grouped mode returns is an estimate of a standard deviation, not a measurement of one, and where the raw values still exist you should use them.

Plain dataset mode has its own quiet failure mode, which is parsing. Values may be separated by commas, spaces, or new lines, so a column pasted straight out of a spreadsheet works. Zeros and negatives are kept as genuine observations rather than treated as missing, and non-numeric text is skipped with a count shown so the omission is never silent. What the calculator cannot do is know what you meant: a decimal comma read as a separator, a thousands separator splitting one number into two, a value pasted twice, or a footnote marker riding along on the end of a cell will all parse into something plausible and change the answer. These are real limitations, not edge cases — check that the reported count matches the number of observations you believe you have before quoting any figure from this page.

When Excel, your TI-84 and this page disagree, check the quartile convention

Cross-checking a result against another tool is good practice, and most mismatches trace to one of three named causes rather than to an arithmetic error anywhere.

The first is the divisor, wearing a different label. In Excel, STDEV.S applies the n−1 sample formula and STDEV.P applies the N population formula; a handheld calculator usually shows both at once as sₓ and σₓ on the same results screen. Reading the wrong one off a device is the single most frequent reason a page like this appears to disagree with a graphing calculator, and it is not a disagreement at all — the two tools computed different statistics because they were asked to.

The second is the quartile convention, and it is genuinely a convention rather than a correct-or-incorrect matter. This page uses type 7, which matches Excel’s QUARTILE.INC. Other widely used definitions — Excel’s own QUARTILE.EXC, Tukey’s hinges, and the several types implemented by statistical software — interpolate differently and will report different values for Q1 and Q3 on the same data. Because the IQR, the box-plot whiskers and the 1.5 × IQR outlier fences are all built from those quartiles, one convention change can move all of them and can flag or unflag a value. It leaves the mean, the variance and the standard deviation completely untouched, so a result that differs in the quartiles but agrees on the SD is behaving exactly as expected.

The third is rounding. Every statistic here is computed at full precision and rounded only for display, so a tool that rounds an intermediate step — or a hand calculation that carried the mean to two decimals before squaring the deviations — will drift in the trailing digits. To locate any of the three, work through the deviation table above, which shows each value, its deviation, its squared deviation and its z-score, and compare row by row; the exported workbook rebuilds the same statistics as live Excel formulas, so you can inspect a disagreement in the cell where it starts. What this page will not do is walk through a specific device’s button sequence, and no restatement of the formulas here overrides the method your course, journal, or regulator requires.

Which calculator should I use? For everyday percentages and reverse percentages, the percentage calculator; for growth between two values, the percentage change calculator; for business margins, the profit margin calculator.

Sources and methodology

The formulas on this page are the ones published in NIST’s e-Handbook of Statistical Methods: the sample variance and standard deviation carry the N − 1 denominator, and the handbook ties that divisor to the degree of freedom spent estimating the mean from the same sample. Quartiles and the IQR use the type-7 convention (matching Excel QUARTILE.INC), and the exported workbook reproduces every statistic as live Excel formulas validated cell by cell against this page’s engine. Calculator Matters is not affiliated with any source listed. Links open in a new tab.

• NIST/SEMATECH e-Handbook of Statistical Methods — 1.3.5.6. Measures of Scale — prints the sample variance as s² = Σ(Yᵢ − Ȳ)²/(N − 1) and the standard deviation as its square root, alongside MAD, average absolute deviation and the IQRSupports: The n−1 (not n) denominator in the sample standard deviation and variance formulas, plus the MAD and IQR definitions. Verified 2026-09-01.
• NIST/SEMATECH e-Handbook of Statistical Methods — 7.2.3. Are the data consistent with a nominal standard deviation? — writes the sample s with the 1/(N − 1) factor and tests it against a known population σ₀ on N − 1 degrees of freedomSupports: The sample s (n−1) versus population σ distinction, with both appearing in the same expression. Verified 2026-09-01.
• NIST/SEMATECH e-Handbook of Statistical Methods — 7.2.6.3. Tolerance intervals for a normal distribution — states that ν is the degrees of freedom used to estimate the standard deviation, and equals N − 1 when the same sample estimates both the mean and the standard deviationSupports: Why one degree of freedom is spent: the mean is estimated from the same sample, which is the mechanism behind the n−1 divisor. Verified 2026-09-01.
• NIST/SEMATECH e-Handbook of Statistical Methods — What are outliers in the data? — detection and the role of robust measuresSupports: Outlier detection guidance and why outliers should be reviewed, not deleted. Verified 2026-06-13.
• OpenStax Introductory Statistics — Measures of the spread of the data — sample vs population standard deviationSupports: Sample (n−1) vs population (N) standard deviation, with worked examples. Verified 2026-06-13.

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Read the guide

There is no dedicated descriptive-statistics guide yet. The sections above cover sample versus population, Bessel's correction, standard error, outlier detection, the coefficient of variation, grouped data, and the empirical rule in full.