Calculate sample and population standard deviation, variance, mean and other descriptive statistics instantly, with detailed formulas, worked examples and step-by-step calculations.
Find the mean, subtract it from every value, square each of those differences, add the squares, divide by n − 1 for a sample or N for a whole population, then take the square root. That result is the standard deviation — the typical distance between a value and the mean, expressed in the same units as your data. Dividing by n − 1 is called Bessel's correction, and it is the right choice for almost all real-world data, because most data sets are samples rather than complete populations.
Paste or type your numbers separated by commas, spaces, tabs or new lines — CSV columns paste in directly. Percentage symbols are removed, so 5% is treated as the number 5, not the decimal 0.05. Dollar signs are removed the same way.
| Statistic | Value |
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Each value's z-score shows how many standard deviations it sits from the mean, using the sample standard deviation selected above. Z flag and IQR flag are genuinely different methods and can disagree on the same value — both are shown rather than merged into one verdict.
| # | Value | Deviation (x−mean) | Z-Score | Z Flag | IQR Flag |
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Standard deviation is a single number that answers a very practical question: how far from average is a typical value in this data set? A small standard deviation means the values huddle close to the mean. A large one means they are scattered widely. Because it is expressed in the same units as the original data — dollars, millimetres, marks, degrees — it can be read directly rather than needing translation.
Consider two Year 10 classes that both averaged 70 on the same test. Class A scored 70, 72, 68, 71, 69, giving a sample standard deviation of about 1.58. Class B scored 50, 90, 60, 80, 70, giving a sample standard deviation of about 15.81. Identical means, completely different stories: one class is uniformly around the average, the other contains students who are struggling badly and students who are well ahead. The mean alone hides that entirely, which is why standard deviation almost always belongs beside it.
The symbol depends on what your numbers represent. A lower-case Greek sigma, σ, is the standard deviation of an entire population. A plain s is the standard deviation of a sample drawn from a larger group. The distinction is not cosmetic — the two are calculated slightly differently, as the next sections explain.
Averages are comfortable and often misleading. Standard deviation is what turns an average into information you can act on.
This is the single most common point of confusion, and choosing wrongly changes your answer.
A population is every member of the group you care about — every student in one classroom, every item in one production batch, all twelve months of last year's electricity bills. If your data covers all of them, divide the sum of squared deviations by N.
A sample is a subset used to estimate something about a larger group you did not fully measure — 200 surveyed customers standing in for 50,000, or 30 tested components standing in for a whole production run. Here you divide by n − 1 instead of n.
Why the subtraction? Because you used the sample's own mean as the reference point, deviations measured from it are slightly too small on average, so dividing by n would systematically understate the true spread. Dividing by n − 1, known as Bessel's correction, removes that bias and makes s² an unbiased estimator of the population variance.
| Feature | Population standard deviation (σ) | Sample standard deviation (s) |
|---|---|---|
| Divide by | N (the count) | n − 1 (Bessel's correction) |
| Symbol | σ (sigma), variance σ² | s, variance s² |
| Mean symbol | μ (mu) | x̄ (x-bar) |
| Use when | You have measured every member of the group | Your data is a subset standing in for a larger group |
| Relative size | Always the smaller of the two | Always slightly larger |
| Spreadsheet function | STDEV.P | STDEV.S |
| Typical use | Classroom marks, one full batch, one complete year | Surveys, experiments, quality testing, research |
The mean, or arithmetic average, is the balance point of a data set: add every value and divide by how many there are. It is the reference point that standard deviation measures distance from, so it has to be calculated first.
The mean is sensitive to extreme values, which is worth knowing when you interpret a standard deviation. One enormous outlier drags the mean towards itself and inflates the standard deviation, so both statistics move at once. The median — the middle value when the data is sorted — is far more resistant, which is why property prices and household incomes are almost always reported as medians. Comparing the mean and median of the same data set is a quick way to detect skew: if they differ noticeably, the distribution is lopsided.
Variance is the average of the squared deviations from the mean — in other words, standard deviation before the square root is taken. Squaring does two useful things: it makes every deviation positive, so values above and below the mean cannot cancel out, and it gives larger deviations disproportionate weight.
The catch is units. If your data is in centimetres, the variance is in square centimetres, which has no physical meaning here. Taking the square root converts it back to centimetres, and that is precisely what standard deviation is for. Variance remains the working currency of statistical theory — it adds neatly across independent variables, which standard deviation does not — while standard deviation is the number you report to people.
| Feature | Variance | Standard deviation |
|---|---|---|
| Definition | Mean of the squared deviations | Square root of the variance |
| Units | Original units, squared | Same as the original data |
| Symbols | σ² or s² | σ or s |
| Easy to interpret? | Not directly — the units are artificial | Yes — read it as a typical distance from the mean |
| Main use | Statistical theory, ANOVA, combining variability | Reporting, risk measures, z-scores, error bars |
| Additive? | Yes, for independent variables | No |
Both versions share the same skeleton and differ only in the divisor.
There is also a computational form that avoids storing every deviation, which is how calculators and spreadsheets have historically done the arithmetic in a single pass:
It gives identical answers to the definition, which is why the calculator above reports Σx² alongside Σx — you can check your own work either way.
Many natural and manufactured measurements cluster symmetrically around their mean, thinning out towards the extremes — the familiar bell curve, known formally as the normal or Gaussian distribution. Heights, measurement errors, and the averages of large samples all tend towards this shape.
A normal distribution is fully described by just two numbers: its mean, which locates the centre, and its standard deviation, which sets the width. Change the mean and the whole curve slides sideways; change the standard deviation and it stretches flat or squeezes tall. That is a remarkable economy — two statistics capture an entire distribution — and it is the main reason standard deviation is so central to statistics.
For data that is approximately normal, the empirical rule gives an immediate sense of scale.
| Range | Proportion of data | Interpretation |
|---|---|---|
| mean ± 1 SD | ≈ 68.27% | Roughly two values in three are this close to average |
| mean ± 2 SD | ≈ 95.45% | Outside this band is genuinely unusual — about 1 in 22 |
| mean ± 3 SD | ≈ 99.73% | Beyond this is about 1 in 370 — often treated as an outlier |
A standard deviation is meaningless in isolation — it only means something relative to the mean and to the context. An SD of 3 cm is enormous for machined bearings and trivial for adult heights.
To compare spread across data sets with different scales or units, divide the standard deviation by the mean and express it as a percentage. That is the coefficient of variation (CV), and because it is unitless it compares directly.
A z-score restates any value as the number of standard deviations it sits from the mean: z = (x − mean) / SD. A positive z is above average, negative is below, and the magnitude tells you how unusual it is. Z-scores let you compare a maths mark with a swimming time, because both are converted to the same scale.
There is no universal threshold. It depends entirely on what the number is for. In manufacturing, small is the whole goal — tight tolerances mean consistent parts. In an investment portfolio, some volatility is the price of return. In a class of students, a very small standard deviation might mean excellent teaching or a test that was too easy to distinguish anyone. Always ask what spread should look like before deciding whether yours is good.
| Mistake | Why it goes wrong | How to avoid it |
|---|---|---|
| Using N when you have a sample | Understates the spread, badly on small data sets | Ask whether your data is the whole group; if not, use n − 1 |
| Forgetting the square root | You have calculated the variance, not the standard deviation | Check the units: SD matches your data, variance does not |
| Averaging the deviations without squaring | Positives and negatives cancel and the total is always zero | Square first, then average, then take the root |
| Rounding the mean before subtracting | Small rounding errors get squared and compound | Carry full precision through and round only the final answer |
| Comparing SDs across different units or scales | An SD in dollars cannot be compared with an SD in kilograms | Use the coefficient of variation instead |
| Applying the 68–95–99.7 rule to skewed data | Those percentages assume an approximately normal shape | Plot the data first, or use quartiles and the IQR |
| Treating a large SD as an error | High variability is often the real finding, not a mistake | Investigate the cause before "cleaning" the data |
| Ignoring outliers | Squaring gives extreme values enormous influence | Check extremes, and report the median and IQR alongside |
| Mixing up STDEV.P and STDEV.S in a spreadsheet | They answer different questions | STDEV.S for samples, STDEV.P for populations |
Standard deviation is the backbone of quality control. A production line measures a critical dimension on sampled units and tracks how far the spread drifts; control charts draw warning limits at two and three standard deviations from the target, and the "six sigma" methodology takes its name from the goal of fitting six standard deviations between the process mean and the nearest tolerance limit. Elsewhere in business it measures demand volatility for inventory planning, variability in delivery times, and consistency of staff performance or customer satisfaction scores.
In finance, the standard deviation of returns is the standard definition of volatility, and therefore of risk. It feeds directly into portfolio theory, where combining assets whose returns move differently reduces the standard deviation of the whole portfolio without necessarily reducing expected return. It also underpins the Sharpe ratio (excess return divided by standard deviation), value-at-risk models, and the volatility inputs in options pricing. Superannuation funds in Australia describe their investment options as "growth" or "conservative" largely on the basis of expected return variability.
Every experimental measurement carries uncertainty, and standard deviation is how that uncertainty is reported. Error bars on a chart are usually one standard deviation or one standard error; the standard error of the mean, s / √n, shrinks as the sample grows, which is the mathematical reason larger studies give more precise estimates. Standard deviation also feeds into confidence intervals, hypothesis tests and the calculation of how large a sample needs to be in the first place.
Standardised testing reports results in standard-deviation units so that cohorts and years can be compared fairly. Medical reference ranges — bone density scores, growth charts, blood test intervals — are defined as a number of standard deviations from a reference mean. Climate records describe an unusually hot month by how many standard deviations it sits above the long-term average. And anyone comparing quotes, delivery estimates or fuel consumption is doing informally what standard deviation does precisely.
x̄ = Σx / nSS = Σ(x − x̄)²σ² = SS / N | Population SD: σ = √(SS / N)s² = SS / (n − 1) | Sample SD: s = √(SS / (n − 1))s² = (Σx² − (Σx)²/n) / (n − 1)σ² = σ × σ, σ = √(σ²)z = (x − mean) / SDCV = SD / mean × 100%SE = s / √nmax − min | IQR: Q3 − Q168% / 95% / 99.7% within 1 / 2 / 3 SD=STDEV.S(range), =STDEV.P(range), =VAR.S(range), =AVERAGE(range)
1) (4+8+6+5+3)/5 = 26/5 = 5.2 2) Deviations −1.2, 2.8, 0.8, −0.2, −2.2; squares 1.44, 7.84, 0.64, 0.04, 4.84 → 14.8 3) √49 = 7 4) 0 — every value equals the mean 5) z = (65 − 50) / 10 = 1.5
1) SS = 14.8, s² = 14.8/4 = 3.7, s ≈ 1.9235 2) Mean 20.2, SS = 148.8; population σ² = 29.76 so σ ≈ 5.4553; sample s² = 37.2 so s ≈ 6.0992 3) Mean 18, SS = 192, s² = 192/7 ≈ 27.4286, s ≈ 5.2372; mode = 23 (appears three times) 4) SE = 8 / √64 = 8 / 8 = 1 5) CV = 3/15 = 20% versus 20/120 ≈ 16.7%, so the first data set is relatively more variable despite its much smaller SD
=STDEV.S(range) for a sample and =STDEV.P(range) for a population. The matching variance functions are =VAR.S(range) and =VAR.P(range), and =AVERAGE(range) gives the mean. Google Sheets uses the same names. The older =STDEV() function is equivalent to STDEV.S. If your figures differ from this calculator, check which of the two you used — that is nearly always the reason.