What standard deviation tells you
Standard deviation measures how far values typically sit from their mean. A small one means the data is bunched together; a large one means it's spread out. Two sections of a course can both average 75 on a test, but if one has a standard deviation of 4 points and the other 15, the students in the first scored close to one another while the second ranged from struggling to near perfect.
It's in the same units as the data, which is what makes it readable. If the scores are in points, the standard deviation is in points; if reaction times are in milliseconds, so is their spread. For data that follows a normal, bell-shaped distribution, about 68% of values fall within one standard deviation of the mean and about 95% within two. That's what a z-score uses: z = (x − x̄) ÷ s counts how many standard deviations a value sits from the mean, so a z of 2 is unusual and a z of 0.3 is ordinary.
Variance is the same idea before the square root: the average squared distance from the mean. It's what the math works with, because variances of independent quantities add, but its units are squared (points², ms²), so it's hard to read on its own. The standard error of the mean, s ÷ √n, answers a different question. It isn't about how spread out the data is, but how precisely the sample's mean estimates the population's. Collect more data and the standard error shrinks; the standard deviation doesn't.
The formulas, and why the sample one divides by n − 1
Both versions follow the same four steps: find the mean, subtract it from every value, square those deviations and add them up, then divide and take the square root. Only the divisor changes. For a population (every member of the group you're describing) divide by N: σ = √(Σ(x − μ)² ÷ N). For a sample (some members, used to learn about the rest) divide by n − 1: s = √(Σ(x − x̄)² ÷ (n − 1)). In Excel and Google Sheets, these are STDEV.P and STDEV.S.
Dividing by n − 1 is called Bessel's correction. In plain terms: a sample's values sit a little closer to their own mean than to the true population mean, because the sample mean is calculated from those same values. So the squared deviations come out slightly too small, and dividing by n − 1 instead of n scales them back up. The correction matters most for small samples. With 5 values it raises the variance by 25%; with 100 values, by about 1%. With a single value, n − 1 is 0 and there is no sample standard deviation at all, because one value tells you nothing about spread.
Here is the calculator's example, 2, 4, 4, 4, 5, 5, 7, 9, worked by hand. The values add up to 40, so the mean is 40 ÷ 8 = 5. The deviations from 5 are −3, −1, −1, −1, 0, 0, 2 and 4. Their squares are 9, 1, 1, 1, 0, 0, 4 and 16, which sum to 32. Treated as a population, the variance is 32 ÷ 8 = 4 and the standard deviation is √4 = 2. Treated as a sample, the variance is 32 ÷ 7 = 4.5714 and the standard deviation is √4.5714 = 2.1381. The standard error of the mean is 2.1381 ÷ √8 = 0.7559.
So which should you use? If the numbers are a subset (30 students surveyed out of a whole school, ten readings from an experiment you could repeat), use the sample formula. Use the population formula only when you have every value you're describing, like the scores of everyone in your own class when that class is all you're reporting on. When in doubt it's usually a sample, which is why the calculator starts there.
Quartiles, IQR and outliers: why calculators disagree
Quartiles split the sorted data into four parts. A quarter of the values sit below Q1, half below the median and three quarters below Q3. The interquartile range, IQR = Q3 − Q1, is the spread of the middle half, and unlike the standard deviation it isn't dragged around by one extreme value. Tukey's 1.5 × IQR rule uses it to flag outliers: anything below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.
The catch is that there's no single agreed way to find a quartile that falls between two values. In 1996, Hyndman and Fan compared nine methods used by statistical software, and the NIST/SEMATECH handbook still lists several in common use. This calculator uses one of them, type 7: the p-th quantile sits at position (n − 1) × p + 1 in the sorted list, interpolating between the values either side. It's the default in R and NumPy, and it's what Excel's QUARTILE.INC and Google Sheets' QUARTILE compute, so the answers match a spreadsheet.
On the example data, type 7 gives Q1 = 4, a median of 4.5 and Q3 = 5.5, so the IQR is 1.5 and the fences are 1.75 and 7.75. That makes 9 an outlier. Type 6, used by Excel's QUARTILE.EXC, Minitab and SPSS, puts Q3 at 6.5 and the upper fence at 10.25, so 9 isn't one. The median-of-halves method in many intro courses and on TI-83 and TI-84 calculators takes the median of each half (4 and 6 here), which puts the upper fence at exactly 9, so 9 isn't beyond it either. None of these is wrong. Check which method your course expects, and say which one you used when it matters.
Common mistakes
Using the wrong formula. Most homework and lab data is a sample, so most answers need n − 1. If your answer is a little smaller than the one in the back of the book, you probably divided by n. Switch between sample and population above and compare the two.
Rounding too early. If you round the mean before subtracting it, every deviation carries the error, and squaring magnifies it. Keep full precision until the final answer. The step-by-step table shows rounded numbers, but each step is computed from the unrounded ones.
Stopping one step short, or going one too far. Variance and standard deviation differ by a square root. In the example, s² = 4.5714 but s = 2.1381. If a question asks for the standard deviation and your answer is in squared units, take the root.
Mixing up standard deviation and standard error. Error bars and "mean ± SE" use the standard error, which is smaller than the standard deviation by a factor of √n. Reporting one as the other misstates the spread. And to compare spread across different units, such as kilograms against centimeters, use the coefficient of variation, s ÷ mean, which has no units. It only makes sense for positive amounts measured from a true zero.