Which standard deviation should you use?
This is the most common source of mismatched answers between calculators. The short version:
- Use the sample standard deviation s (divide by n − 1) when your numbers are a sample and you want to say something about the wider group they came from. That covers nearly all lab measurements, surveys and quality-control checks.
- Use the population standard deviation σ (divide by n) only when you have every member of the group and don't care about anyone outside it, such as the marks of everyone in one class.
For large datasets the two are almost identical. For small ones they differ noticeably: with the 10 numbers above, s = 9.01 but σ = 8.54. If a textbook or spreadsheet gives you a different answer from Kalkuli, this is the first thing to check. In Excel, STDEV.S is the sample version and STDEV.P is the population version.
I've written a full walk-through of the calculation, with a table you can follow by hand, in how to calculate standard deviation by hand.
What each figure means
| Figure | What it tells you |
|---|---|
| Mean (x̄) | The total divided by the count. Pulled towards outliers. |
| Median | The middle value once sorted. Better than the mean for skewed data like house prices or salaries. |
| Mode | The most common value, if any value appears more than once. |
| Standard deviation | The typical distance of a value from the mean, in the same units as the data. |
| Variance | Standard deviation squared. Useful in calculations, harder to interpret. |
| Standard error (SE) | s / √n: how much the mean itself would vary if you repeated the sampling. It shrinks as n grows. |
| Q1, Q3 and IQR | The values a quarter and three-quarters of the way up the sorted data, and the gap between them, which covers the middle half. |
| Outliers | Values more than 1.5 × IQR below Q1 or above Q3 (Tukey's rule). |
There are at least nine published ways to calculate quartiles, and they disagree for small datasets. Kalkuli interpolates between ranked values, which matches Excel's QUARTILE.INC, Google Sheets' QUARTILE and R's default. Some school textbooks use the "median of each half" method, which can give slightly different Q1 and Q3.
Permutations vs combinations
Both count the ways to choose r items from n. The difference is whether order matters.
- Permutations, nPr = n! / (n − r)!: gold, silver and bronze among 8 runners is 8 × 7 × 6 = 336, because the same three runners in a different order is a different result.
- Combinations, nCr = n! / (r! (n − r)!): a 3-person committee from 8 people is 336 ÷ 3! = 56, because the order you pick them in doesn't matter.