Which calculator for which situation
- Summarising data. Mean and standard deviation for central tendency and spread; geometric mean when values multiply, as growth rates and returns do.
- Comparing values. Z-score measures how many standard deviations a point sits from the mean — which is what lets you compare quantities on different scales.
- Counting possibilities. Combinations and permutations: permutation when order matters, combination when it does not.
- Probability and sampling. Probability for simple and compound events; sample size for how many responses a survey needs.
Three readings that change the conclusion
A mean without spread says little. Two sets with identical means can be completely different. Standard deviation tells you whether values are clustered or scattered — and it is why "an average wait of 15 minutes" can mean fifteen minutes for everyone, or five for some and forty for others.
A mean without a median misleads on skewed data. In income, house prices and waiting times, a few very high values pull the mean upward. It is common for most of the sample to sit below the mean — in those cases the median describes the typical case better.
A percentage without the absolute number is half a fact. "Grew 200%" may be from 2 to 6. Every relative change needs the absolute figure beside it to be interpretable.
Sample size: why 385 keeps appearing
For a large population, a 5% margin of error and 95% confidence, the required size is 385 responses — and it barely changes whether the population is 50,000 or 50 million. It is a counter-intuitive and correct result: precision is determined by the sample's size, not by the fraction of the population it represents.
Halving the margin of error, however, is expensive: it requires four times the sample. A 2.5% margin needs roughly 1,537 responses.
Frequently asked questions
Population or sample standard deviation? Use the sample version (dividing by n−1) when your data is a sample of something larger — which is almost always. The population version (dividing by n) applies only when you hold every element.
What does 95% confidence mean? That if you repeated the same sampling procedure many times, about 95% of the intervals constructed would contain the true value. It does not mean there is a 95% chance the value lies in this particular interval — the distinction sounds subtle and is the source of most misreadings.
Combination or permutation? Ask whether reordering creates a different outcome. A race podium: yes, permutation. Cards in a hand: no, combination.
Does correlation prove causation? No. Two variables can move together by chance, through a common cause, or through reverse causality. Correlation is a starting point for investigation, never a conclusion.