Three averages, three different claims
The mean is the total divided by the count, the median is the middle value when sorted, and the mode is the most common value. On symmetric data they roughly agree. On skewed data they diverge, and which one is quoted becomes a rhetorical choice.
Income is the classic case. A handful of very high earners pull the mean well above the median, so mean income describes a person who does not exist while median income describes the person in the middle. House prices, wealth, response times and company sizes are all skewed the same way.
When a headline quotes an average without saying which, assume it was chosen to support the point being made. Asking for the median is usually the single most useful question you can put to a statistic.
Spread matters as much as centre
Two datasets can share a mean and describe completely different realities. Standard deviation measures typical distance from the mean, and without it an average is half a description.
For roughly normal data, about sixty-eight per cent of values fall within one standard deviation of the mean, ninety-five per cent within two and ninety-nine point seven within three. That is what makes a z-score meaningful: it converts any measurement into how many standard deviations it sits from the centre.
The caveat is the word normal. Incomes, house prices and wait times are not normally distributed, and applying normal-distribution percentiles to them produces confident nonsense.
Population or sample
Dividing by the number of values gives the population standard deviation; dividing by one fewer gives the sample estimate. The second is larger, and it is the right choice whenever your data represent a wider group. Which is almost always.
The correction exists because a sample's own mean sits closer to its own values than the true population mean does, so the raw calculation systematically understates the true spread. Subtracting one from the denominator compensates.
Use the population version only when you genuinely have every member of the group you care about: every student in one class, every machine on one line, every transaction in one day.
Judging a sample
For a large population, roughly 385 responses give a five per cent margin of error at ninety-five per cent confidence, and about 1,067 give three per cent. Precision improves with the square root of the sample, so halving the margin of error costs four times as many responses.
Population size barely matters once it is large. A well-drawn sample of a thousand describes a country of twenty-five million about as precisely as a city of two hundred thousand, which people find counterintuitive and which is nonetheless true.
How the sample was drawn matters far more than its size. A biased sample of ten thousand is worse than a random sample of five hundred, because size reduces random error and does nothing at all about systematic error.
The errors that survive good arithmetic
Survivorship bias analyses only what remains: successful companies, surviving aircraft, funds that were not closed. The missing cases usually carry the information you actually needed.
Correlation is not causation, and the reason is rarely that the relationship is fake. It is that a third variable drives both, or the causality runs the other way. Ice cream sales and drownings rise together because both follow summer.
Regression to the mean explains a great deal of apparent effectiveness. Extreme measurements tend to be followed by less extreme ones for purely statistical reasons, so any intervention applied at a low point appears to work. Without a control group you cannot tell the treatment from the tendency.
Questions people ask
- When should I use the median instead of the mean?
- Whenever the data are skewed or contain outliers. Income, house prices, wait times, company sizes. The median describes the typical case in those distributions, while the mean is pulled toward the extreme tail and can describe nobody. On symmetric data the two agree and the mean is fine.
- What is the difference between population and sample standard deviation?
- The sample version divides by one fewer than the count, which corrects for the fact that a sample understates the true spread. Use it whenever your data represent a larger group, which is nearly always. The population version is correct only when you have measured every member of the group you care about.
- How big does my survey sample need to be?
- About 385 responses for a five per cent margin of error at ninety-five per cent confidence, or 1,067 for three per cent, for any large population. Precision scales with the square root of the sample, so improvements get expensive quickly. How representative the sample is matters more than how large it is.
- What does a z-score of 2 mean?
- The value sits two standard deviations above the mean, which for normally distributed data places it above roughly 97.7 per cent of the distribution. For skewed data such as incomes, that percentile interpretation does not hold, even though the z-score can still be calculated.
- Why does a margin of error not depend on population size?
- Because precision comes from the number of independent observations, not from what fraction of the population they represent. Once a population is large, sampling a thousand people from it tells you about as much whether the population is a hundred thousand or fifty million. The correction only matters for small, finite populations.
Sources
- Australian Bureau of Statistics: understanding statistics. Reference on sampling, averages and statistical literacy