Average Calculator

Three averages that answer different questions

The mean is the sum divided by the count. It uses every value and is pulled by extreme ones.

The median is the middle value when sorted. It is unaffected by how extreme the extremes are, which makes it the right choice for skewed data.

The mode is the most frequent value, and it is the only one that works for categorical data. A dataset can have no mode, one, or several.

When the mean misleads

Income is the standard example. A small number of very high incomes pull the mean well above the median, so mean income describes almost nobody.

UK median household income sits substantially below the mean for exactly this reason, and reporting that uses the mean makes the typical household appear better off than it is.

The general rule: use the median when the distribution is skewed or contains outliers, and the mean when it is roughly symmetric. Reporting both, and the gap between them, is more informative than either alone.

Weighted means

Where values represent different quantities, a plain mean is wrong. Averaging a price of £2 for 100 units and £8 for 10 units gives £5, when the actual average price paid is £2.55.

The weighted mean multiplies each value by its weight, sums, and divides by the total weight. It is the correct calculation whenever the values do not represent equal amounts.

This appears constantly and is constantly done wrong: average prices across different volumes, average grades across different credit hours, average returns across different amounts invested. Each needs weighting and each is frequently averaged plainly.

The spread matters as much

Two datasets can share a mean and be nothing alike. Values of 50, 50, 50 and values of 0, 50, 100 both average 50.

Standard deviation measures typical distance from the mean, and it is what distinguishes them. Range gives the crude version; the interquartile range gives a version robust to outliers.

An average reported without any measure of spread is half a description. It is the most common omission in casual data reporting and the one that most often leads people to conclusions the data does not support.

Averages of averages

Averaging a set of averages is wrong unless the groups are the same size, and it produces Simpson's paradox in the worst cases.

If class A averages 70% across 10 students and class B averages 80% across 40, the overall average is not 75%. It is 78%, weighted by the number of students.

Simpson's paradox is the extreme form: a trend that appears in every group can reverse when the groups are combined, purely from differing group sizes. It has produced genuinely misleading conclusions in published research, and the defence against it is always to work from the underlying counts rather than from summary averages.

The geometric mean deserves a mention because there is one common situation where the arithmetic mean is simply the wrong tool. For rates of change compounding over time, investment returns being the obvious case, the arithmetic mean overstates the outcome. Returns of plus 50% then minus 50% average zero arithmetically and leave you with 75% of what you started with. The geometric mean, the nth root of the product of the growth factors, gives the rate that would actually have produced the observed result, and it is the honest figure for any sequence of proportional changes.