Describing data: averages, spread and skew

In 2025, the middle household in the United States had an income of 87,460 dollars before tax. The mean, the figure most people call “the average”, was 126,700 dollars. Both come from the same yearly US Census Bureau survey. Yet they are almost 40,000 dollars apart. So which one tells you what a typical household earns?

To find out, shrink the problem. The table shows 20 invented households in Riverton, a made-up town, lined up from poorest to richest. Incomes are in thousands of dollars a year, so 29 means 29,000 dollars.

The mean sits above the median because of a few big incomes at the top, such as 80,000 and 104,000 dollars. Twelve of the 20 households earn less than the mean. Data like these are skewed: most values bunch together, and a long tail stretches out to one side. A tail at the top, as here, is positive skew (skew to the right). Incomes usually skew this way: in the US survey, the mean has topped the median every year since 1967.

An outlier, a value far from all the others, drags the mean towards it, because the mean uses the size of every value. It barely moves the median, because the median only asks which household stands in the middle.

The median cuts the line in half. Quartiles cut it into four equal groups, like the rows of the table. The lower quartile is the income a quarter of the way up the line: 39,000 dollars. The upper quartile is three-quarters of the way up: 56,000 dollars. Deciles cut the line into ten equal groups, here of two households each; the poorest group is the bottom decile. Each group holds the same number of households, not the same share of income.

Spread tells you how far apart the incomes are. The interquartile range is the gap between the two quartiles: 56,000 − 39,000 = 17,000 dollars. It shows how spread out the middle half of households is.

The variance measures spread around the mean. Take each income’s distance from the mean, square it, and average the squares. Squaring makes the units “squared dollars”, which nobody can picture. So economists usually quote the variance’s square root, the standard deviation. It is back in dollars and shows roughly how far a typical income sits from the mean.

Spread is closely tied to inequality: how unevenly total income is shared. Add up Riverton’s incomes from the poorest household up. The poorest four-fifths earn 69 percent of the total, so the richest fifth get 31 percent. A Lorenz curve plots this running (cumulative) share of income against the running share of households.

The more unevenly income is shared, the further the Lorenz curve sags below the diagonal line of perfect equality, where everyone earns the same.