Home Prices Fell 2%? Mix Effects and Simpson's Paradox

"Home prices fell 2% this year." Headlines like this seem straightforward, yet they hide a trap. The average sale price can fall even when every neighborhood's prices rise. This effect, close to Simpson's paradox, is one of the most important things to understand about price indices.

Two regions, both rising

Take two regions. Last year the expensive one sold 40 homes at an average of 800 thousand, and the cheaper one sold 60 homes at 400 thousand. The overall average was (40 × 800 + 60 × 400) / 100 = 560. This year both regions' prices rose 5%, to 840 and 420. But sales shifted: the expensive region sold only 20 homes and the cheap one 80.

The new overall average is (20 × 840 + 80 × 420) / 100 = 504, a fall of 10%, even though every home in both regions became 5% more expensive. The descriptive statistics calculator will reproduce either average from a list of sales.

What is happening

Central banks have noted that median prices are affected by the mix of properties sold, and that this makes seasonal swings in the data (Reserve Bank of Australia). Statistical agencies therefore use methods that hold the mix constant: hedonic regression, repeat sales and stratification (Australian Bureau of Statistics).

The average moved because the mix of homes sold changed, not because prices fell.

The five-number summary calculator shows how the median and quartiles shift when cheap sales dominate, a quick way to detect that the composition changed.

Simpson's paradox with proportions

The same reversal appears with rates. The famous Berkeley admissions data showed a pattern against women in aggregate, which disappeared, and even slightly reversed, within departments, because women applied more to more selective ones (Bickel, Hammel and O'Connell, 1975).

In housing, imagine the share of properties selling quickly. Pooled across a city, a policy might look to raise the odds, while within every neighborhood it lowers them. The odds ratio calculator gives each neighborhood's odds ratio, and the chi-square test for independence tests each table. Compare the subgroup results before trusting the pooled one.

How to read a headline

Ask whether the index is mix-adjusted, how many transactions it rests on, and whether prices changed within comparable groups. A fall in the average with rises in every region is a warning sign.

Median versus mean

Replacing the mean with the median helps with outliers, but it does not remove the mix problem. If expensive homes are a smaller share of sales this year, the median falls as well. That is why stratified or hedonic indices are preferred when the goal is to measure the price of a typical home, rather than the typical price of homes that happened to sell.

Sample size is another limit. A quiet month with few sales can produce swings that reverse the next month, so analysts smooth over several months or use repeat sales of the same property.

Also check the period. A 2% annual change can hide a seasonal pattern and sharp movements in the last months, and small local markets can swing wildly on a few sales. A single national figure answers a question that most buyers and sellers are not asking: they care about their street.