I said "average" not "mean". The median is a form of average. BLS looks at mostly median households in the survey dataset, but also considers means of some ag

I said "average" not "mean". The median is a form of average. BLS looks at mostly median households in the survey dataset, but also considers means of some aggregates, when choosing weightings of goods in the basket. I suspect in the specific case you're talking about -- housing -- using the mean would overweight housing in the basket.