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All Statistics questions

Question 12 of 20 Fundamentals Interpretation

When is the median a better summary than the mean?

Short answer

When data are skewed or contain extreme values, the mean is pulled toward the tail while the median, the middle value, is not. For roughly symmetric data the two agree and the mean is the more efficient summary.

Why

The NIST Engineering Statistics Handbook explains that for skewed data the mean is pulled in the direction of the skew, and that extreme values in the tails distort the mean but not the median, because the median depends on ranks. Biological data are often right-skewed. Viral loads, cytokine concentrations, hospital length of stay and read counts all have a long upper tail.

Suppose nine patients stay 3, 3, 4, 4, 5, 5, 6, 7 and 60 days. The mean is about 10.8 days, longer than eight of the nine stays, while the median of 5 days describes a typical patient. Which summary is right depends on the question, not only the shape.

A hospital budgeting beds needs the mean, because total bed-days equal the mean times the number of patients, and the long stay really costs money. A patient asking how long they will probably stay needs the median. Report both, with the range or interquartile range, when the data are skewed.

The median is robust but not free: it uses less of the information in the data, so for genuinely symmetric data its estimate is less precise than the mean. A log transformation and a geometric mean are another option for multiplicative data.

A strong answer covers

  • Mean is pulled toward the tail of skewed data
  • Median depends on ranks and resists outliers
  • Many biological measures are right-skewed
  • The question decides the summary, totals need the mean
  • Report spread such as the interquartile range

Common mistakes

  • Deleting extreme but genuine values so the mean looks typical
  • Assuming the median is always better, even when a total such as bed-days is what matters