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Explain what measures of central tendency show

CFA® Level I · Quantitative Methods · Statistical Characteristics of Asset Returns · QM.2.a

The idea

The mean, median and mode summarise where a distribution sits, and they disagree in informative ways. The arithmetic mean uses every observation, which also makes it sensitive to outliers. The median — the middle value once sorted — is robust to extremes, so a mean far above a median signals right skew. The mode is the most frequent value and is the only one that works for nominal data. For returns compounded over time, the geometric mean is the correct summary; the arithmetic mean overstates realised performance whenever returns vary.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which measure of central tendency is least affected by extreme outliers?

  1. The median — correct
  2. The geometric meanWhy not: It is dragged badly by extremes — a single return of −100% takes it to −100% however good the other years were.
  3. The arithmetic meanWhy not: The most affected, not the least: it uses every observation's magnitude, so one extreme value pulls it directly.

The median depends only on the middle observation's position, so a single extreme value barely moves it.

A harder one

A return distribution has a mean of 8% and a median of 5%. This most likely indicates:

  1. negative (left) skewWhy not: The right relationship, read backwards. Left skew pulls the mean *below* the median; here the mean is above it.
  2. positive (right) skew — correct
  3. a symmetric distributionWhy not: Symmetry would put the mean and median together. A 3-point gap is the evidence against it.

Large positive outliers drag the mean above the median. Mean > median is the signature of right skew.

The hardest tier

For a data set with any variation in its values, the geometric mean is:

  1. less than the arithmetic mean — correct
  2. greater than the arithmetic meanWhy not: Backwards. The inequality runs the other way for any set of non-negative numbers, and it is why compounded returns lag the simple average.
  3. equal to the arithmetic meanWhy not: Only when every value is identical. The question stipulates variation, which is exactly the condition that opens a gap.

The geometric mean is at most the arithmetic mean, with equality only when every value is identical. This is why compounded returns lag the simple average.

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