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Work out quantiles and explain what they show

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

The idea

Quantiles cut a distribution into equal-frequency parts: quartiles into four, deciles into ten, percentiles into a hundred. The median is the 50th percentile. The interquartile range, the third quartile less the first, describes the spread of the middle half and ignores the tails entirely, which makes it robust to outliers in a way standard deviation is not. A box plot draws exactly these figures. Quantiles are how performance is usually ranked in practice — a manager in the first quartile is being described by position, not by distance from a mean.

Formulas

Position of a percentile in a data set

Ly = (n + 1) × (y ÷ 100)

Ly
the position in the list, counting from the smallest
n
the number of observations
y
the percentile wanted

The data must be sorted first, and the answer is a position rather than a value. When it lands between two observations, interpolate between them — the position 7.5 means halfway between the seventh and the eighth.

A practice question

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

Which quantile is the median?

  1. The 25th percentileWhy not: That is the first quartile — a quarter of the observations below it, not half.
  2. The 50th percentile — correct
  3. The 75th percentileWhy not: That is the third quartile. Three-quarters of observations fall below it.

Half the observations fall below the median, which is the 50th percentile, the second quartile and the fifth decile — the same point under three names.

A harder one

The first quartile of a return series is 3% and the third quartile is 11%. What is the interquartile range?

  1. 14%Why not: 11 + 3 — the quartiles added. A range is a difference between two points, not a total.
  2. 7%Why not: The midpoint of the two quartiles, (11 + 3) ÷ 2. That describes where the middle half sits, not how wide it is.
  3. 8% — correct

The interquartile range is Q3 − Q1 = 11% − 3% = 8%, the spread of the middle half of the observations. It ignores the tails, which is what makes it resistant to outliers.

The hardest tier

Why might an analyst prefer the interquartile range to the standard deviation when describing a fund's returns?

  1. It is unaffected by a handful of extreme observations — correct
  2. It always produces a smaller numberWhy not: Not always, and size is not the reason. A distribution with a wide middle and thin tails can have an interquartile range exceeding its standard deviation.
  3. It accounts for skewness automaticallyWhy not: It does not measure skewness at all — it is a single width. Skewness needs its own statistic, or the gap between mean and median.

Standard deviation squares deviations, so one extreme month can dominate it. The interquartile range depends only on rank order in the middle of the distribution and is unmoved by the tails.

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