The twenty quarterly returns of Worked example 3 average +2.67% and have a median of +4.20%. The mean sits 1.53 points below the median, and the reason is the single −21.40% quarter dragging it down. That relationship runs one way: a left tail pulls the mean below the median, a right tail pushes it above.
The mode is the value that appears most often, and on a continuous series it is usually useless. All twenty of these returns are distinct, so the series has no mode at all. Group them into ten-point buckets centred on zero and two buckets tie at eight observations each, while shifting every edge by five points leaves a single bucket of ten. The modal bucket is an artefact of where you drew the edges. Use the mode for categorical data: ratings, sectors, recommendation types. Leave it alone for returns.
What are quartiles and percentiles?
Quantiles divide a sorted series into equal-count pieces: quartiles into four, quintiles into five, deciles into ten, percentiles into a hundred. On the twenty-quarter series, the first quartile is −1.925%, the median +4.20%, the third quartile +8.725%, and the interquartile range 10.65 points. The first decile is −7.69% and the ninth is +12.75%.
One caution that costs people real money. There is no single definition of a sample quantile, and different software interpolates differently between the two straddling observations. The figures above use linear interpolation, which is what a spreadsheet's percentile function and the standard array libraries do by default. A different convention moves the first quartile by a few tenths of a point on a sample this small. Say which convention you used, or your risk report will not reconcile with the one it is compared against.
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What is a box plot?
A box plot is the quantile family drawn. The box runs from the first to the third quartile with a line at the median; the whiskers extend to the most extreme observations within 1.5 interquartile ranges of the box, here −1.925 − 1.5(10.65) = −17.90 and 8.725 + 1.5(10.65) = +24.70. Only one observation falls outside, the −21.40% quarter, and it is plotted as its own point. The construction tests nothing. It is a convention for making the eye stop on a value worth investigating.