Quartiles, Percentiles, and the Five-Number Summary

The median tells you the middle value, but it says nothing about how spread out the rest of the data is. Two data sets can share the same median and look completely different otherwise — one tightly packed, one wildly scattered. Quartiles fix that by splitting the sorted data into four chunks of roughly equal size, so you see where the bulk of the data sits, not just its center.

To find quartiles, sort the data first — this step never skips. The median splits the data into a lower half and an upper half. Q1 is the median of the lower half; Q3 is the median of the upper half. If the overall median is a value, you generally leave it out of both halves before taking their medians. Put the minimum, Q1, median, Q3, and maximum together and you get the five-number summary, a five-number sketch of the whole distribution's shape and spread. Percentiles generalize this idea: the kth percentile is the value below which k percent of the data falls. The 90th percentile of a data set means 90 percent of the values in that set are at or below it — it has nothing to do with correctness or a score out of 100.

This topic sets up everything that follows in the unit. The interquartile range, IQR = Q3 - Q1, measures spread and feeds directly into the rule for flagging outliers. The five-number summary is also exactly what a boxplot draws, so once you can find these five numbers by hand, reading or sketching a boxplot is just connecting dots.