Box-plot features from five-number summaries

The race's five-number summary is minimum 21, lower quartile 24, median 29, upper quartile 35 and maximum 58 minutes, in that order. Values can match but cannot decrease. The median is the third entry. For an even count, average the two middle observations; the result need not be recorded. Here we use numerical summaries and descriptions of box-plot features, without assessing drawing or reading a completed diagram.

A five-number summary lists minimum, Q1, median, Q3 and maximum in that order, so its third entry is the median. In this elementary box-plot convention, Q1 and Q3 are box ends, the median marks its line, and the extremes are whisker tips. IQR = Q3 - Q1; range = maximum - minimum. For the race, these are 35 - 24 = 11 and 58 - 21 = 37 minutes. Coincident values allow a zero-length whisker or a median line at a box end.

Last year's numerical summary was 26, 32, 36, 40, 50 minutes. This year's median is 29 rather than 36, so its median finishing time is seven minutes shorter. Last year's IQR is 40 - 32 = 8, smaller than this year's 11. Compare median location and middle spread separately. Equal medians can accompany different IQRs. Lower race times mean quicker finishes; higher test-score medians mean higher typical scores. An IQR does not establish every extreme result.

For the junior summary 12, 14, 16, 24, 30 minutes, the lower central span is median - Q1 = 2 and the upper is Q3 - median = 8. Divide 8 by 2: the upper span is four times as wide. A wider upper span can suggest right skew; a wider lower span can suggest left skew. Neither establishes tail lengths or the full distribution's shape. A centred median equals (Q1 + Q3) / 2, but use that calculation only when centring is given.

This helper shows the central gaps on a scale, not a box-and-whisker diagram.

Quartiles mark approximate quarter ranks, not guaranteed counts in value intervals. In an ideal four-quarter rank model, the two outer quarters make half. Actual counts depend on list size, ties and boundaries. A bare five-number summary omits the group size, individual gaps, clusters and mean. Matching summaries need not mean matching observations. A longer whisker describes a wider value interval, not a larger count or equal counts across groups.

For raw lists, use medians of ordered halves, excluding the central observation when the full count is odd. Average a half's middle pair when needed. Seven observations use positions 1, 2, 4, 6, 7 for the summary. For n = 4k distinct observations, Q1 averages positions k and k + 1; Q3 averages 3k and 3k + 1. Exactly k observations are strictly above Q3 and 2k strictly between the quartiles. Ties can invalidate these exact counts.

Check that units match before comparing summaries. Multiply hours by 60 to obtain minutes. A planned pair of box plots needs the same value units and the same units per centimetre of paper. Equal paper widths mean equal value widths only under the same scale. To convert a value span to a paper length, divide the span by the number of value units represented by one centimetre.

If every observation is multiplied by the same positive factor, every summary value and every difference is multiplied by that factor. This includes the median, IQR, range and whisker lengths. Multiplying all observations by 2 doubles their IQR. Adding the same constant instead moves the summary values but leaves these widths unchanged.

Symmetry of the observations requires matching values and counts at equal distances on either side of the centre. Equal quartile gaps do not establish that match. Even symmetric spacing of all five summary values is insufficient: unreported observations between the cuts can break the matching pairs. Check the raw list before claiming symmetry.

A left whisker's numerical length is Q1 - minimum; a right whisker's is maximum - Q3. Together with the two central spans, these lengths add to the full range. For the race, they are 3, 5, 6 and 23 minutes. Compare two spans by subtracting their lengths, or divide one length by another for a ratio. A zero length means coincident endpoints, not necessarily an empty data group.