I keep tripping over box plots like they’re tiny skateboards. I think I get the five-number summary idea, but the whiskers are messing with me: do they always stretch all the way to the actual min and max, or do they stop at 1.5×IQR from the quartiles and leave the faraway points as dots? If I’m only shown a box plot image with no caption, how can I tell which convention was used? Also, when there’s an even number of data points, are Q1 and Q3 the medians of the halves including the overall median, or excluding it-and how much does that choice shift the edges of the box? If two box plots have the same median but very different whiskers, is it fair to say one dataset is clearly more spread out, or is that a trap? Right now box plots feel like a bento box where the noodles sometimes count as whiskers and sometimes as rebellious outliers. Could someone explain the rules I should assume and how to read these consistently?
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3 Responses
Whiskers stop short when the plot uses Tukey’s rule: they extend only to the most extreme data within 1.5×IQR of Q1 and Q3, and any points beyond that are shown as separate markers. Another common convention is to run whiskers all the way to the sample minimum and maximum. If there’s no caption, look for outlier points: when you see dots beyond the whiskers, it’s usually the 1.5×IQR version; if there are no dots and the whiskers touch the extremes, that must be the min–max style. Quartiles add another layer: there are multiple definitions in use. With an even number of observations, a classroom rule takes Q1 and Q3 as the medians of the lower and upper halves including the overall median, while software often uses interpolated quantiles; this changes the box edges a little but not much in practice. If n is odd, some methods still include the median in both halves, others don’t. When two box plots share the same median but have very different whiskers, it is fair to say one set looks more spread out; longer whiskers usually signal a larger standard deviation. Still, focus on the box length (the IQR) for the core spread, because long whiskers can be driven by a few extreme values while the middle 50% stays tight.
Great question-box plots come in two main flavors, which is why the whiskers sometimes “stop short.” In the Tukey (most common) convention, the box spans Q1 to Q3, and whiskers extend only to the most extreme data points within 1.5×IQR of each quartile; anything beyond is shown as individual points (outliers). In the five-number-summary convention, the whiskers go all the way to the actual minimum and maximum, with no outlier dots. If you’re given only an image, dots beyond the whiskers are a strong clue it’s the Tukey rule; without dots, you can’t be certain, so read the box (IQR) first and treat whiskers cautiously. About quartiles: there are multiple definitions; with an even sample size, “including vs excluding the median” makes no difference to the halves, but different percentile algorithms (simple medians-of-halves vs interpolated quartiles) can nudge Q1/Q3 slightly-typically by at most about one data step-so the box edges can shift a little. A quick example: data 1, 2, 2, 3, 4, 5, 6, 20. Median = (3+4)/2 = 3.5; lower half 1,2,2,3 gives Q1 = (2+2)/2 = 2; upper half 4,5,6,20 gives Q3 = (5+6)/2 = 5.5; IQR = 3.5. Tukey fences are 2 − 1.5×3.5 = −3.25 and 5.5 + 1.5×3.5 = 10.75, so the upper whisker stops at 6 and 20 is a plotted outlier; under the min–max rule the upper whisker would reach 20. Finally, when two box plots share the same median, call one “more spread out” only if its box (IQR) is clearly wider; longer whiskers alone may just mean heavier tails or a few outliers, not uniformly greater variability.
Short answer: many box plots use Tukey’s rule (whiskers go to the last data point within 1.5×IQR; anything farther is a dot), but some go to min/max-if you don’t see dots, you can’t tell without a caption, so assume nothing; quartiles also vary (most split halves excluding the overall median), which only shifts the box by about one data rank, and you can call one dataset “more spread out” only if both plots use the same rule and scale. Good refresher here: https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/a/box-and-whisker-plots-review – Hope this helps!