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what do lines that double back on themselves mean box plots

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cgenes on 13 May 2019
Commented: mura0087 on 24 May 2021
i'm using anova1 and it produces plots like this
it's a dumb question: but what do the lines mean that double back on themselves? red line in median i thin whiskers are outliers and the boxes should be 75 and 25 percentiles i thought

Accepted Answer

Adam Danz
Adam Danz on 13 May 2019
Edited: Adam Danz on 5 Jan 2021
It's not a dumn question. Matlab's documentation can be thin when it comes to statistical methods - even when those methods are engrained into their built-in functions.
The notches on the sides of a box plot can be interpreted as a comparison interval around the median values. The height of the notch is the median +/- 1.57 x IQR/sqrt(n) where IQR is the interquartile range defined by the 25th and 75th percentiles and n is the number of data points [1]. If the notches of two boxes do not overlap, there is strong evidence that their medians "significantly differ"[2], a phrase that is working its way out of statistics [3].
The reason why your notches extend beyond the 25th and/or 75th percentiles is due to the uncertainty of the true median value. This often happens if your sample size is small (because you're dividing by the sqrt(n) to calculate the notch height).
image above: [2]
To apply that to your box plots:
Update: a pictoral explanation of boxplot notches has been added to Matlab's documentation [4,5].

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