Chief Executive Officer summary
The history is much longer and more complicated than many people think it is.
Executive summary
The history of what Tukey called box plots is tangled up with that of what are now often called dot or strip plots (dozens of other names) and with representations of the empirical quantile function.
Box plots in widely current forms are best known through the work of John Wilder Tukey (1970, 1972, 1977).
But the idea of showing the median and quartiles as basic summaries -- together often but not always with dots showing all values -- goes back at least to the dispersion diagrams (many variant names) introduced by the geographer Percy Robert Crowe (1933). These were staple fare for geographers and used in many textbooks as well as research papers from the late 1930s on.
Bibby (1986, pp.56, 59) gave even earlier references to similar ideas taught by Arthur Lyon Bowley (later Sir Arthur) in his lectures about 1897 and to his
recommendation (Bowley, 1910, p.62; 1952, p.73) to use minimum and maximum and 10, 25, 50, 75 and 90% points as a basis for graphical summary.
Range bars showing extremes and quartiles are often attributed to Mary Eleanor Spear (1952) but in my reading fewer people cite Kenneth W. Haemer (1948). Haemer's articles on statistical graphics in the American Statistician around 1950 were inventive and have critical bite and remain well worth re-reading. (Many readers will be able to access them through jstor.org.) In contrast Spear's books (Spear 1969 is a rehash) were accessible and sensible but deliberately introductory rather than innovative or scholarly.
Variants of box plots in which whiskers extend to selected percentiles are commoner than many people seem to think. Again, equivalent plots were used by geographers from the 1930s on.
What is most original in Tukey's version of box plots are first of all criteria for identifying points in the tails to be plotted separately and identified as deserving detailed consideration -- and as often flagging that a variable should be analysed on a transformed scale. His 1.5 IQR rule of thumb emerged only after much experimentation. It has mutated in some hands to a hard rule for deleting data points, which was never Tukey's intent. A punchy, memorable name -- box plot -- did no harm in ensuring much wider impact of these ideas. Dispersion diagram in contrast is rather a dull and dreary term.
The fairly long list of references here is, possibly contrary to appearances, not intended to be exhaustive. The aim is just to provide documentation for some precursors and alternatives of the box plot. Specific references may be helpful for detailed queries or if they are in to close to your field. Conversely, learning about practices in other fields can be salutary. The graphical -- not just cartographical -- expertise of geographers has often been underestimated.
More details
Hybrid dot-box plots were used by Crowe (1933, 1936), Matthews (1936), Hogg (1948),
Monkhouse and Wilkinson (1952), Farmer (1956), Gregory (1963), Hammond and McCullagh (1974), Lewis (1975), Matthews (1981), Wilkinson (1992, 2005), Ellison (1993, 2001), Wild and Seber (2000), Quinn and Keough (2002), Young et al. (2006) and Hendry and Nielsen (2007) and many others. See also Miller (1953, 1964).
Drawing whiskers to particular percentiles, rather than to data points within so many IQR of the quartiles, was emphasised by Cleveland (1985), but anticipated by Matthews (1936) and Grove (1956) who plotted the interoctile range, meaning between the first and seventh octiles, as well as the range and interquartile range. Dury (1963), Johnson (1975), Harris (1999), Myatt (2007), Myatt and Johnson (2009, 2011) and Davino et al. (2014) showed means as well as
minimum, quartiles, median and maximum. Schmid (1954) showed summary graphs with median, quartiles and 5 and 95% points. Bentley (1985, 1988), Davis (2002), Spence (2007, 2014) and Motulsky (2010, 2014, 2018) plotted whiskers
to 5 and 95% points. Morgan and Henrion (1990, pp.221, 241), Spence (2001, p.36), and Gotelli and Ellison (2004, 2013, pp.72, 110, 213, 416) plotted whiskers to 10% and 90% points. Harris (1999) showed examples of both 5 and 95% and 10 and 90% points. Altman (1991, pp.34, 63) and Greenacre (2016) plotted whiskers to 2.5% and 97.5% points. Reimann et al. (2008, pp.46-47) plotted whiskers to 5% and 95% and 2% and 98% points.
Parzen (1979a, 1979b, 1982) hybridised box and quantile plots as quantile-box plots. See also (e.g.) Shera (1991), Militký and Meloun (1993), Meloun and Militký (1994). Note, however, that the quantile box plot of Keen (2010) is just a box plot with whiskers extending to the extremes. In contrast, the quantile box plots of JMP are evidently box plots with
marks at 0.5%, 2.5%, 10%, 90%, 97.5%, 99.5%:
see Sall et al. (2014, pp.143-4).
Here are some notes on variants of quantile-box plots.
(A) The box-percentile plot of Esty and Banfield (2003) plots the same information differently, plotting data as continuous lines and producing a symmetric display in which the vertical axis shows quantiles and the horizontal axis shows not plotting position $p$, but both
min($p, 1 - p$) and its mirror image $-$min($p, 1 - p$). Minor detail: in their paper plotting positions are misdescribed as "percentiles". See also Martinez et al. (2011, 2017), which perpetuates that confusion.
The idea of plotting min($p, 1 - p$) (or its percent equivalent) appears independently in (B) "mountain plots" (Krouwer 1992; Monti 1995; Krouwer and Monti 1995; Goldstein 1996) and in (C) plots of the "flipped empirical distribution function" (Huh 1995). See also Xue and Titterington (2011) for a detailed analysis of folding a distribution function at any quantile.
From literature seen by me, it seems that none of these threads -- quantile-box plots or the later variants (A) (B) (C) -- cites each other.
!!! as at 3 October 2018 details for some references need to be supplied in the next edit.
Altman, D.G. 1991.
Practical Statistics in Medical Research.
London: Chapman and Hall.
Bentley, J.L. 1985.
Programming pearls: Selection.
Communications of the ACM 28: 1121-1127.
Bentley, J.L. 1988.
More Programming Pearls: Confessions of a Coder.
Reading, MA: Addison-Wesley.
Bibby, J. 1986.
Notes Towards a History of Teaching Statistics.
Edinburgh: John Bibby (Books).
Bowley, A.L. 1910.
An Elementary Manual of Statistics.
London: Macdonald and Evans. (seventh edition 1952)
Cleveland, W.S. 1985. Elements of Graphing Data.
Monterey, CA: Wadsworth.
Crowe, P.R. 1933.
The analysis of rainfall probability: A graphical method and its application to European data.
Scottish Geographical Magazine 49: 73-91.
Crowe, P.R. 1936.
The rainfall regime of the Western Plains.
Geographical Review 26: 463-484.
Davis, J.C. 2002.
Statistics and Data Analysis in Geology.
New York: John Wiley.
Dickinson, G.C. 1963.
Statistical Mapping and the Presentation of Statistics.
London: Edward Arnold. (second edition 1973)
Dury, G.H. 1963.
The East Midlands and the Peak.
London: Thomas Nelson.
Farmer, B.H. 1956.
Rainfall and water-supply in the Dry Zone of Ceylon.
In Steel, R.W. and C.A. Fisher (eds)
Geographical Essays on British Tropical Lands.
London: George Philip, 227-268.
Gregory, S. 1963. Statistical Methods and the Geographer.
London: Longmans. (later editions 1968, 1973, 1978; publisher later Longman)
Grove, A.T. 1956. Soil erosion in Nigeria. In Steel, R.W. and C.A. Fisher (eds)
Geographical Essays on British Tropical Lands.
London: George Philip, 79-111.
Haemer, K.W. 1948. Range-bar charts.
American Statistician 2(2): 23.
Hendry, D.F. and B. Nielsen. 2007.
Econometric Modeling: A Likelihood Approach.
Princeton, NJ: Princeton University Press.
Hogg, W.H. 1948.
Rainfall dispersion diagrams: a discussion of their advantages and
disadvantages.
Geography 33: 31-37.
Ibrekk, H. and M.G. Morgan. 1987.
Graphical communication of uncertain quantities to nontechnical people.
Risk Analysis 7: 519-529.
Johnson, B.L.C. 1975.
Bangladesh. London: Heinemann Educational.
Keen, K.J. 2010.
Graphics for Statistics and Data Analysis with R.
Boca Raton, FL: CRC Press. (2nd edition 2018)
Lewis, C.R. 1975.
The analysis of changes in urban status: a case study in Mid-Wales and the
middle Welsh borderland.
Transactions of the Institute of British Geographers
64: 49-65.
Martinez, W.L., A.R. Martinez and J.L. Solka. 2011.
Exploratory Data Analysis with MATLAB.
Boca Raton, FL: CRC Press.
Matthews, H.A. 1936.
A new view of some familiar Indian rainfalls.
Scottish Geographical Magazine 52: 84-97.
Matthews, J.A. 1981.
Quantitative and Statistical Approaches to Geography: A Practical Manual.
Oxford: Pergamon.
Meloun, M. and J. Militký. 1994.
Computer-assisted data treatment in analytical chemometrics.
I. Exploratory analysis of univariate data.
Chemical Papers 48: 151-157.
Militký, J. and M. Meloun. 1993.
Some graphical aids for univariate exploratory data analysis.
Analytica Chimica Acta 277: 215-221.
Miller, A.A. 1953.
The Skin of the Earth.
London: Methuen. (2nd edition 1964)
Monkhouse, F.J. and H.R. Wilkinson. 1952.
Maps and Diagrams: Their Compilation and Construction.
London: Methuen. (later editions 1963, 1971)
Morgan, M.G. and M. Henrion. 1990.
Uncertainty: A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis.
Cambridge: Cambridge University Press.
Myatt, G.J. 2007.
Making Sense of Data: A Practical Guide to Exploratory Data Analysis and Data Mining.
Hoboken, NJ: John Wiley.
Myatt, G.J. and Johnson, W.P. 2009.
Making Sense of Data II: A Practical Guide to Data Visualization, Advanced Data Mining Methods, and Applications.
Hoboken, NJ: John Wiley.
Myatt, G.J. and Johnson, W.P. 2011.
Making Sense of Data III: A Practical Guide to Designing Interactive Data Visualizations.
Hoboken, NJ: John Wiley.
Ottaway, B. 1973.
Dispersion diagrams: a new approach to the display of carbon-14 dates.
Archaeometry 15: 5-12.
Parzen, E. 1979a.
Nonparametric statistical data modeling.
Journal, American Statistical Association 74: 105-121.
Parzen, E. 1979b.
A density-quantile function perspective on robust estimation.
In Launer, R.L. and G.N. Wilkinson (eds) Robustness in Statistics.
New York: Academic Press, 237-258.
Parzen, E. 1982.
Data modeling using quantile and density-quantile functions.
In Tiago de Oliveira, J. and Epstein, B. (eds)
Some Recent Advances in Statistics. London: Academic Press,
23-52.
Quinn, G.P. and M.J. Keough. 2002.
Experimental Design and Data Analysis for Biologists.
Cambridge: Cambridge University Press.
Reimann, C., P. Filzmoser, R.G. Garrett and R. Dutter. 2008.
Statistical Data Analysis Explained: Applied Environmental Statistics with R.
Chichester: John Wiley.
Sall, J., A. Lehman, M. Stephens and L. Creighton. 2014.
JMP Start Statistics: A Guide to Statistics and Data Analysis Using JMP.
Cary, NC: SAS Institute.
Shera, D.M. 1991.
Some uses of quantile plots to enhance data presentation.
Computing Science and Statistics 23: 50-53.
Spear, M.E. 1952. Charting Statistics.
New York: McGraw-Hill.
Spear, M.E. 1969. Practical Charting Techniques.
New York: McGraw-Hill.
Tukey, J.W. 1970.
Exploratory data analysis. Limited Preliminary Edition. Volume I.
Reading, MA: Addison-Wesley.
Tukey, J.W. 1972.
Some graphic and semi-graphic displays.
In Bancroft, T.A. and Brown, S.A. (eds)
Statistical Papers in Honor of George W. Snedecor.
Ames, IA: Iowa State University Press, 293-316.
(also accessible at http://www.edwardtufte.com/tufte/tukey)
Tukey, J.W. 1977.
Exploratory Data Analysis.
Reading, MA: Addison-Wesley.
Wild, C.J. and G.A.F. Seber. 2000.
Chance Encounters: A First Course in Data Analysis and Inference.
New York: John Wiley.
It is possible to manipulate this plot and change what the characteristics represent. Everything I'm going to write represents the standard depiction of this type of plot
Box plot basics
This plot tells you about the distribution, variance, and allows you to compare groups in terms of distribution and variance.
For a box plot, the horizontal line within each box is the median of the group.
It is a safe bet that the X in each box represents the average value of the group. The box size represents the range of values in the interquartile range.
If you put the values in the group in order by value, and split the group evenly four ways, you would create quantiles. The middle two quantiles are the interquartile range. The quartile is the single value at the threshold between quantiles.
The legs or the lines that extend through the center of each box, with perpendicular end caps, represent the normally distributed range of the group. When you see values or asterisks above or below these lines, these are considered outliers.
The method used to determine the outlying values below the line is to take the lowest value in the 'box' (the value that represents the first quartile) and subtract the 1.5 * IQR. At the top end, you add 1.5 * IQR to the largest value in the 'box', the value that represents the 3rd quartile.
Your Box Plot
Where's the Box?
Looking at the visual you provided, the first group, Absence/Veluwe 2 - doesn't have a box. It is probably due to very few values in the group or very little or no difference between the values in the group (relative to the other groups).
If the values in this group were 14.11104, 14.156493680, and 14.3496 there would be no difference reflected here because the other groups have much larger differences in their values (relatively speaking).
Box Size and Variance: Part 1
Do you see how the boxes vary in size? This may indicate that the variance is heterogeneous with enough significance that it could cause a problem in analysis methods that rely on least-squares methods. The outliers would cause issues as well.
Distribution
When you see the mean and median are approximately equal then your data is leaning towards a normal distribution. Combine that with arms that are approximately the same length and the mean and median centered in the box and normal distribution is extremely likely.
Again - relative to the data you're working with - if all the numbers are in the trillions, differences of 100 aren't a big deal - but a 100 here - whew!-
The legs of Presence-Luttenbergerven are pretty approximately equal in length. The median is approximately centered, the average looks pretty close - it's a pretty safe bet that the data in this group is normally distributed.
Box Size and Variance: Part 2
When you look at Presence-Veluwe 2 The legs are a little uneven, but it is a pretty safe bet that this group is normally distributed, as well.
When you look at all three groups that represent Presence, the ranges are about 20, 20, and 15 - it's a pretty safe bet that there is homogeneity - or equal variance between the groups.
The box in this plot represents the center values, so if the range is fairly even, but the boxes are not centered about the arms, the median and mean are uncentered, or vary quite a bit this could be due to variance, distribution, or both.
Please note, I'm not an artist!
Best Answer
It indicates that the 25th percentile is about 21, 75th percentile about 30.5. And the lower and upper limits of the notch are about 18 and 27.
A common reason is that your distribution is skewed or sample size is low. The notch's boundary is based on:
$median \pm 1.57 \times \frac{IQR}{\sqrt{n}}$
If the distance between median and the 25th percentile and the distance between median and the 75th percentile are extremely different (like the one at the right) and/or the sample size is low, the notch will be wider. If it's wide enough that the notch boundary is more extreme than the 25th and 75th percentiles (aka, the box), then the notched box plot will display this "inside out" shape.