Boxplots and Beyond III: Violin Plots

This post is the third in a series of four on boxplots and closely related data visualization techniques for comparing subsets of a dataset, or comparing different datasets that we hope or expect to be similarly distributed.  The previous two posts in this series have dealt with the basic boxplot, simple variations like log transformations and variable-width boxplots, and the procedures used to identify and represent outliers in a boxplot display, both traditionally and in the face of significant asymmetry.  All of the boxplot variations discussed so far retain the basic “box” structure, however, and this is one of the inherent limitations of this data display: their visual simplicity can hide a lot of important details about how the data values are distributed.  A much more flexible extension of the basic boxplot is the violin plot, constructed by combining the concept of the boxplot with that of nonparametric density estimates.  Both boxplots and nonparametric density estimates are discussed in Exploring Data, but the idea of putting them together into a single data display was not.  The violin plot discussed in this post is one way of achieving this combination, and my next post will discuss beanplots, also built from these same components.  Violin plots are supported in R both as part of the lattice graphics package and via the add-on package wvioplot.  Because I find it somewhat easier to use, this post discusses the second of these options.

I have previously discussed the Old Faithful geyser data frame faithful, included in the datasets package in the base R installation, in part because it provides a nice, real example of a dataset with a bimodal distribution of values.  In fact, this data frame includes two variables – the waiting times between successive eruptions discussed last time, and the duration of each eruption – and it turns out that both of these variables exhibit strongly bimodal distributions.  This point is illustrated clearly in the four plots shown above.  The upper two plots are scatter plots of the observed values of the variables themselves, while the bottom two plots are nonparametric density estimates generated via the density command in R, using its default options.  As in my previous discussion of the waiting time data from this dataset, the bimodal character of these variables is clear from the two pronounced peaks seen in these density estimates.  Further, this bimodal character is also clearly evident in the scatter plots of the data values themselves, particularly in the duration data which shows a gap between approximately 2.5 and 3.0 where there are almost no points at all.  The main reason for considering this example here is that bimodality represents one of those potentially significant distributional details that is generally lost in boxplots.  This point is clearly illustrated in the figure shown below, which presents side-by-side boxplots of the waiting time divided by 20 on the left, and the duration of the eruptions on the right.  (The waiting times were rescaled simply to make the two boxplots fit reasonably in the same figure.)  Both plots suggest significant distributional asymmetry, but neither one gives any hint of the strongly bimodal character of these variables.

For comparison, the two figures shown below were both generated using the wvioplot package, which generates plots that may be viewed as boxplots whose “boxes” have been curved to reflect the estimated distribution of values over the observed data range.  The plot on the left uses the default options for wvioplot, and one of the points of this example is that these default options are not always much more informative than the basic boxplot.  This is certainly the case here: neither of the left-hand violin plots is at all suggestive of the bimodality we know to be a key characteristic of these variables.  The right-hand violin plots represent are slightly customized, generated by specifying two options.  First, note that the default violin plots on the left are both clipped, exhibiting hard limits at the smallest and largest values that appear in the data.  Clipping is the default, but specifying “clip = F” overrides this behavior, which is not typical for nonparametric density estimates: instead of abruptly dropping to zero, these estimates typically decay smoothly to zero and, as a result, extrapolate somewhat beyond the observed data range.  Specifying “clip = F” causes the density estimates used in the violin plots to exhibit this smooth decay behavior.

The second, more important, difference between the two sets of violin plots shown in the above pair of figures is that the right-hand pair specifies a different bandwidth parameter for these nonparametric density estimators.  In Chapter 8 of Exploring Data, I discuss the trade-offs inherent in selecting the bandwidth parameter for nonparametric density estimators: making the bandwidth too small yields a high-variance, “noisy” estimate with a lot of spurious “details,” while making the bandwidth too large yields low-variance but highly biased estimates that can miss critical details that are present.  For the violin plots generated by the wvioplot package, this bandwidth is specified by the adjust parameter, with the default value “adjust = 3” giving the results shown on the left.  The plots on the right were generated using the smaller value, “adjust = 1”, which reduces the bandwidth of the density estimator enough to see the bimodal character of the two variables.

One of the key points of this example is that, while the added capability inherent in the violin plot relative to the simpler boxplot can convey extremely useful details that we might otherwise miss, we can’t always rely on the default settings for the procedure to show us these details.  The basic source of this difficulty is that more complex procedures like nonparametric density estimators (at least, more complex compared with the five-number summary on which the basic boxplot is based) have tuning parameters, and simply using the default options for these parameters doesn’t always lead to the results we want.  For this reason, it is a good idea to spend a bit of time exploring the key options in using these procedures.

Boxplots and Beyond – Part II: Asymmetry

In my last post, I discussed boxplots in their simplest forms, illustrating some of the useful options available with the boxplot command in the open-source statistical software package R.  As I noted in that post, the basic boxplot is both useful and popular, but it does have its limitations.  One of those limitations is that the standard boxplot outlier rule is not appropriate for highly asymmetric data.  Specifically, the standard rule declares points to be outliers if they lie a fixed distance – typically, 1.5 times the interquartile distance (IQD) – beyond the quartiles.  While this rule is appropriate for symmetric, approximately Gaussian data distributions, highly asymmetric situations call for an outlier detection rule that treats upward-outliers and downward-outliers differently.  For a strongly right-skewed distribution, for example, flagging any point more than 1.5 IQD’s above the upper quartile may be too liberal, declaring too many points to be upward-outliers, while flagging any point more than 1.5 IQD’s below the lower quartile may be too conservative, declaring too few points to be downward-outliers.  To handle these cases, it is necessary to incorporate some measure of skewness into the outlier detection rule. 

One of the points I discuss in Chapter 7 of Exploring Data is that the standard skewness estimator - defined as a normalized third moment - is extremely sensitive to outliers, worse even than the standard deviation in this respect.   This point is illustrated in the above figure, which gives a standard boxplot summary comparing four different skewness estimates for two different cases.  The boxplots on the left present results for these skewness measures each applied to 1000 statistically independent standard Gaussian data samples, each of length n = 100, while the boxplots on the right present the corresponding results for exactly the same data sequences, but with each Gaussian data sample contaminated by a single additive outlier, lying 8 standard deviations above the mean.  The boxplots designated “Mo” in these comparisons correspond to the standard moment-based estimator, while the other boxplots correspond to three other skewness estimators, discussed in the next paragraph.  Two points are immediately evident from these results: first, that the moment estimator exhibits much higher variability than the other three estimators considered here, and second, that the moment estimator is extremely outlier-sensitive.  In particular, note that a single outlier in a dataset of size 100 causes the median estimate to shift from about zero, corresponding to the correct nominal result, to a large enough value to suggest asymmetry comparable to the J-shaped exponential distribution (for which the skewness is 2), all on the basis of a single anomalous data point.

Two of the other three skewness estimators compared in these boxplots are Hotelling’s skewness measure, designated “Ho,” and Galton’s skewness measure, designated “Ga.”  Both of these alternatives to the standard moment-based skewness measure are discussed in Chapter 7 of Exploring Data, which presents similar comparisons to those presented here for these three asymmetry measures.  Hotelling’s measure is equal to the difference between the mean and the median of the data, divided by the estimated standard deviation, and it can be shown to lie between -1 and +1.  The boxplot comparisons presented here show that this measure is much less outlier-sensitive than the standard skewness measure, although this estimate does exhibit a slight positive bias (slight is the operative word here: the mean value for the contaminated “Mo” estimates is 2.36, while that for the “Ho” estimate is 0.05).  Galton’s skewness measure is based on sample quartiles and is closely related to the fourth measure discussed next.  Specifically, both of these measures are based on the following function: let x and y be any two values from the data sample satisfying the condition that x is smaller than the sample median m, which is in turn smaller than y.  The function h(x,y) on which these measures are based is defined as the ratio of two computed values: the numerator is (y – m) – (m – x), and the denominator is y – x.  If we take x as the lower quartile and y as the upper quartile, we obtain Galton’s skewness measure, designated “Ga” in the boxplots.  The other skewness measure is the medcouple, defined as the median of all values of h(x,y) computed from points satisfying the condition that x < m < y.  Note that both the Galton and medcouple measures also lie between -1 and +1, since this is true for h(x,y) for any admissible values of x and y.  Detailed comparison of the “Ga” and “Mo” boxplots shows that both of these estimators exhibit small bias – i.e., their median values are both approximately zero, as they should be – and that the medcouple exhibits a slightly smaller range of variation than Galton’s skewness measure does.

Procedures in R to compute the standard skewness measure, Hotelling’s measure, and Galton’s measure are available from the companion website for Exploring Data.  The medcouple is not discussed in the book, but it is available in the R add-on package robustbase, which also includes a procedure to construct the adjusted boxplot described next.  There, rather than using 1.5 interquartile distances to set both upper and lower outlier limits as in the standard boxplot rule, the adjbox procedure in the robustbase package makes this distance depend on the medcouple value computed from the data.  Specifically, both outlier detection limits have the same general form, but different specific parameters: in both cases, the nominal distance of 1.5 IQD’s is multiplied by a scale factor computed from the medcouple value.  For lower outliers, this scale factor is exp(-3.5 Mc), while for upper outliers, this scale factor is exp(+4Mc).  A detailed discussion of this outlier detection rule and the rationale behind it is presented both in the paper, “An Adjusted Boxplot for Skewed Distributions,” by Ellen Vanderviere and Mia Huber, published in the proceedings of the 2004 COMPSTAT symposium, and from the technical report cited in the R documentation for the adjbox procedure.

The above figure compares two sets of boxplots for the UScereal dataset discussed last time, from the MASS package in R.  Specifically, both boxplots summarize the reported potassium content for breakfast cereals from 6 different manufacturers, each identified by a one-letter designation.  Both boxplots are generated using the log = “y” and varwidth options to obtain boxplots whose width is proportional to the square root of the number of records in each subsample.  The difference in these two sets of boxplots is that the one on the left uses the standard outlier identification algorithm, which is the default for the boxplot command in base R, while the boxplot series on the right uses the modified outlier detection rule just described, based on the medcouple skewness measure and available via the adjbox command in the robustbase add-on package.  The points declared outliers for each boxplot are shown as solid circles for emphasis (simply specify “pch = 16” in either boxplot command), and these points illustrate the difference between the two boxplot results.  Specifically, the left-hand group of boxplots only show upward outliers, one for manufacturer G (General Mills) and two for manufacturer K (Kellogs), while the right-hand boxplots only show downward outliers, three for General Mills, one for Kellogs, and one for Quaker Oats (manufacturer Q).  The question remains which of these conclusions is more reasonable, and this question will be revisited in subsequent posts using two additional boxplot extensions: violin plots and bean plots, both of which incorporate nonparametric density estimates to give a more detailed picture of the data.  For now, it is enough to note three things: first, an alternative to the standard boxplot is available that provides distinct detection rules for upper and lower outliers; second, that this alternative approach often finds different outliers than the standard boxplot rule does; and third, there is evidence to suggest that this modified boxplot does indeed perform better in the face of significant distributional asymmetry.  For a more detailed discussion of this point, refer to the technical report cited above.

Finally, it is worth emphasizing that the robustbase package contains a lot more than just the adjbox procedure discussed here, including multivariable outlier detection procedures like adjOutlyingness, robust multivariate location and scale estimation procedures like covMcd, and robust fitting procedures like lmrob and glmrob for linear models and generalized linear models (specifically, a robust logistic regression procedure for binomial data, and a robust Poisson regression procedure for count data), among others.  The biggest drawback of this package is that the documentation is incomplete, so it is important to check out the references cited there to get a clear idea of what some of these procedures do and how they do it.  That said, it is important to emphasize two points.  First, if R were a commercially supported software package, this level of documentation would be inexcusable, but R is not a commercial product: it is the freely-available end result of the loosely coordinated efforts of a very large group of unpaid volunteers.  The second point is that, despite the limitations of the documentation, the robustbase package provides implementations of procedures that could, in principle, be built on the basis of their published descriptions by anyone who wanted to.  The experience of doing that would be educational and intellectually rewarding, but it would also be a lot of work that most of us would simply never do, so we would not have any of the neat goodies that the authors have made available to us.  In the end, while I would strongly encourage them to finish the documentation, more importantly, I offer my heartfelt appreciation for their releasing the package so I can use it.

 

Boxplots and Beyond - Part I

Boxplots are a simple and reasonably popular way of summarizing the range of variation of a real-valued variable across different subsets of data.  Typical examples might include diastolic blood pressure across a group of patients, broken down by gender and smoking status, or the breaking strength of material samples broken down by material type and manufacturing process.  Like all simple summaries, boxplots have their limitations, and these have motivated a number of useful variations on the theme.  The basic boxplot is introduced in Chapter 1 of my book Exploring Data in Engineering, the Sciences and Medicine, and it is used fairly extensively throughout the rest of the book.  This post is the first in a series of four that briefly describe some useful extensions of the basic boxplot that are not covered in the book, including variable width boxplots, boxplots based on a robust asymmetry measure, violin plots, and beanplots, all easily constructed using procedures from the open-source R programming language.

As an example, the above figure shows the simplest form of the boxplot summary, constructed from the UScereal dataset included in the MASS package in R, which characterizes 65 cereals from 6 different manufacturers, based on the information included on their FDA-mandated package labels.  The six manufacturers are abbreviated "G" for General Mills, "K" for Kellogs, "N" for Nabisco, "P" for Post, "Q" for Quaker Oats, and "R" for Ralston Purina, and the above boxplot summarizes the potassium content for one cup of each cereal, based on the label information.  (The metadata for this dataset list the units for this quantity as "grams," but it seems more likely that the units are actually milligrams; otherwise, the largest value in this dataset would be almost a kilogram, which seems like an awful lot of potassium for one cup of breakfast cereal.  This highlights one of the other points I discuss in Exploring Data, that metadata is not always accurate.)  For those not familiar with the basic boxplot, it is based on Tukey's 5-number summary for each data subset, plotted as follows:

1. the sample minimum, defining the horizontal line at the bottom of each plot;
2. the lower quartile, defining the lower limit of the box in each figure;
3. the sample median, represented by the heavy line inside each box;
4. the upper quartile, defining the upper limit of each box;
5. the sample maximum, defining the horizontal line at the top of each plot.

Although the figure shown above represents the simplest form of the boxplot, it is probably not the most commonly used, and it is not the default boxplot generated in R.  Instead of always marking the sample minimum and maximum with horizontal lines as in the plot above, it is more common to put these lines at the limits of the nominal range of the data inferred from the upper and lower quartiles, denoting any points that fall outside this range as open circles.  This is illustrated in the boxplot shown below, which was constructed from the same dataset as that shown above, but using the default boxplot command in R.

In Chapter 7 of Exploring Data, I discuss the problem of outlier detection, using either the well-known "three-sigma edit rule" (which, ironically enough, often fails to detect outliers because they inflate the standard deviation used to classify points as outliers, making this classification less likely), or the more robust Hampel identifier that is analogous but replaces the outlier-sensitive mean with the outlier-resistant median, and replaces the extremely outlier-sensitive standard deviation with the highly robust MADM scale estimate.  Unfortunately, I do not discuss the boxplot outlier detection rule in Exploring Data, but I do discuss it at some length in my other book, Mining Imperfect Data.  The basic idea is that the interquartile distance (IQD) - i.e., the difference between the upper and lower quartiles, corresponding to the width of the central box in the boxplot - is used to determine the nominal range of data variation.  Like the MADM scale estimate, the IQD is less outlier-sensitive than the standard deviation, so it provides a more reliable basis for detecting outliers.  In the typical boxplot representation - like the example shown above - the upper end of the nominal data range is defined as the upper quartile plus 1.5 times the IQD, and the lower end of the nominal data range is defined as the lower quartile minus 1.5 times the IQD.  If the observed range of data values falls within these limits, the horizontal lines at the top and bottom of the boxplots correspond to the sample maximum and sample minimum as described above.  If any points fall outside this range, however, they are plotted as open circles and the horizontal lines correspond to the most extreme data values that fall within this nominal range.  In cases where the data distribution is markedly asymmetric, this approach may not be appropriate, and the next post in this series will describe a better alternative from the R package robustbase.

One of the topics I discuss at some length in Exploring Data is the use of transformations in exploratory data analysis, both to simplify some problem formulations and to potentially give more informative views of a dataset.  One transformation that is often extremely useful when the data values span a wide range is the logarithm.  The plot above shows the same boxplot as before, but now with the log="y" plotting option specified to give a logarithmic scaling to the y-axis.  Note that the appearance of this plot is quite different, giving greater visual emphasis to the range from 20 to 500 where most of the data values lie, while the previous plot gave equal emphasis to the range of values above 500, occupied by only a few points.  Logarithmic transformations are not always useful or even feasible - the R boxplot command returns an error message and generates no plot if the data includes zeros or negative values - but in the right circumstances they can be very informative.

Finally, another extremely useful boxplot option is varwidth="T" which causes the widths of the components of a boxplot to be drawn with variable size, larger for boxplots based on more data and smaller for boxplots based on less data.  The above figure shows the effect of adding this option to the previous boxplot.  Here, the width of each boxplot is drawn proportional to the square root of the number of data values on which it is based.  Thus, it is clear from this boxplot that two of the manufacturers (G and K) are more widely represented in these 65 cereals than the others, while manufacturer N has the smallest representation.  (In fact, of the 65 cereals, the representations of the different manufacturers are 22 for G, 21 for K, 3 for N, 9 for P, and 5 each for Q and R.)  Note that the results of designed experiments are often fairly evenly balanced, so that the different subsets being compared are of about the same size.  Where the variable width option becomes particularly useful is in the analysis of historical datasets that were not collected on the basis of a designed experiment, where the sizes of the different subsets compared may vary substantially from one to another.  The practical advantage of variable width boxplots like the one shown above is that they draw your attention to any substantial size differences that may exist between the data subsets being compared, differences that may not be obvious from the outset.  The rationale for the square root scaling used by the varwidth option is that the variability of asymptotically normal estimators - such as the median and the upper and lower quartiles - exhibits a standard deviation that decreases inversely with the square root of the sample size.  Thus, the square root of the sample size may be taken as a rough measure of "strength of evidence" in comparing these estimators or displays like boxplots that are based on these estimators.  Also, in characterizing very large datasets it may happen that the sizes of the subsets being compared vary by several orders of magnitude, and in such cases, making the boxplot widths proportional to the square root of sample size rather than proportional to sample size itself gives a much more reasonable display.  On the other hand, arbitrary scalings are possible with the R boxplot command, explicitly specifying the relative width of each boxplot via the width parameter.

That's all for now.  Subsequent posts in this sequence will discuss the alternative outlier detection strategy employed in the robustbase package (along with some of the other useful goodies in that package), an extension called violin plots that combines the idea of the boxplot with a nonparametric density estimate (these are available both in the wvioplot add-on package and as part of the lattice package included with base R installations), and another similar extension called beanplots, available via the beanplot add-on package.

The Art of Exploratory Data Analysis

This blog is about the art of exploratory data analysis, which is also the subject of my new book, Exploring Data in Engineering, the Sciences, and Medicine (http://www.oup.com/us/ExploringData).  This art is appropriate in situations where you are faced with an existing dataset that you want to understand better.  As Stanford University statistics professor Persi Diaconis describes it in "Theories of data analysis: From magical thinking through classical statistics" (Chapter 1 of Exploring Data Tables, Trends, and Shapes, D.C. Hoaglin, F. Mosteller, and J.W. Tukey, eds., Wiley, 1985), this art involves the following activities:

We look at numbers and try to find patterns.  We pursue leads suggested by background information, imagination, patterns perceived, and experience with other data analyses.

Later posts will discuss various things in more detail, but the objective of this first post is to set the stage for these later discussions, emphasizing the following three points.  First, data analysis is an exercise in approximation rather than a purely mathematical activity that leads to definite, correct answers.  In particular, any answer other than “4” to the question, “how much is 2 + 2?” is simply wrong (barring tricks, of course, like casting the problem in base 3 arithmetic).  Similarly, other mathematical questions like “what is the square root of 3?” or “what is the probability of drawing a sample from a Gaussian distribution that lies 2 or more standard deviations away from the mean?” also have clear, correct answers.  In contrast, data analysis questions like, “how is the waiting time between eruptions related to the duration of eruptions for the Old Faithful geyser in Yellowstone National Park?” do not have similarly well-defined answers.  Certainly, we can provide quantitative characterizations that attempt to describe this relationship (for example, the product-moment correlation coefficient between these values computed from a set of 272 observations is approximately 0.901), but this can usually be done in more than one way (e.g., the Spearman rank correlation coefficient between these same numbers is somewhat smaller, at 0.778), and however we compute these numbers they are generally only partial characterizations.  The practical consequence of this difference between data analysis problems and purely mathematical problems is the necessity of somehow dealing with unexplainable variation in the data.

This need for approximation in analyzing real-world data has been widely recognized and it motivates the widespread use of probability and statistics in analyzing data.  This is not the only possible approach - set-theoretic methods based on the “unknown but bounded” uncertainty model are also possible, for example, although they are not nearly as popular or as well developed as statistical methods - and there are even those who are unwilling to accept uncertainty in describing real-world data.  One colorful example is Alfred William Lawson’s Principle of Zig-Zag-and-Swirl, discussed briefly in my book and in more detail in L.D. Henry’s biography of Lawson, Zig-Zag-and-Swirl (see the Other Interesting Books section of The Exploring Data Store at the end of this post for details).  Lawson dabbled in many things, from playing and managing minor league baseball to writing a Utopian novel (the late Martin Gardner characterized it as “the worst work of fiction ever published”).  He is credited with introducing the term “aircraft” into general use not long after the Wright brothers first flight, and he obtained the first U.S. airmail contract before his company fell into receivership.  He is of interest here because he attempted to develop his own laws of physics, advocating the development of something he termed “Supreme Mathematics” to deal with the complexity of the real world.  As an illustrative example, he asked us to consider “a germ, moving across a blood corpuscle in the body of a man who is walking down the aisle of a flying airplane.”  To describe the true complexity of the germs path, including the man’s motion in the airplane, the airplane’s flight over the earth, the earth’s rotation about the sun, and so forth, Lawson argued that ordinary mathematics was inadequate and needed to be replaced by his Supreme Mathematics.  The fundamental difficulty is that, if we are really attempting to describe anything interesting about the germ, we probably don’t care at all about whatever eighteenth-order influence Jupiter’s gravity may have on the germ’s motion.  If we are willing to live with approximations, we can avoid the necessity of Supreme Mathematics.

To proceed, then, we need to make working assumptions about the nature of our approximations, providing practical descriptions of the uncertainty inherent in our data.  As noted, this data approximation problem can be approached in a number of different ways, but the most popular approach is via probability theory, describing our data uncertainty with some sort of random variable model.  To make this idea useful, we need to adopt a specific random variable model, which brings us to the second key aspect of data analysis considered here: exactly how do we describe data uncertainty?  Historically, the Gaussian distribution has been adopted so extensively as a data characterization that it has frequently been simply assumed without question.  In fact, a contemporary of Poincare’s once observed that:

Experimentalists tend to regard it as a mathematical result that data values obey a Gaussian distribution, whereas mathematicians tend to regard it as an experimental result.

In practice, this assumption sometimes represents a reasonable approximation of reality, but sometimes it is horribly inadequate.  This point is important because adopting the Gaussian distribution as a working assumption leads to a lot of very useful data analysis techniques (ordinary least squares regression methods, to name only one example), but in cases where the Gaussian distribution is a poor approximation, the results obtained using these methods can be extremely misleading. 

The third key point of this post is that simple graphical tools for assessing the reasonableness popular working assumptions like this one are extremely useful in exploratory data analysis, as are alternative analysis approaches that perform better when these assumptions are inappropriate.  Future posts will describe some of these tools and alternatives in greater detail, but the following example provides a useful illustration of both how we might assess real data distributions and the fact that these distributions can sometimes be very non-Gaussian.  Both of these points are illustrated in the four plots in the figure below.  The upper two plots show nonparametric density estimates, computed from two different datasets.  The curves in these plots are estimates of the probability density function appropriate to these datasets, if we regard each record in the dataset as a statistically independent random sample, drawn from some unknown probability distribution.  (If you are familiar with histograms, think of these curves as “smoothed histograms.”)

The upper left plot shows the density estimated from the measurements of the carapace length of 200 crabs, from the crabs dataset included in the MASS package that is part of the open-source R programming language discussed briefly at the end of this post.  In this case, the estimated probability density appears reasonably similar to the “bell-shaped curve” that characterizes the Gaussian probability distribution, so this working assumption may be fairly reasonable here.  This assumption is further supported by the bottom left plot, which shows the normal quantile-quantile (Q-Q) plot constructed from these 200 measurements.  The construction and interpretation of these plots is described in detail in my book and in various other places (see, for example, the Wikipedia entry on Q-Q plots at http://en.wikipedia.org/wiki/Q-Q_plot), but the key point here is that they represent an informal graphical test of the assumption that the Gaussian distribution is a reasonable approximation.  If the data values fall approximately on the reference line - as in this plot - the Gaussian assumption may be a reasonable basis for analysis.  Conversely, the plot on the lower right shows marked deviations from this reference line, suggesting that the Gaussian distribution may not be reasonable there.  This plot was constructed from 272 measurements of the waiting times between eruptions of the Old Faithful geyser in Yellowstone National Park, and the nonparametric density estimate shown in the upper right plot was also constructed from this data sequence.  (These numbers were taken from the faithful dataset that is included as part of the base installation of the R programming language.)  This upper plot shows two pronounced peaks – one at approximately 50 minutes and the other at approximately 80 minutes – suggesting that the distribution of waiting times is bimodal, consistent with the “kink” seen in the Q-Q plot below it.  The point here is that, while the Gaussian distribution may be reasonable as an approximate description of the crab carapace length data, it is not at all reasonable for the Old Faithful waiting time data.

In future posts, I plan to discuss some of these ideas in more detail, illustrating them with examples.  In no particular order, a few of the topics that appear on my radar are:

• Metadata (the information describing the contents of a dataset that is all too often missing, incomplete, or incorrect);
• Boxplots, modified boxplots, violinplots, and beanplots - useful tools for characterizing the range of variation of a numerical variable over different data subgroups;
• Various types of data anomalies, including outliers, inliers, missing data, and misalignment errors: what they are, how to detect them, and what to do about them;
• Interestingness measures as useful characterizations of categorical variables;
• Data transformations and the things they can do, both expected and unexpected, sometimes good and sometimes very bad;
• And anything else related to exploratory data analysis that strikes me as interesting along the way.

The results presented here were obtained using the open-source software package R, which I like a lot because it is both freely available and fabulous in the range of computational tools it provides.  For details, documentation, and download instructions, go to the Comprehensive R Archive Network (CRAN) at the following Web site:

            http://cran.r-project.org/

I should emphasize, however, that this blog is not about the R programming language per se, but rather about the kinds of data analysis that R supports extremely well. 

I hope you find this blog useful and interesting, and I welcome your comments and questions.  I can’t promise to respond right away, but I do promise to carefully consider any questions and comments that come my way and to respond to them when and how I can.