Graphical summaries of data #

Many powerful approaches to data analysis communicate their findings via graphs. These are an important counterpart to data analysis approaches that communicate their findings via numbers or tabless.

Here we will illustrate some of the most common approaches for graphical data analysis. Throughout this discussion, it is important to remember that graphical data analysis methods are subject to the same principles as non-graphical methods. A graph can be either informative or misleading, just like any other type of statistical result. To understand whether a graph is informative, we should consider the following:

• Every graph should provide insight into the specific research question that is the overall goal of the data analysis.

• The graph is constructed using a sample of data, but the purpose of the graph is to learn about the population that the sample represents.

• What statistical principal or concept is the graph based on?

• What are the theoretical properties of any numerical summaries that are shown in the graph?

Almost every statistical graphic conveys a statistical concept that can be defined in a non-graphical manner. Graphs may show associations, location, dispersion, tails, conditioning, or almost any other statistical feature of the data or population. Graphs make it easier for the viewer to digest such information, but when interpreting a graph it is always important to keep in mind the specific statistical concept on which the graph is based.

Statistical graphics have an aesthetic dimension that is usually not evident when presenting findings through, say, tables. Our goal here is to focus on the content of graphs, not their aesthetic properties. Very crude graphs that have deep content are much more informative than beautiful graphs that convey only superficial content. In recent years, the field of infographics has grown rapidly. There is no sharp line dividing infographics from statistical graphs, however in general, the former tend to convey relatively simple insights in an aesthetically engaging way, while the latter aim to convey deeper and more subtle insight, with less focus on presentation.

Challenges and limitations of graphs #

One of the main challenges in statistical graphics is to fit the greatest amount of useful information into a single graph, while allowing the graph to remain interpretable. More complex graphs can suffer from overplotting, in which the plot elements are so crowded on the page that they fall on top of each other. This can limit the legibility of plots formed from large datasets unless a great deal of preliminary summarization of the data is performed.

Another challenge that arises in graphing complex datasets is that most graphs are two-dimensional, so that they can be viewed on a screen (or printed on paper). Some graphing techniques extend to three dimensions, but many datasets have a natural dimensionality that is much greater than 2 or 3. A few methods for graphing work around this, but require more effort from the person viewing the graph.

Boxplots #

Boxplots are a graphical representation of the distribution of a single quantitative variable. A boxplot is based on a set of quantiles calculated using a sample of data. Below is an example of a single boxplot, drawn horizontally, showing the distribution of income values based on a sample of 100 individuals.

The “box” in a boxplot (shaded blue above) spans from the 25th to the 75th percentiles of the data, with an additional line drawn cross-wise through the box at the median. “Whiskers” extend from either end of the box, and are intended to cover the range of the data, excluding “outliers”.

The concept of an outlier is extremely problematic and no generically useful definition of outliers has been proposed. For the purpose of drawing a boxplot, the most common convention is to plot the upper (right-most) whisker at the 75th percentile plus 1.5 times the IQR, or to the greatest data value less than this quantity. Analogously, the lower (left-most) whisker is drawn at the 25th percentile minus 1.5 times the IQR, or to the least data value greater than this quantity. Finally, individual points sometimes called “fliers” are drawn corresponding to any value that falls outside the range spanned by the whiskers. A single box-plot, as above, is often drawn horizontally, but may also be drawn vertically.

There are many alternative ways of defining the locations of the whiskers in a boxplot. The approach described above is most common, and is chosen so that with “light tailed” distributions, well under 1% of the data will fall outside of of the whiskers.

The boxplot above shows a right-skewed distribution. This is evident because the upper whisker is further from the box than the lower whisker. Also, within the box, the median is closer to the lower side of the box than to the upper side of the box. Overall, the lower quantiles are more compressed, and the upper quantiles are more spread out, which is a feature of right-skewed distributions.

Side-by-side boxplots #

Boxplots are commonly used to compare distributions. A “side-by-side” or “grouped” boxplot is a collection of boxplots drawn for different subsets of data, plotted on the same axes. These subsets usually result from a stratification of the data, according to some stratifying factor that partially accounts for the heterogeneity within the population of interest. For example, below we consider boxplots showing the distribution of income, stratified by sex.

Histograms #

A histogram is a very familiar way to visualize quantitative data. A histogram is constructed by breaking the range of the values into bins and counting the number (or proportion) of observations that fall into each bin. The shape of a histogram shows visually how likely we are to observe data value in each part of the range. We are more likely to observe values where the histogram bars are higher, and less likely to observe values where the histogram bars are lower.

Histograms closely resemble “bar charts”, but with the added statistical aspect that the goal is to capture the density at each possible point in the population. “Density” is a measure of how commonly we observe data “near”, rather than “at” a point. For example, the density of household incomes at 45,000 USD would not be the exact number or frequency of households with this income. Instead, it reflects the frequency of households that have an income near 45,000 USD.

The width of the histogram bars (or “bins”) is a parameter of the method. There are “rules of thumb” for setting the width of the bins, but no single such rule is used universally. Two such rules are “Scott’s rule”, which uses bins of width \(3.49\cdot\sigma/n^{1/3}\) , where \(\sigma\) is the standard deviation, and \(n\) is the sample size (number of observations), and the “Freedman-Diaconis rule”, which uses bins of width \(2\cdot{\rm IQR}/n^{1/3}\) . Note that in both cases, the bins are narrower when the sample size is larger. Narrower bins allow us to see the finer scale structure of the distribution, but if the sample size is small, narrow bins produce unstable results because each bin contains only a very few observations. The above rules for calculating the bin width aim to compromise between these two issues.

A histogram can be used to assess almost any property of a distribution. The common measures of location and dispersion can be judged from visual inspection of the histogram. As always, we should remember that features of the histogram may not always reflect features of the population from which the data were sampled. For example, a histogram may show two modes (i.e. is bimodal) even when the underlying distribution only has one mode (i.e. is unimodal). Moreover, the number of modes in a histogram can change as the bin width is varied.

Histograms are easy to communicate about, but may not be effective when working with small samples, where they can accentuate non-generalizable features of the sample (i.e. characteristics of the sample that are not present in the population). This is reflected in the following mathematical fact. For many statistics, if we wish to reduce the error relative to the population value of the statistic by a factor of two, we need to increase the sample size by a factor of four. In the case where we are aiming to estimate a density, in order to reduce the error by a factor of two, we need to increase the sample size by a factor of eight.

With a sufficiently large collection of representative data, the histogram should closely match the population’s probability density function (PDF). The PDF is usually a smooth curve, rather than a series of steps as in a histogram. This fact inspired the development of a modified version of a histogram that presents us with a smooth curve instead of a series of steps. This technique is called kernel density estimation (KDE). It produces graphs such as shown below.

Kernel density estimates may provide a somewhat more accurate estimation of the underlying density function compared to a histogram. But like a histogram, they can be unstable and produce artifacts. For example, the KDE above shows positive density for negative income values, even though all of the income values used to fit the KDE were positive (in some cases, income can take a negative value, but in this case no such values were present). More advanced KDE methods not used here can mitigate this issue.

One advantage of using a KDE rather than a histogram is that it is easier to overlay multiple KDEs on the same axes for comparison without too much overplotting. This might allow us to compare, say, the distributions of female and male incomes as follows.

Quantile plots #

A quantile plot is a plot of the pairs \((p, q_p)\) , where \(q_p\) is the p’th quantile of a collection of quantitative values. Since \(p\) can be any real number between 0 and 1, the graph of these pairs constitutes a function. By construction, this must be a non-decreasing function. A quantile plot contains essentially the same information as a histogram, but is represented in a very different way. Note that unlike the histogram, for which the bin width is a parameter that must be selected, there is no such parameter in the quantile plot. Arguably, the quantile plot is a more stable and informative summary of a sample, especially if the sample size is moderate. However most people are more comfortable interpreting histograms than quantile functions.

As an example, the following plot shows simulated systolic blood pressure values for a sample of females and a sample of males. In this case, at every probability point \(p\) , the blood pressure quantile for males is greater than the blood pressure quantile for females, indicating that male blood pressure is “stochastically greater” than female blood pressure.

Below is another example that shows two quantile functions, but in this case the quantile functions cross. As a result, there is no “stochastic ordering” between the data for females and for males. Also note that the quantile curve for females is steeper than the curve for males, indicating that the female blood pressure values are more dispersed than those for the males.

Quantile-quantile plots #

A quantile-quantile plot, or QQ plot, is a plot based on quantiles that is used to compare two distributions. Recall that a quantile plot plots the pairs \((p, q_p)\) for one sample. A QQ plot plots the pairs \((q^{(1)}_p, q^{(2)}_p)\) , where \(q^{(1)}_p\) are the quantiles for the first sample, and \(q^{(2)}_p\) are the quantiles for the second sample. In a QQ-plot, the value of p is “implicit” – each point on the graph corresponds to a specific value of p, but you cannot see what this value is by inspecting the graph.

As an example, suppose we are comparing the number of minutes of sleep during one night for teenagers and adults. This might give us the following QQ-plot:

The above QQ-plot shows us that teenagers tend to sleep longer than adults, and this is especially true at the upper end of the range. The QQ-plot approximately passes through the point (600, 800), meaning that for some probability p, 600 is the p’th quantile for adults and 800 is the p’th quantile for teenagers.

The slope of the curve in the QQ-plot reflects the relative levels of dispersion in the two distributions being compared. Since the slope of the curve in the above QQ-plot is greater than that of the diagonal reference line, it follows that the values plotted on the vertical axis (teenager’s values) are more dispersed than the values plotted on the horizontal axis (adult’s values).

An important property of a QQ-plot is that if the plot shows a linear relationship between the quantiles, then the two distributions are related via a location/scale transformation. That is, there is a linear function \(a + bx\) that maps one distribution to the other. In the example above, there is a substantial amount of curvature in the graph, so it does not seem to be the case that the sleep durations for adults and teenagers are related via a location/scale transformation.

Dot plots #

Dot plots display quantitative data that are stratified into groups. One axis of the plot is used to display the quantitative measure, and the other axis is used to separate the results for different groups. A series of parallel “guide lines” are used to show which points belong to each group. Dot plots are often used to display a collection of numerical summary statistics in visual form. Sometimes people say that dot plots are used to “convert tables into graphs”. Due to overplotting, dot plots are less commonly used to show raw data. The example below shows how dot plots can be used to display the median age stratified by sex, for people living in each of eleven countries.

The plot above shows that the median age for females is greater than the median age for males in every country. This is mainly due to females having longer life expectancies than males. We also see that some countries have much lower median ages for both sexes compared to other countries. Countries that have recently had high birth rates, such as Ethiopia and Nigeria, tend to have much lower median ages than countries with lower birth rates, such as Japan.

Scatterplots #

A scatterplot is a very widely-used method for visualizing bivariate data. They have many uses, but the most relevant for us is to plot the joint (empirical) distribution of two quantitative values. As an example, suppose that we observe paired data values giving the annual minimum and annual maximum temperature at a location. We could view these data with a scatterplot, placing, say, the minimum temperature value on the horizontal (x) axis, and the maximum temperature value on the vertical (y) axis. The number of points is the sample size, here being the number of locations for which temperature data are available. A possible graph of this type is shown below.

Several characteristics of the relationship between minimum and maximum temperature are evident from the plot above. The maximum temperature at each location is at least as large as the minimum temperature. There is a positive association in which locations with a lower minimum temperature tend to have a lower maximum temperature compared to places with a higher maximum temperature, but there is a lot of scatter around this trend. Warmer places tend to have a smaller range between their minimum and maximum temperatures. Concretely, locations on the equator and at low elevation, such as Singapore, have relatively constant temperature throughout the year. Locations near the center of large continents, like Winnipeg, Canada, can have extremely cold winters and also rather hot summers. Coastal regions that are far from the equator, such as Dublin, Ireland, have mild winters and cool summers.

To aid in interpreting a scatterplot, it is useful to plot a smooth curve that runs through the center of the data. This is called scatterplot smoothing, and can be accomplished with several algorithms, one of which is known as lowess. The population analogue of a scatterplot smooth is the conditional mean, or conditional expectation, denoted \(E[Y|X=x]\) , for the conditional mean of \(Y\) given \(X\) . The conditional mean is a function of \(x\) , and can be evaluated at any point \(x\) in the domain of \(X\) . The conditional mean is (roughly speaking), the average of all values of \(Y\) whose corresponding value of \(X\) is near \(x\) .

The plot below adds the estimated conditional mean (orange curve) to the scatterplot of temperature data discussed above. The conditional mean curve is increasing, showing that, as noted above, a location with lower annual minimum temperature tends on average to have a lower annual maximum temperature (relative to other locations).

Time series plots #

Some data have a serial structure, meaning that the values are observed with an ordering. Very often, such observations are made over time, which gives us time series or longitudinal data. Sometimes we observe a single time series over a long period of time, such as the value of a commodity in a market recorded every day over many years. Other times, we observe many short time series recorded irregularly. We may plot these time series together, leading to what is sometimes called a “spaghetti plot”. For example, in a study of human growth, we may observe measurements of the body weight of research subjects at various ages, giving us the spaghetti plot below:

Parallel coordinate plots #

Scatterplots in the plane are limited to two dimensions. Various techniques have been developed to overcome this limitation, one of which is the parallel coordinate plot. A parallel coordinate plot places the coordinate axes for the multiple dimensions as parallel lines, rather than as perpendicular lines. Using parallel lines means that data for far more than two or three variables can be placed on a single page.

Below is an example of a parallel coordinates plot, showing four attributes of a set of ten countries. A scatterplot of these points would live in four-dimensional space, which is quite challenging to visualize directly. Note that the attributes are converted to Z-scores, which is common in a parallel coordinates plot when the variables being plotted fall in very different ranges. The plot shows us that the life expectancies for females and for males are quite similar – the country with the highest life expectancy for females also has the highest life expectancy for males, and the country with the lowest life expectancy for females also has the lowest life expectancy for males. There is also a substantial positive relationship between the economic status of a country, as measured by its gross domestic product (GDP) and life expectancy. However no relationship is evident between GDP and population, or between either of the life expectancy variables and population.

Mosaic plots #

The graphs above primarily use quantitative data. A mosaic plot is a plot that is used with nominal data. Specifically mosaic plots are used when the units of analysis are cross-classified according to two nominal factors. In the example below, people with cancer are cross-classified by their biological sex, and by the type of cancer that they have:

The width of each box in the mosaic plot corresponds to the relative overall prevalence of the corresponding cancer type. The heights of the boxes correspond to the sex-specific prevalences. Based on this graph, we see that digestive, lung, and breast cancers are much more common than, say, oral and endocrine cancers. The mosaic plot also shows us that while breast and endocrine cancers are more common in females, the other cancer types are more common in males.

An important property of a mosaic plot is that the area of each box is proportional to the number of units that fall into the box. Thus, we can see that the area of the female breast cancer box is larger than the the combined areas of the female and male lung cancer boxes. Thus, there are more cases of breast cancer in females than the combined cases of lung cancer for both sexes.