This is great for a paper or technical report, but it takes effort to discern patterns; I’d much rather see a few plots, like achievement vs. year by grade and gender.
Uses of graphics
There is a broad distinction between:
exploratory graphics, which are intended to be seen only by analysts; and
presentation graphics, which are intended to be seen by an audience.
Exploratory graphics are made quickly in large volumes, and usually not formatted too carefully. Think of them like the pages of a sketchbook.
Presentation graphics are made slowly with great attention to detail. Think of them as exhibition artworks.
The two are not mutually exclusive: an especially helpful exploratory graphic is often worth developing as a presentation graphic to help an audience understand ‘what the data look like’.
Elements of statistical graphics
Statistical graphics are actually quite simple. They consist of the following four elements:
Axes
References for all other graphical elements.
Geometric objects
Points, lines, curves, filled regions, etc.
Aesthetic attributes
Color, shape, size, opacity/transparency.
Text
Labels, legends, and titles.
Axes
We are all familiar with axes. The word axis literally means axle: an axis is an object that other things turn around.
In statistical graphics, axes establish positional references for locating any geometric object – line, point, polygon – on the graphic.
Geometric objects
Geometric objects are the things depicted on a plot, whatever those may be; typically points, lines, polygons, and shapes.
Aesthetic attributes
For us, aesthetics will mean qualities of geometric objects, like color or transparency.
Primary aesthetics in statistical graphics are:
Shape (for points)
Color
Size
Opacity/transparency
Text
Text is used to label axes, objects, legends, and specify titles.
Text may seem innocuous, but it is what creates story – text gives a plot its plot!
Statistical graphics are mappings
Statistical graphics are mappings of dataframe columns and attributes to graphical elements: axes, geometric objects, and aesthetic attributes.
For a simple example, consider the following time series of Cuba’s population by year:
Mappings:
population \(\longrightarrow\) y coordinate of axis;
year \(\longrightarrow\) x coordinate of axis;
observations \(\longrightarrow\) line
Mapping columns to aesthetics
Now consider aggregated populations by global region and year:
Mappings:
population \(\longrightarrow\) y
year \(\longrightarrow\) x
region \(\longrightarrow\) color
observations \(\longrightarrow\) line (groupwise by color)
Using aesthetics
The ability to map variables to the elements of a graphic is essential because it means we can display more than two variables at a time by leveraging aesthetic attributes.
For example, in lab you’ll begin with this scatterplot:
Each point represents a country in a particular year. The graphic shows that life expectancy increases with GDP per capita.
Using aesthetics
In the lab you’ll add aesthetic mappings step by step until arriving at this plot:
This figure displays the same x-y relationship as before, but together with time, continental region, and population.
Verges on too complex.
Using graphics for discovery
Further incorporating sex shows that GDP per capita is associated with differential life expectancy gaps between men and women:
In other words, on average women outlive men by longer in wealtheir countries.
Maybe just by an additional 2-3 years in wealthier countries
Clear pattern but lots of variation for a given GDP/capita
Altair
Altair, a python library, creates graphics exactly as described above: mapping columns of a dataframe to graphical elements.
It has a somewhat idiosyncratic syntactical pattern involving a “chart”, “marks”, and “encodings”:
Altair syntax
Example handle
Operation
Chart
alt.Chart(df)
Coerces a dataframe df to a chart object
Mark
mark_point()
Specifies a geometric object
Encoding
encode(x = ..., y = ..., color = ...)
Maps columns of df to objects and aesthetics
Basic use of syntax
A chart specification, mark(s), and encodings are chained together to make a graphic.
Code
alt.Chart( popregion ).mark_line( ).encode( x ='Year:T', y ='Population', color ='Region').properties( width =500, height =100).configure_axis( labelFontSize =16, titleFontSize =16).configure_legend( labelFontSize =16, titleFontSize =16)
Choice of scale
The choice of scales for each mapping can either reveal or obscure patterns in data.
When population is mapped onto a logarithmic rather than linear scale, rates of increase become evident in less populous regions:
Code
alt.Chart( popregion).mark_line( ).encode( x ='Year:T', y = alt.Y('Population', scale = alt.Scale(type='log')), # change axis scale color ='Region').properties( width =350, height =100).configure_axis( labelFontSize =16, titleFontSize =16).configure_legend( labelFontSize =16, titleFontSize =16)
Note: scale is adjusted at the encoding level by alt.Y(...); every encoding channel has an analogous function, e.g., alt.X(...), alt.Color(...), alt.Shape(...), etc., with optional scale arguments.
Common statistical graphics
Broadly, the most common statistical graphics can be divided according to the number of variables that form their primary display. The uses listed below are not exclusive, just some of the most common.
One-variable graphics are used to visualize distributions.
Two-variable graphics are used to visualize relationships.
Three-variable graphics are used to visualize spatial data, matrices, and a collection of other data types.
Most graphics you’ll encounter are grouped one- or two-variable graphics with superpositions of geometric objects differentiating observed from inferred values – e.g., scatterplots with points color-coded by another (grouping) variable and trend lines.
Single-variable graphics
Single-variable graphics usually display the distribution of values of a single variable.
Common single-variable graphics
Histograms and smoothed density plots show shape but depend on arbitrary binning/smoothing parameters.
CDF and quantile plots show the distribution exactly but are harder to interpret.
Histograms
Histograms show the relative frequencies of values of a single variable.
Code
alt.Chart( popcountry.loc["1970"]).mark_bar().encode( x = alt.X('log(Population)', bin= alt.Bin(maxbins =50)), y ='count()').properties( height =150, title ='National populations in 1970').properties( width =500, height =300).configure_axis( labelFontSize =16, titleFontSize =16).configure_legend( labelFontSize =16, titleFontSize =16).configure_title( fontSize =16)
The main advantage of the histogram is it shows the shape of a distribution.
Bin widths
The main downside is that the shape depends on bin width, which is an arbitrary parameter.
Code
alt.Chart( popcountry.loc["1970"]).mark_bar().encode( x = alt.X('log(Population)', bin= alt.Bin(maxbins =10)), y ='count()').properties( height =150, title ='National populations in 1970').properties( width =350, height =100).configure_axis( labelFontSize =16, titleFontSize =16).configure_legend( labelFontSize =16, titleFontSize =16).configure_title( fontSize =16)
Code
alt.Chart( popcountry.loc["1970"]).mark_bar().encode( x = alt.X('log(Population)', bin= alt.Bin(maxbins =50)), y ='count()').properties( height =150, title ='National populations in 1970').properties( width =350, height =100).configure_axis( labelFontSize =16, titleFontSize =16).configure_legend( labelFontSize =16, titleFontSize =16).configure_title( fontSize =16)
Always experiment with multiple bin widths to ensure you don’t overlook any important details such as outliers, multiple modes, etc.
Binning at left is too coarse, obscures outlying values
Binning at right is good
Density plots
Denisty plots are smoothed histograms – we’ll discuss further next week. They also require some arbitrary choices that affect the appearance.
Density plots with different smoothing kernels and bandwidths
More single-variable graphics
Grouped single-variable graphics allow visualization of multiple distributions.
Grouped single-variable graphics
Boxplots
Boxplots display data quantiles and outliers, conveying skewness and range.
Due to their compactness, they are useful for comparing multiple distributions.
Many boxplots
Single-variable graphics are not necessarily limited to univariate data; one might want to compare distributions using the same single-variable displays shown groupwise.
