Announcement
Look At The Data!

"You can observe a lot by watching." --Yogi Berra

The first thing to do with any data set is look at it. If it fits on a single page, look at the raw data values. Plot them: histograms, dot plots, box plots, schematic plots, scatterplots, scatterplot matrices, parallel plots, line plots. Brush the data, lasso them. Use all of your software's capabilities. If there are too many observations to display, work with a random subset. The most familiar and, therefore, most commonly used displays are histograms and scatterplots. With histograms, a single response (measurement, variable) is divided into a series of intervals, usually of equal length. The data are displayed as a series of vertical bars whose heights indicate the number of data values in each interval. With scatterplots, the value of one variable is plotted against the value of another. Each subject is represented by a point in the display. Dot plots (dot density displays) of a single response show each data value individually. They are most effective for small to medium sized data sets, that is, any data set where there aren't too many values to display. They are particularly effective at showing how one group's values compare to another's. When there are too many values to show in a dotplot, a box plot can be used instead. The top and bottom of the box are defined by the 75-th and 25-th percentiles of the data. A line through the middle of the box denotes the 50-th percentile (median). Box plots have never caught on the way many thought they would. It may depend on the area of application. When data sets contain hundreds of observations at most, it is easy to display them in dot plots, making graphical summaries largely necessary. However, the box plots make it easy to compare medians and quartiles, and they are indispensible when displaying large data sets. One problem with box plots is that they always give the impression that data are unimodal. The plots to the left display the duration of erruptions of Old Faithful Geyser at Yellowstone National Park taken during two one-week periods. One thing it shows is it that Old Faithful is only "kind of faithful". The duration times can vary not only vary considerably but also they are bimodal. There's the 2 minute version, give or take 15 seconds, and the 4 minute 15 second version, give or take 30 seconds. Printing a box plot on top of a dot plot has the potential to give the benefits of both displays. While I've been flattered to have some authors attribute these displays to me, I find them not to be as visually appealing as the dot and box plots by themselves...unless the line thicknesses and symbol sizes are just right. The diagram to the left isn't too bad. Parallel coordinate plots and line plots (also known as profile plots) are ways of following individual subjects and groups of subjects over time. Most numerical techniques make assumptions about the data. Often, these conditions are not satisfied and the numerical results may be misleading. Plotting the data can reveal deficiencies at the outset and suggest ways to analyze the data properly. Often a simple transformation such as a log, square root, or square can make problems disappear.

The diagrams to the left display the relationship between homocysteine (thought to be a risk factor for heart disease) and the amount of folate in the blood. A straight line is often used to describe the general association between two measurements. The relationship in the diagram to the far left looks decidedly nonlinear. However, when a logarithmic transformation is applied to both variables, a straight line does a reasonable job of describing the decrease of homocysteine with increasing folate.

What To Look For:
A Single Response

The ideal shape for the distribution of a single response variable is symmetric (you can fold it in half and have the two halves match) with a single peak in the middle. Such a shape is called normal or a bell-shaped curve. One looks for ways in which the data depart from this ideal.

• Are there outliers--one or two observations that are far removed from the rest? Are there clusters as evidenced by multiple peaks?
• Are the data skewed? Data are said to be skewed if one of the tails of a histogram (the part that stretches out from the peak) is longer than the other. Data are skewed to the right if the right tail is longer; data are skewed to the left if the left tail is longer. The former is common, the latter is rare. (Can you think of anything that is skewed to the left?) Data are long-tailed if both tails are longer than those of the ideal normal distribution; data are short-tailed if the tails are shorter. Usually, a normal probability plot is needed to assess whether data are short or long tailed.
• Is there more than one peak? If data can be divided into categories that affect a particular response, the response should be examined within each category. For example, if a measurement is affected by the sex of a subject, or whether a subject is employed or receiving public assistance, or whether a farm is owner-operated, the response should be plotted for men/women, employed/assistance, owner-operated/not separately. The data should be described according to the way they vary from the ideal within each category. It is helpful to notice whether the variability in the data increases as the typical response increases.

Many Responses

The ideal scatterplot shows a cloud of points in the outline of an ellipse. One looks for ways in which the data depart from this ideal.

• Is the cloud of points something other that elliptically shaped, that is, does it look like something other than a football or a circle?
• Are there outliers, that is, one or two observations that are removed from the rest? It is possible to have observations that are distinct from the overall cloud but are not outliers when the variables are viewed one at a time! Are there clusters, that is, many distinct clouds of points?
• Do the data seem to be more spread out as the variables increase in value?
• Do two variables tend to go up and down togther or in opposition (that is, one increasing while the other decreases)? Is the association roughly linear or is it demonstrably nonlinear?

Comment If the departure from the ideal is not clear cut (or, fails to pass what L.J. Savage called the "Inter-Ocular Traumatic Test"--It hits you between the eyes!), it's not worth worrying about. For example, consider this display which shows histograms of five different random samples of size 20, 50, 100, and 500 from a normal distribution. By 500, the histogram looks like the stereotypical bell-shaped curve, but even samples of size 100 look a little rough while samples of size 20 look nothing like what one might expect. The moral of the story is that if it doesn't look worse than this, don't worry about it!