Dot plot, stem-and-leaf, probability plot and ogive

Why more than a histogram

A histogram and a box plot (see basic statistics) are the usual first look at a set of data. The pictures here answer other questions. Where exactly is each value? Is the distribution close enough to normal for a normal-based calculation? What share of values falls below a given point?

Dot plot

A dot plot puts one dot per value along a number line, stacking repeats. Nothing is grouped, so gaps, clusters, outliers and rounding (values piling up on whole numbers) are plain to see. It works best for small and moderate data sets.

Stem-and-leaf display

Each value is split into a stem (the leading digits) and a leaf (the next digit). With a leaf unit of 0.1, the value 16.4 has stem 16 and leaf 4; later digits are dropped, not rounded. Turned on its side the display is a histogram that keeps the actual numbers, so you can still read the minimum, maximum and median from it. When there are too few stems, each stem is split over two lines (leaves 0–4 and 5–9) or five. The depth column counts values from the nearer end; the line with the median shows its own count in parentheses.

Normal probability plot

The sorted values are plotted against the percentage of a normal distribution expected below each one. This tool uses Blom’s plotting positions, (i − 0.375)/(n + 0.25), and a percent axis scaled so a normal distribution plots as a straight line. The line drawn is the normal distribution with the sample mean and standard deviation.

• Points along the line: consistent with a normal distribution.
• An arc: skewness. A long right tail bends the top of the plot to the right.
• An S shape: tails heavier or lighter than normal.
• One or two points off at the end: outliers.
• Steps or a break in slope: rounding, or a mixture of two processes.

The Anderson-Darling statistic puts a number on the fit, weighting the tails, where normality matters most for capability and for variables sampling. A p-value below 0.05 is evidence that the data are not normal. A large p-value is not proof of normality, especially with few values. The plot correlation r is the correlation between the values and their normal scores; close to 1 means a straight plot. It is the basis of the Ryan-Joiner test.

Cumulative frequency and the ogive

A frequency table groups the values into classes and counts each one. Adding the counts down the table gives the cumulative frequency. Plotting the cumulative percent at each upper class boundary gives the ogive, from which you can read the share of values below any point, or the value below which a given share falls. Reading across at 50% estimates the median from the grouped data, L + ((n/2 − F)/f) × w.

On the exam

The CQE Body of Knowledge asks you to construct and interpret frequency distributions, including cumulative ones (VI.A.6), and probability plots for normal and other distributions (VI.A.7). The CSSBB lists normal probability plots, frequency distributions and cumulative frequency distributions among the graphical methods (V.D.4), and the CQPA covers the advantages and limits of data plotting (III.B.5). Expect to read a stem-and-leaf display, to judge normality from a probability plot, and to read a percentile from an ogive.

Other tools and templates