Graphical Representations for One Quantitative Variable

A list of 30 daily high temperatures tells you very little until you can see where the values pile up and where they thin out. That pattern, which values occur and how often, is the variable's distribution. For a categorical variable, topic 1.4 showed it with bars in any order. A quantitative variable's values sit on a number line, so its graphs keep them in numerical order, smallest to largest, and that order is what lets you see clusters, gaps and extremes. Three graphs do the job: dotplots, stem-and-leaf plots and histograms.

Dotplots

A dotplot draws a number line for the variable and puts one dot above each observation's value. Equal (or nearly equal) values stack up in a column.

A youth soccer team scored these numbers of goals in its 22 games: three games with 0 goals, six with 1, seven with 2, three with 3, two with 4 and one with 6.

Every observation is its own dot, so you can recover the whole data set from the graph and count anything you need: the team scored 2 or more goals in $7 + 3 + 2 + 1 = 13$ of its 22 games. The empty spot at 5 matters too. The axis is a number line, so 5 gets its place even though no game had 5 goals, and the game with 6 goals stands apart from the rest. (Topic 1.6 gives the words for features like that.)

To build one: draw and label a number line covering the smallest to the largest value with an even scale, then place one dot per observation, stacking repeats evenly so that column heights compare honestly. The number line can also run vertically, with the dots lined up beside it instead of stacked above it. Dotplots work best for small data sets, up to a few dozen values; with hundreds of dots the stacks become unreadable.

Stem-and-leaf plots

A stem-and-leaf plot (stemplot) also keeps every value, but writes it with digits instead of dots. Each value is split into a stem (all but its last digit) and a leaf (its last digit, or the last digit you keep if you round first). Here are the lengths, in minutes, of the 15 episodes of a podcast's first season:

23, 31, 35, 38, 41, 42, 42, 44, 47, 49, 52, 55, 58, 61, 76

The stems are the tens digits, 2 through 7, written in a column from smallest to largest with a vertical line to their right. Each leaf goes beside its stem, in increasing order:

Key: $4|1 = 41$ minutes.

Turned on its side, a stemplot is a graph: the row of leaves is a bar whose length is the number of values in that stem's range. The 40s row is longest, so 40 to 49 minutes is the most common range, with 6 of the 15 episodes, and you can still read every exact value. Three rules keep it honest:

• Include a key. $4|1$ could mean 41 minutes, 4.1 hours or 410 people. Without the key the graph has no units and no scale.
• List every stem in the range, even if it has no leaves. If no episode had been in the 60s, the stem 6 would still appear, with nothing beside it, so the gap shows.
• Use one digit per leaf, and line the leaves up. Values with more digits than you need can be rounded or truncated first: 247 seconds could go on stem 24 with leaf 7 (key $24|7 = 247$ seconds), or be rounded to 250, counted in tens as 25, and placed on stem 2 with leaf 5 (key $2|5 = 250$ seconds). The key tells the reader which, and the rounded plot no longer shows the 7.

When almost all the values share two or three stems, the plot is too squashed to show anything. Split stems fix this by giving each stem two rows: leaves 0 to 4 on the first row and 5 to 9 on the second. Worked Example 3 below uses them.

Histograms

A histogram groups values into intervals of equal width, called bins (or classes), and draws a bar over each bin whose height is the number of observations in it (frequency) or the fraction of all observations in it (relative frequency). The bins sit along the number line in order and share their edges, so the bars touch. A gap between bars means an empty bin, which is information, just like the empty spot at 5 goals in the dotplot.

Building one takes two decisions and a rule.

• Where to start, and how wide to make the bins. Pick a convenient starting value at or below the minimum and a convenient width, such as 1, 2, 5 or 10 units, that gives roughly 5 to 15 bins. Every bin must have the same width.
• The boundary rule. A value exactly on an edge belongs to the bin it starts. With bins $50 \le x < 55$ and $55 \le x < 60$, a 55 goes in the second. Each value lands in exactly one bin, so the frequencies add to $n$.
• Frequency or relative frequency. A relative frequency histogram divides every bar's height by $n$. Its bars have exactly the same shape; only the vertical scale changes. Relative frequency is the choice when you'll compare data sets of different sizes.

Most histograms put the bins on the horizontal axis with vertical bars, but you can also run the bins up the vertical axis with horizontal bars; the meaning is the same.

A histogram gives up the exact values, unless each bin is so narrow that it can hold only one possible value (bins 1 unit wide for whole-number data), in which case it is a dotplot drawn with bars. From the bar over 70 to 75 you know how many values are in that range but not what they are. That's the price of being able to show 30 or 30,000 values in the same tidy picture.

Bin width changes the picture

Here are the daily high temperatures for the 30 days of April in one city, in °F:

47, 50, 52, 53, 54, 54, 55, 56, 57, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 71, 72, 73, 73, 74, 75, 76, 78, 81

These are two histograms of the same 30 values, one with bins 10 degrees wide and one with bins 5 degrees wide, both starting at 45.

The wide bins show one hump. The narrow bins show two clusters, a cooler group of days in the 50s and a warmer group in the low 70s, with fewer days in between. Neither graph is wrong: both follow the rules, and the data are the same. Wide bins average over detail, so features narrower than a bin disappear. Bins that are too narrow have the opposite problem: with only 30 values and bins 1 degree wide, most bars would be 0, 1 or 2 tall and the picture would be noise. When you draw a histogram, try more than one width and pick the one that shows the pattern without the noise; when you read one, remember that a different bin choice might show something this one hides.

Worked examples

Where students lose points