Topic 1.5 Representing a Quantitative Variable with Graphs: construct and interpret dotplots, stem-and-leaf plots, and histograms for a quantitative variable, and choose an appropriate display.

What this topic is asking

The College Board (Topic 1.5) wants you to display a single quantitative variable with a dotplot, a stem-and-leaf plot (stemplot), or a histogram, to construct each correctly, and to choose the display that best suits the data set's size and purpose.

Dotplots

Because a dotplot keeps every value, you can read the data straight off it: the height of a stack is the frequency of that value. For more than roughly $50$ values, the stacks grow tall and the plot becomes hard to read, which is when a histogram takes over.

Stem-and-leaf plots

A stem-and-leaf plot splits each number into a stem (the leading digit or digits) and a leaf (usually the final digit). For data like $42, 45, 47, 51, 58$, the stems are $4$ and $5$ and the leaves are written beside them ($4 \mid 2\,5\,7$ and $5 \mid 1\,8$). Like a dotplot, it keeps every value, but it also reveals shape by how long each row is, and it works well up to perhaps a hundred values. A back-to-back stemplot shares one stem column to compare two groups, a neat way to display two distributions side by side.

Histograms

The price of grouping is that you lose the individual values: once data are binned, you cannot recover the exact numbers. Conventionally a value that lands on a boundary goes into the bin on its right, so each value falls in exactly one bin.

Choosing a display and a bin width

The right display depends on the data set's size and your goal. For a small set (say, under $30$ values), a dotplot or stemplot is excellent because it keeps every value, so the reader sees the raw data and the shape together. For a large set (hundreds or thousands of values), a histogram is the only practical choice, because plotting every point would be illegible; the histogram trades individual values for a clear overall shape. The subtle skill is choosing the bin width. Too few, very wide bins smooth the data so much that real clusters and gaps disappear and the distribution looks falsely simple. Too many, very narrow bins produce a spiky, jagged display where random wobble looks like structure. A sensible middle ground (often somewhere around five to a dozen bins, depending on $n$) shows the genuine shape. Because the same data can look unimodal under one bin width and bumpy under another, the College Board treats bin width as a real analytical decision, and an exam answer that notes how bin width affects the apparent shape demonstrates exactly the understanding being assessed.

What every quantitative display lets you see

Whichever display you choose, the payoff is that you can now describe the distribution's shape (symmetric, skewed, number of peaks), spot outliers and gaps, and locate the rough center and spread by eye, which is precisely what the next topic, describing distributions, formalises. A good display turns a list of numbers into a picture in which these features jump out, which is why the College Board puts display before description: you cannot describe a shape you have not drawn. When you construct any of these displays under exam conditions, label the axes (including units), use a consistent scale, and for a stemplot include a key explaining what a stem and leaf represent, because unlabeled displays lose easy marks even when the data are plotted correctly.

Try this

Q1. A data set has only $12$ values and you want to keep every exact value visible. Which display is most appropriate? [1 point]

• Cue. A dotplot (or stem-and-leaf plot), because both preserve every individual value for a small data set.

Q2. Explain why a histogram's bars touch but a bar graph's do not. [2 points]

• Cue. Histogram bins are adjacent intervals of a continuous number line, so the bars are contiguous; bar-graph categories are distinct, so gaps separate them.