AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-5)
Unit 1 · Topic 1.5
1.5 Graphical Representations for One Quantitative Variable
For a quantitative variable, you need graphs that keep the numbers in order: dotplots, stemplots and histograms. Each shows how the values are distributed, and each has strengths. With histograms, the choice of bin width can change the picture.
Key terms
- dotplot
- stem-and-leaf plot
- histogram
- bin width
- distribution
Dotplots
A dotplot puts a dot above a number line for each value. Values that are the same (or nearly the same) stack on top of each other. Dotplots show every data point, so they work best for small to medium data sets, say up to about 50 values.
You can read individual values, count how many are above or below a cutoff and see the overall shape at a glance.
Stem-and-leaf plots
A stem-and-leaf plot (stemplot) splits each number into a stem (everything but the last digit) and a leaf (the last digit). Stems go in a column from smallest to largest, and each leaf is written beside its stem, also in order. Always include a key, like "7 | 3 means 73 points."
A stemplot keeps every actual value while showing the shape, like a sideways histogram. If too many leaves pile onto a few stems, you can split stems: one row for leaves 0–4 and another for 5–9.
A back-to-back stemplot shares one column of stems between two groups, with one group's leaves going left and the other's going right. It's a quick way to compare two small data sets.
Histograms
A histogram groups values into intervals of equal width called bins, then draws a bar for each bin. The bar's height is the frequency (count) or relative frequency in that bin. The bars touch because the number line is continuous: one bin ends where the next begins.
Decide in advance which bin a boundary value goes in. The usual convention is that a bin like 70 to 80 includes 70 but not 80, so 80 goes in the next bin.
Bin width matters. Very wide bins hide detail, such as two separate peaks merging into one. Very narrow bins make the graph jagged and hard to read. Try more than one width before deciding what shape the data have. Histograms can also be drawn with bins on the vertical axis and horizontal bars.
Which graph to use
None of these work for categorical data. If the variable is categorical, use a bar chart or pie chart.
| Graph | Shows individual values? | Best for |
|---|---|---|
| Dotplot | Yes | Small data sets, quick comparisons |
| Stemplot | Yes | Small data sets, keeps exact values |
| Histogram | No | Large data sets, overall shape |
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Make a stemplot and a histogram
Fifteen quiz scores: 62, 67, 71, 73, 75, 78, 78, 81, 84, 85, 86, 88, 90, 93, 97. Make a stemplot, then give the bar heights for a histogram with bins 60–<70, 70–<80, 80–<90 and 90–<100.
Show the solutionHide the solution
- Step 1: Use the tens digit as the stem and the ones digit as the leaf. Key: 7 | 3 means 73.
- Step 2: Stem 6: leaves 2 7. Stem 7: leaves 1 3 5 8 8. Stem 8: leaves 1 4 5 6 8. Stem 9: leaves 0 3 7.
- Step 3: For the histogram, count the values in each bin. 60–<70: 62, 67, so 2. 70–<80: 71, 73, 75, 78, 78, so 5. 80–<90: 81, 84, 85, 86, 88, so 5. 90–<100: 90, 93, 97, so 3.
- Step 4: Check: 2 + 5 + 5 + 3 = 15.
Answer: Stemplot rows: 6 | 2 7; 7 | 1 3 5 8 8; 8 | 1 4 5 6 8; 9 | 0 3 7 (key 7 | 3 = 73). Histogram heights: 2, 5, 5, 3.
- Example 2Calculator allowed
Trap: bin width hides a feature
A histogram of waiting times at a clinic uses bins 20 minutes wide and shows one peak. With bins 5 minutes wide, the same data show two clear peaks, near 10 minutes and near 35 minutes. Which description should you trust?
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- Step 1: Both graphs show the same data. Wide bins can lump two clusters into one bar.
- Step 2: The narrower bins reveal that waits cluster in two places, which might mean two kinds of patients (walk-ins and appointments, for example).
- Step 3: The bimodal shape is a real feature the wider bins hid.
Answer: Trust the narrower-bin histogram: the distribution is bimodal, with peaks near 10 and 35 minutes. The wide bins hid that.
Common mistakes
- Leaving out the key on a stemplot, so a reader can't tell whether 7 | 3 means 73, 7.3 or 730.
- Writing leaves out of order, or dropping repeated values (both 78s need a leaf).
- Drawing gaps between histogram bars as if it were a bar chart.
- Counting a boundary value in two bins, or in the wrong bin.
On the exam
- You might be asked to sketch a histogram or dotplot from data. Label the axes with the variable and units and use a sensible, evenly spaced scale.
- Questions also ask what a given graph does or doesn't show. Remember that a histogram can't tell you an exact individual value.
Connected topics
Videos
Check yourself
4 questions on 1.5 Graphical Representations for One Quantitative Variable. Pick an answer to see if you got it, and why.
A histogram shows the heights of 80 sunflowers in bins of width 20 cm. Which of the following can't be found exactly from the histogram alone?
Two histograms are made from the same data on 200 commute times. One uses bins 2 minutes wide and the other uses bins 20 minutes wide. Which statement is true?
| Stem | Leaves |
|---|---|
| 3 | 2 5 8 |
| 4 | 0 1 4 4 7 9 |
| 5 | 2 3 3 6 |
| 6 | 1 8 |
| 7 | (no leaves) |
| 8 | 4 |
Invented data: wait times, in minutes, for 16 patients at a clinic. Key: 3 | 2 means 32 minutes.
What is the median wait time?
Which of the following best describes the distribution of wait times?
0 of 4 answered