Choosing a display for a question
Bar graph, circle graph, dot plot, stem-and-leaf plot, histogram or box plot. Which one answers the question you were asked about your data?
Where you are 12 in Data displays
When you reach for it
You have data for a Statistical questions and variability and want to draw it. Six displays are on offer. None is wrong, but each shows some features of the data and hides others, so the choice rests on two things: what kind of data you have, and what you want to find out. Each display has its own lesson, linked below; this one teaches how to choose. All the data are made up.
The procedure
First, what kind of data
Ask whether each value names a group or is a number (Types of data: categorical, numerical, discrete, continuous). The answer splits the displays in two, because a display for numbers is built on a number line, and categories have no place on one. Favorite colors cannot go on a dot plot, and 120 potato masses would make a bar graph with a bar for nearly every potato.
Displays for categories
A bar graph gives each category a bar whose height is its count, all read against one scale. Bars side by side on one baseline are easy to compare, so a bar graph answers “which is most common?” and “how many more?”. It shows counts, not shares.
A circle graph gives each category a sector sized by its share of the whole, so it answers “what share?” at a glance, but hides the counts unless the total is given. It fits only when the parts make one whole: every case lands in exactly one category.
Displays for numbers
A dot plot puts one dot over each value on a number line. For a small set it shows everything: every value, the most common one, gaps, clusters and shape. It works best when values repeat on a short scale, so that dots stack, and it crowds as the set grows.
A stem-and-leaf plot also keeps every value, digit for digit, but groups the values in rows by their leading digits. It suits a small set spread out with few repeats, and two groups can share one column of stems.
A histogram (drawing one, reading one) counts the values in bins of equal width. It suits a large set, or data that arrive already grouped in a table. It shows shape at any size and gives up the single values.
A box plot draws five numbers: the minimum, Q_1, the median, Q_3 and the maximum. It shows center and spread at a glance and looks just as simple for 1{,}000 values as for 10, so several groups on one scale compare quickly. It hides single values, the count, gaps, clusters and peaks.
Let the question choose
The question names the feature you need. “Which is most common?” needs counts side by side: a bar graph, or a dot plot for numbers. “What share?” needs a whole: a circle graph. “How spread out?” and “Which group is typically larger?” need center and spread: box plots on one scale, or dot plots for small sets. “What shape?” needs a dot plot or stem-and-leaf plot for a small set, a histogram for a large one. When two displays could both answer, take the one where the answer is read off rather than worked out.
Worked examples
Lost property: most common, or what share?
Kelda sorted the 40 items in her school’s lost property box: 13 water bottles, 12 jumpers, 9 hats and 6 lunch boxes. (a) Which kind of item is left behind most often? (b) What share of the box is water bottles? Compare a bar graph and a circle graph.
SolutionThe kinds of item are categories. Both displays fit: each item is of exactly one kind, and 13 + 12 + 9 + 6 = 40, so the four parts make one whole.
(a) The bars stand on one baseline against one scale, so the water bottles’ bar plainly reaches one count above the jumpers’. The circle graph’s sectors are \displaystyle \frac{13}{40} \times 360^\circ = 117^\circ and \displaystyle \frac{12}{40} \times 360^\circ = 108^\circ, only 9^\circ apart; without the printed percents the eye could hardly tell them apart. The bar graph answers (a).
(b) The circle graph shows the water bottles as a little under a third of the circle, labeled 32.5\%. The bar graph gives only the count; you must add every bar to get 40 and divide, \displaystyle \frac{13}{40} = 0.325. The circle graph answers (b).
Bus stop: the most common wait
Kilian timed his wait at the bus stop on 20 school mornings, in whole minutes: 5, 3, 7, 5, 2, 11, 4, 6, 5, 8, 4, 3, 6, 5, 12, 7, 4, 2, 6, 5. Which wait happens most often, and was any wait far from the rest? Compare a dot plot with a histogram of bin width 4 minutes.
SolutionThe waits are a small set of whole minutes on a short scale, so values repeat and dots stack.
The dot plot answers both parts. The tallest stack, five dots, is at 5 minutes, the most common wait. Nothing sits over 9 or 10, and the waits of 11 and 12 minutes stand apart beyond that gap.
In the histogram each bin includes its left edge and not its right, so the waits of 4, 8 and 12 minutes go in the bins starting there. The bins starting at 0, 4, 8 and 12 minutes hold 4, 13, 2 and 1 waits.
The tallest bar says that 13 waits were from 4 to under 8 minutes, but not which minute was most common, and the gap at 9 and 10 vanishes inside the bar from 8 to 12. The histogram shows the overall shape, mostly short waits with a thin tail to the right, but loses what the question asked.
Sunflowers: where the heights crowd
Klara measured the 16 sunflowers in the school garden, in centimeters: 74, 58, 90, 67, 81, 70, 52, 76, 64, 83, 69, 97, 73, 61, 85, 78. In which ten-centimeter stretch do the most heights lie, and how tall was the tallest? Compare a dot plot with a stem-and-leaf plot.
SolutionNo two heights are equal, so the dot plot is a single row of 16 dots spread over 45 centimeters. Nothing stacks, and it is hard to see where the heights crowd.
A stem-and-leaf plot with the tens digit as the stem groups the heights in rows ten centimeters wide, like a histogram on its side, and still keeps every digit.
| Stem | Leaves |
|---|---|
| 5 | 2\;\;8 |
| 6 | 1\;\;4\;\;7\;\;9 |
| 7 | 0\;\;3\;\;4\;\;6\;\;8 |
| 8 | 1\;\;3\;\;5 |
| 9 | 0\;\;7 |
Key: 7 \mid 3 means 73 cm.
The longest row is stem 7: 5 of the 16 sunflowers are from 70 to 79 centimeters. The last leaf gives the tallest exactly, 97 cm, and counting down the leaves, the 8th and 9th heights are 73 and 74, so the median is 73.5 cm. The dot plot holds the same values but leaves the grouping to the reader.
Potatoes: one group or two?
Kornelia’s garden club weighed all 120 potatoes from its harvest. The table counts them in bins that include their left edge; the five-number summary is 62, 156, 228, 299, 391 grams.
| Mass (grams) | Potatoes |
|---|---|
| 50 to 100 | 6 |
| 100 to 150 | 22 |
| 150 to 200 | 26 |
| 200 to 250 | 12 |
| 250 to 300 | 24 |
| 300 to 350 | 22 |
| 350 to 400 | 8 |
(a) Do the masses form one group or two? (b) What mass splits the lighter half of the potatoes from the heavier half? Compare a histogram with a box plot.
Solution120 values are too many to draw one by one, so the table gives a histogram and the summary a box plot.
(a) The histogram has two peaks, from 150 to 200 grams and from 250 to 300 grams, with a dip between: only 12 potatoes lie from 200 to 250 grams. The masses form two groups, perhaps from two kinds of potato, though no display can say why. The box plot shows none of this; its box looks like any other, and the two peaks are invisible. Its median line at 228 even sits in the dip, at a mass few potatoes have. The histogram answers (a).
(b) The box plot gives the median directly: 228 grams. The histogram only narrows it to a bin: 6 + 22 + 26 = 54 potatoes are under 200 grams and 54 + 12 = 66 under 250, so the 60th and 61st masses lie in the bin from 200 to 250, but where in it the histogram cannot say. The box plot answers (b).
Courgettes: which variety is more reliable?
Katrin grew 12 plants of each of two courgette varieties and counted the courgettes picked from each plant over the summer, sorted here.
Variety A: 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 12.
Variety B: 1, 2, 4, 5, 7, 8, 9, 11, 13, 14, 16, 18.
Which variety typically gives more courgettes per plant, and which gives a more predictable crop? Compare dot plots with box plots on one scale.
SolutionWith 12 values, the median is halfway between the 6th and 7th values, Q_1 between the 3rd and 4th, and Q_3 between the 9th and 10th. The five-number summaries are
The box plots answer both questions at a glance. The median lines sit at the same place, 8.5, so neither variety typically gives more. A’s box is 10 - 7.5 = 2.5 courgettes wide and B’s is 13.5 - 4.5 = 9, more than three times as wide: A’s crop is far more predictable. The dot plots show the same, and with 12 plants each they are a fair choice too. With 200 plants of each they would crowd, while the box plots would look just as simple.
Bus stop ages: three questions, three displays
Kelda noted the ages of the 24 people waiting at Kilian’s bus stop one morning, sorted: 7, 9, 11, 12, 12, 14, 15, 15, 15, 16, 17, 22, 28, 31, 34, 35, 41, 44, 47, 52, 58, 63, 69, 74. Choose a display for each question. (a) What share of the people were under 18? (b) What was the middle age, and how spread out were the ages? (c) Was there a cluster, or one most common age?
Solution(a) Ages are numbers, but “under 18” turns them into categories: 11 under 18, 11 from 18 to 64, and 2 aged 65 or over. Every person is in exactly one group, so a circle graph fits, with sectors \displaystyle \frac{11}{24} \times 360^\circ = 165^\circ, 165^\circ and \displaystyle \frac{2}{24} \times 360^\circ = 30^\circ. Its first sector, a little under half the circle, answers (a).
(b) Center and spread: a box plot. The median is halfway between the 12th and 13th ages, \displaystyle \frac{22 + 28}{2} = 25. The lower half gives \displaystyle Q_1 = \frac{14 + 15}{2} = 14.5 and the upper half \displaystyle Q_3 = \frac{44 + 47}{2} = 45.5, so the IQR is 45.5 - 14.5 = 31 years.
(c) Single values matter: a stem-and-leaf plot.
| Stem | Leaves |
|---|---|
| 0 | 7\;\;9 |
| 1 | 1\;\;2\;\;2\;\;4\;\;5\;\;5\;\;5\;\;6\;\;7 |
| 2 | 2\;\;8 |
| 3 | 1\;\;4\;\;5 |
| 4 | 1\;\;4\;\;7 |
| 5 | 2\;\;8 |
| 6 | 3\;\;9 |
| 7 | 4 |
Key: 1 \mid 5 means 15 years.
The row for stem 1 holds 9 of the 24 people, a cluster of teenagers, and the leaf 5 appears three times in it: 15 is the most common age. The box plot hides both; no box plot shows a most common value.
Boundaries
One variable at a time. Two numbers for each case, such as a sunflower’s height and its number of leaves, need a scatter plot; two categories for each case need a two-way table; a value followed over time needs a line graph.
A circle graph needs one whole. If a case can sit in two categories, the sectors mean nothing (Exercise 15).
Grouped data stay grouped. From a table of bins only a histogram can be drawn; the single values are gone.
“Small” and “large” are a judgment. The test is whether every value can still be seen: a stem-and-leaf plot of 60 two-digit values still works, a dot plot of 60 masses to the gram does not.
A good choice drawn badly still misleads. Every display needs a title, labeled axes and units (Reading a graph: titles, labels and units), and an honest scale (Misleading graphs).
Where this goes next
Box plots side by side are the subject of Comparing box plots, and the shapes a histogram or dot plot shows have names, such as skewed. The same habit, question first, later chooses which number to report: a mean or a median for center, the Mean absolute deviation or the IQR for spread.
Exercises
Choosing for a question
For each, name the display you would choose and say why.
- A garden club recorded the crop grown in each of its 40 beds. Question: what share of the beds grow beans?
- A bus company has 600 waiting times from one stop. Question: are most waits short, with a few long ones?
- The heights of the 15 sunflowers in each of two beds, every height kept visible. Question: which bed grew taller sunflowers?
- The number of items handed in to lost property on each of 18 days, whole numbers from 0 to 9. Question: how many items in a day is most common?
- The masses of 80 potatoes from each of four plots. Question: which plot has the heaviest typical potato, and which the most even sizes?
- The type of each bus stopping at a stop in one day: single-decker, double-decker or minibus. Question: which type comes most often?
Two displays of one data set
Kelda also counted the items handed in to lost property on each of 18 school days: 3, 5, 4, 6, 5, 2, 7, 5, 4, 9, 5, 6, 3, 4, 5, 8, 6, 5.
- From the dot plot, which number of items was handed in most often, and on how many days?
- Count the days in histogram bins of width 3 starting at 0, each bin including its left edge. Which bar is tallest? Could the histogram answer Exercise 7?
- Find the five-number summary. Which of the three displays, dot plot, histogram or box plot, shows that exactly 6 days had 5 items?
A large data set
A bus company recorded 160 waits at one stop, in minutes, in bins that include their left edge.
| Wait (minutes) | Passengers |
|---|---|
| 0 to 5 | 48 |
| 5 to 10 | 56 |
| 10 to 15 | 30 |
| 15 to 20 | 16 |
| 20 to 25 | 6 |
| 25 to 30 | 4 |
- Which of the four displays for numbers can be drawn from this table, and which cannot? Why?
- In which bin does the median wait lie? Is the shape roughly symmetric, or does it have a long tail on one side?
- What fraction of the waits were 15 minutes or more? Which shows that fraction more directly: the histogram, or a circle graph with two sectors, “under 15” and “15 or more”?
Bar graph or circle graph
The garden club’s harvest by crop, in kilograms: potatoes 24, carrots 12, beans 9, onions 9, lettuce 6.
- Find the total, and the angle of each sector if the harvest is drawn as a circle graph.
- How many more kilograms of potatoes than carrots were there? What percent of the harvest was potatoes? Which display shows each answer at a glance?
- Kilian asked 25 classmates which garden jobs they would help with, and each could choose more than one: watering 14, weeding 9, planting 11, harvesting 16. Can a circle graph show these results? What should be drawn instead?
Reasoning
- Explain why a box plot cannot answer “which value is most common?”.
- Explain why a histogram is often the better choice for 500 values, even though it hides every single value.
- A stem-and-leaf plot uses the tens digit as its stem. Explain why the lengths of its rows match the bars of a histogram with bins of width 10 starting at a multiple of ten.
- Two data sets have the same five-number summary. Name a display that might tell them apart, and say what it could show.
- Explain why a circle graph needs parts that make one whole, while a bar graph does not.
Answers
- A circle graph: the crops are categories, each bed grows one crop, and the 40 beds make one whole, so the beans’ sector shows their share.
- A histogram: 600 values are too many to draw one by one, and the bars show the shape.
- A back-to-back stem-and-leaf plot: two small groups share one column of stems and every height is kept.
- A dot plot: small whole numbers on a short scale stack up, and the tallest stack is the most common count.
- Four box plots on one scale: 80 values a plot are too many to draw one by one, and each box plot shows its median (typical mass) and IQR (how even) side by side.
- A bar graph: the types are categories, and the tallest bar is the most common.
- 5 items, on 6 days: the tallest stack.
- The bins starting at 0, 3, 6 and 9 hold 1, 11, 5 and 1 days, so the bar from 3 to 6 is tallest. No: it holds the days with 3, 4 and 5 items together.
- Sorted, there are 18 values; the median is halfway between the 9th and 10th, both 5. Q_1 is the middle of the lower nine, 4, and Q_3 the middle of the upper nine, 6. Summary 2, 4, 5, 6, 9. Only the dot plot shows the six days at 5.
- Only a histogram. A dot plot, a stem-and-leaf plot and an exact box plot need the single values, which the table does not hold.
- The median is halfway between the 80th and 81st waits. 48 are under 5 minutes and 48 + 56 = 104 under 10, so it lies in the bin from 5 to 10. The counts 56, 30, 16, 6, 4 fall away to the right: a long tail on the right, not symmetric.
- 16 + 6 + 4 = 26 waits, so \displaystyle \frac{26}{160} = \frac{13}{80}, about 16\%. The circle graph shows it directly; the histogram needs adding and dividing.
- Total 24 + 12 + 9 + 9 + 6 = 60 kg, so each kilogram gets 360^\circ \div 60 = 6^\circ: potatoes 144^\circ, carrots 72^\circ, beans 54^\circ, onions 54^\circ, lettuce 36^\circ.
- 24 - 12 = 12 kg more, read from the bar graph. \displaystyle \frac{24}{60} = 40\%, read from the circle graph.
- No. 14 + 9 + 11 + 16 = 50, not 25: a classmate can be counted in several jobs, so the counts are not parts of one whole. Draw a bar graph, one bar per job.
- A box plot draws five numbers found by position in the sorted list. A value that appears many times looks no different from one that appears once.
- A dot plot of 500 values is a wall of dots and a stem-and-leaf plot has hundreds of leaves. A histogram turns them into a handful of bars whose heights show shape, crowding and gaps.
- The stem 7 collects every value from 70 to 79, exactly the bin from 70 to 80 that includes its left edge and not its right. The number of leaves in a row is the count in that bin, so the rows are the bars turned a quarter turn, with the values kept.
- A dot plot, or a histogram for a large set. The same five numbers can come from data with a gap or two peaks, or from data spread evenly; a display that draws the values or their counts shows which.
- A circle graph shares out one full turn, 360^\circ, among the parts, so the parts must add up to one total with each case counted once. Each bar of a bar graph is measured against the scale on its own, so the counts need not add up to anything.