Chapter 10 · Statistics: samples and populations
Choosing the Right Data Display
See one data set as a dot plot, a histogram and a box plot, and learn to pick the display that best answers your question.
A data display is a picture of a data set. Different displays answer different questions. Before you choose one, ask what you want to know. In this lesson, you will see one data set drawn three ways.
A survey recorded how many minutes each of students waited for the bus one morning. Here are the wait times, in minutes, in order from least to greatest:
A dot plot (also called a line plot) puts one dot for each value above that value on a number line. Because every value gets its own dot, you can see:
- each wait time exactly, from to minutes;
- each cluster, a group of values close together: waits from to minutes, and waits from to minutes;
- each gap, a stretch with no values: no student waited , , or minutes;
- the mode, the value that occurs most often: minutes, with dots.
A histogram sorts the values into equal bins, intervals of the measured quantity that touch, and draws one bar for the count in each bin. Here the bins start at and are each minutes wide: to , to , to and to . Each bin includes its lower edge but not its upper edge, so goes in the to bin and goes in the to bin. The last bin also includes its upper edge, so goes there too. Sorting the ordered list:
- to : , so students;
- to : , so students;
- to : no values, so students, drawn as a thin flat bar;
- to : , so students.
The counts add to . The histogram shows how many values fall in each bin and the overall shape, but it hides the separate values inside each bin.
A box plot shows only the five-number summary: the minimum, lower quartile, median, upper quartile and maximum. With values, the median is the 8th value in order, which is . Leave it out and split the rest into two halves of values:
- lower half : its middle (4th) value is the lower quartile, ;
- upper half : its middle (4th) value is the upper quartile, .
So the summary is . The range is minutes. The interquartile range (IQR), the length of the box, is minutes. The box plot hides each separate value, how many values there are, and any gaps.
Key concept
Let the question choose the display
Before you choose a display, name what you need to know. Use a dot plot when you need separate values, clusters, gaps or the mode, and the data set is small enough to draw a dot for each value. Use a histogram when you need how many values fall in equal intervals, or the overall shape. Use a box plot when you need the median, quartiles, range or IQR. Every display hides something, so the best choice is the one that shows what your question asks for.
| Display | Shows | Hides |
|---|---|---|
Dot plot | Each value, clusters, gaps, the mode | Nothing about the values, but gets crowded with many values |
Histogram | Counts in equal bins, the overall shape | The separate values inside each bin |
Box plot | Five-number summary, range, IQR | Each value, the number of values, gaps |
Worked example
A question about one value
Using the bus wait times, which display best answers Which wait time was most common? What is the answer?
Name what the question asks for: the value that occurs most often, the mode. To find it, you need to see how many times each separate value occurs.
The dot plot gives each value its own stack, so compare the stack heights. The stack at minutes has dots, and every other stack has dots or fewer.
Check what the others hide. The histogram's to bin has the most values, , but it does not show which value inside that bin repeats. The box plot never shows how often a value occurs; its is the lower quartile, not a mode.
Answer
The dot plot is the best choice, and it shows that the most common wait time was minutes.
Worked example
A question about a count in an interval
Which display best answers How many students waited less than minutes? What is the answer?
This asks for a count of values in an interval: from minutes up to, but not including, minutes. Histogram bins are intervals like this.
The bins to and to together hold every wait under minutes, because a value of would belong in the to bin. Add their counts.
Check the others. You could count the dots from to on the dot plot, but only dot by dot, which gets slow with many values. The box plot hides the number of values, so it cannot give this count.
Answer
The histogram answers it most directly: students waited less than minutes.
Worked example
A question about center and spread
A bus planner asks: What was the typical wait, and how spread out was the middle half of the waits? Which display best answers this, and what are the answers?
Here the typical wait means the median, and the spread of the middle half is the IQR. Both come from the five-number summary, which is exactly what a box plot shows.
The line inside the box marks the median.
The box runs from the lower quartile to the upper quartile, so its length is the IQR. Subtract the lower quartile from the upper quartile.
Check the others. On the dot plot you would have to count to the 8th dot and then work out each quartile yourself. The histogram hides the values: counting values in the first bin places the 8th value in the to bin, but it cannot tell you that value is .
Answer
The box plot is the best choice: the median wait was minutes, and the middle half of the waits has an IQR of minutes.
Takeaway: One data set can be drawn in several ways, and each display shows some things and hides others. First decide what your question needs: separate values and the mode, a count in an interval, or the median and spread. Then choose the dot plot, histogram or box plot that shows it.