12 2.4 Box Plots
Box plots (also called box-and-whisker plots or box-whisker plots) give a good graphical image of the concentration of the data. They also show how far the extreme values are from most of the data. A box plot is constructed from five values: the minimum value, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum value. We use these values to compare how close other data values are to them.
To construct a box plot, use a horizontal or vertical number line and a rectangular box. The smallest and largest data values label the endpoints of the axis. The first quartile marks one end of the box and the third quartile marks the other end of the box. Approximately the middle 50 percent of the data fall inside the box. The “whiskers” extend from the ends of the box to the smallest and largest data values. The median or second quartile can be between the first and third quartiles, or it can be one, or the other, or both. The box plot gives a good, quick picture of the data.
Note:You may encounter box-and-whisker plots that have dots marking outlier values. In those cases, the whiskers are not extending to the minimum and maximum values. |
Consider, again, this dataset.
1 1 2 2 4 6 6.8 7.2 8 8.3 9 10 10 11.5
The first quartile, Q1 is 2.
The median, Q2 is 7.
The third quartile, Q3 is 9.
The smallest value is 1.
The largest value is 11.5.
The following image shows the constructed box plot.
The two whiskers extend from the first quartile to the smallest value and from the third quartile to the largest value. The median is shown with a dashed line.
Note:It is important to start a box plot with a scaled number line. Otherwise the box plot may not be useful. |
Example 1
The following data are the heights of 40 students in a statistics class.
59 60 61 62 62 63 63 64 64 64 65 65 65 65 65 65 65 65 65 66 66 67 67 68 68 69 70 70 70 70 70 71 71 72 72 73 74 74 75 77
Construct a box plot with the following properties; the calculator instructions for the minimum and maximum values as well as the quartiles follow the example.
- Minimum value = 59
- Maximum value = 77
- Q1, First quartile = 64.5
- Q2, Second quartile or median= 66
- Q3, Third quartile = 70
- Each quarter has approximately 25% of the data.
- The spreads of the four quarters are 64.5 – 59 = 5.5 (first quarter), 66 – 64.5 = 1.5 (second quarter), 70 – 66 = 4 (third quarter), and 77 – 70 = 7 (fourth quarter). So, the second quarter has the smallest spread and the fourth quarter has the largest spread.
- Range = maximum value – the minimum value = 77 – 59 = 18
- Interquartile Range: IQR = Q3 – Q1 = 70 – 64.5 = 5.5.
- The interval 59–65 has more than 25% of the data so it has more data in it than the interval 66 through 70 which has 25% of the data.
- The middle 50% (middle half) of the data has a range of 5.5 inches.
TI-Calculator:
To find the minimum, maximum, and quartiles:
- Enter data into the list editor (Pres STAT 1:EDIT). If you need to clear the list, arrow u