The following are the distances (in miles) to the nearest airport for 17 families:
8, 16, 16, 17, 17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 32, 36.
Notice that the numbers are ordered from least to greatest.
Make a box-and-whisker plot for the data.
Answer (1)
Find the minimum value: 8.
Find the maximum value: 36.
Find the median (Q2): 25.
Find the first quartile (Q1): 17.
Find the third quartile (Q3): 29.5.
The box-and-whisker plot is constructed using these values.
Explanation
Understanding the Data We are given a set of distances to the nearest airport for 17 families: 8, 16, 16, 17, 17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 32, 36. Our goal is to create a box-and-whisker plot for this data. This involves finding the minimum, maximum, median (Q2), first quartile (Q1), and third quartile (Q3) of the data set.
Finding the Minimum Value The minimum value in the data set is 8 miles. This will be the left whisker.
Finding the Maximum Value The maximum value in the data set is 36 miles. This will be the right whisker.
Finding the Median (Q2) Since there are 17 data points, the median (Q2) is the (17+1)/2 = 9th value. The 9th value is 25.
Finding the First Quartile (Q1) The first quartile (Q1) is the median of the lower half of the data (excluding the median). The lower half is: 8, 16, 16, 17, 17, 22, 23, 24. The median of this set is the average of the 4th and 5th values, which are 17 and 17. Therefore, Q1 = (17+17)/2 = 17.
Finding the Third Quartile (Q3) The third quartile (Q3) is the median of the upper half of the data (excluding the median). The upper half is: 26, 27, 28, 29, 30, 32, 32, 36. The median of this set is the average of the 4th and 5th values, which are 29 and 30. Therefore, Q3 = (29+30)/2 = 29.5.
Creating the Box-and-Whisker Plot Now we have all the values needed to create the box-and-whisker plot:
Minimum: 8 Q1: 17 Median (Q2): 25 Q3: 29.5 Maximum: 36
The box-and-whisker plot will have a box extending from 17 to 29.5, with a line at 25 indicating the median. The whiskers will extend from 8 to 17 on the left and from 29.5 to 36 on the right.
Examples
Box-and-whisker plots are useful for visualizing the distribution of data. For example, a teacher can use a box-and-whisker plot to visualize the distribution of test scores in a class. This allows the teacher to quickly see the median score, the range of scores, and the spread of the scores. This can help the teacher identify students who are struggling and students who are excelling. Similarly, in business, a box-and-whisker plot can be used to visualize the distribution of sales figures, customer satisfaction scores, or other key performance indicators. This allows businesses to quickly identify trends and outliers, and to make data-driven decisions.
