Drawing a Box-and-Whisker Plot In Exercises 15–18,
(a) find the five-number summary


2 7 1 3 1 2 8 9 9 2 5 4 7 3 7 5 4
2 3 5 9 5 6 3 9 3 4 9 8 8 2 3 9 5

Verified step by step guidance

1. Step 1: Organize the data set in ascending order. This helps in identifying the minimum, maximum, and quartiles easily. The given data set is: 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9.
2. Step 2: Identify the minimum value and the maximum value in the data set. The minimum value is the smallest number in the ordered data set, and the maximum value is the largest number.
3. Step 3: Find the median (Q2). The median is the middle value of the data set when arranged in ascending order. If the data set has an odd number of values, the median is the middle value. If the data set has an even number of values, the median is the average of the two middle values.
4. Step 4: Determine the first quartile (Q1) and third quartile (Q3). Q1 is the median of the lower half of the data (excluding the overall median), and Q3 is the median of the upper half of the data (excluding the overall median).
5. Step 5: Use the five-number summary (minimum, Q1, median, Q3, maximum) to construct the box-and-whisker plot. Draw a number line, plot the five-number summary points, and create a box from Q1 to Q3 with a line at the median. Extend whiskers from the box to the minimum and maximum values.

Key Concepts

Five-Number Summary

The five-number summary is a descriptive statistic that provides a quick overview of a dataset. It consists of the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values. This summary helps in understanding the distribution and spread of the data, making it easier to identify outliers and the overall range.

Box-and-Whisker Plot

A box-and-whisker plot, or box plot, is a graphical representation of the five-number summary. It displays the minimum, Q1, median, Q3, and maximum values, with a box representing the interquartile range (IQR) and 'whiskers' extending to the minimum and maximum values. This visualization helps in comparing distributions and identifying potential outliers in the data.

Quartiles

Quartiles are values that divide a dataset into four equal parts, providing insights into the distribution of the data. The first quartile (Q1) is the median of the lower half, the second quartile (Q2) is the overall median, and the third quartile (Q3) is the median of the upper half. Understanding quartiles is essential for constructing box-and-whisker plots and analyzing data spread.