Drawing a Box-and-Whisker Plot In Exercises 15–18,
(b) draw a box-and-whisker plot that represents the data set.


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Verified step by step guidance

1. Organize the data set in ascending order. This will help in identifying the minimum, maximum, median, and quartiles. The ordered 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. Identify the five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. The minimum is the smallest value, the maximum is the largest value, the median is the middle value, and Q1 and Q3 are the medians of the lower and upper halves of the data, respectively.
3. Calculate the interquartile range (IQR) to check for potential outliers. The formula for IQR is: $\mathrm{IQR}=Q_3-Q_1$. Any data point below $Q_1-1.5\times \mathrm{IQR}$ or above $Q_3+1.5\times \mathrm{IQR}$ is considered an outlier.
4. Draw a number line that includes the range of the data. Mark the five-number summary (minimum, Q1, median, Q3, and maximum) on the number line. Use a box to represent the interquartile range (from Q1 to Q3) and draw a vertical line inside the box at the median. Extend whiskers from the box to the minimum and maximum values that are not outliers.
5. If there are any outliers, plot them as individual points beyond the whiskers. Label the box-and-whisker plot appropriately to ensure clarity and accuracy.

Key Concepts

Box-and-Whisker Plot

A box-and-whisker plot is a graphical representation of a data set that displays its minimum, first quartile (Q1), median, third quartile (Q3), and maximum values. The 'box' shows the interquartile range (IQR), which contains the middle 50% of the data, while the 'whiskers' extend to the smallest and largest values within 1.5 times the IQR from the quartiles. This plot is useful for visualizing the distribution, central tendency, and variability of the data.

Quartiles

Quartiles are values that divide a data set into four equal parts, providing insights into the distribution of the data. The first quartile (Q1) is the median of the lower half of the data, 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 a box-and-whisker plot, as they determine the boundaries of the box and the placement of the median.

Interquartile Range (IQR)

The interquartile range (IQR) is a measure of statistical dispersion that represents the range between the first quartile (Q1) and the third quartile (Q3). It is calculated as IQR = Q3 - Q1 and indicates the spread of the middle 50% of the data. The IQR is particularly useful for identifying outliers and understanding the variability within a data set, making it a critical component in the construction of box-and-whisker plots.