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: {1, 2, 2, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9}.
2. Identify the five-number summary: (1) the minimum value, (2) the first quartile (Q1), (3) the median (Q2), (4) the third quartile (Q3), and (5) the maximum value. Use the ordered data to calculate these values.
3. Calculate Q1 (the median of the lower half of the data, excluding the overall median) and Q3 (the median of the upper half of the data, excluding the overall median).
4. Draw a number line that includes the range of the data. Plot the five-number summary values (minimum, Q1, median, Q3, and maximum) on the number line.
5. Construct the box-and-whisker plot: Draw a box from Q1 to Q3 with a vertical line at the median. Extend whiskers from the box to the minimum and maximum values.

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, median, third quartile, and maximum. 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 helps visualize the distribution, central tendency, and variability of the data.

Quartiles

Quartiles are values that divide a data set into four equal parts, each containing 25% 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, calculated as the difference between the third quartile (Q3) and the first quartile (Q1). It represents the range within which the central 50% of the data lies, providing insight into the spread and variability of the data set. The IQR is crucial for identifying outliers and is used to determine the length of the box in a box-and-whisker plot.