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


39 36 30 27 26 24 28 35 39 60 50 41 35 32 51

Verified step by step guidance

1. Step 1: Organize the data set in ascending order. The given data set is: 39, 36, 30, 27, 26, 24, 28, 35, 39, 60, 50, 41, 35, 32, 51. Arrange these values from smallest to largest.
2. Step 2: Identify the five-number summary of the data set. This includes the minimum value, first quartile (Q1), median (Q2), third quartile (Q3), and maximum value. Use the ordered data to calculate these values.
3. Step 3: Calculate the interquartile range (IQR) by subtracting Q1 from Q3. The formula is: $\mathrm{IQR}=Q_3-Q_1$. This will help determine the range of the box in the plot.
4. Step 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. 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.
5. Step 5: Check for outliers using the 1.5 * IQR rule. Any data point below $Q_1-\; 1.5\; imes\mathrm{IQR}$ or above $Q_3+\; 1.5\; imes\mathrm{IQR}$ is considered an outlier. Adjust the whiskers accordingly if outliers are present.

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 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 crucial for identifying outliers and is used to determine the length of the box in a box-and-whisker plot, providing insight into the variability of the data set.