Using Technology to Find Quartiles and Draw Graphs In Exercises 23–26, use technology to draw a box-and-whisker plot that represents the data set.


Hourly Earnings The hourly earnings (in dollars) of a sample of 21 employees at a consulting firm
25.89 27.09 31.76 28.28 26.19 27.43 24.06
25.61 22.56 29.76 18.01 23.66 38.24 37.27
32.70 31.12 25.87 15.06 23.12 30.62 19.85

Verified step by step guidance

1. Step 1: Organize the data set in ascending order. This is necessary to calculate quartiles and create the box-and-whisker plot. Arrange the hourly earnings values from smallest to largest.
2. Step 2: Identify the minimum, maximum, median (Q2), first quartile (Q1), and third quartile (Q3). Use the following definitions: Q1 is the median of the lower half of the data (excluding the overall median), Q2 is the median of the entire data set, and Q3 is the median of the upper half of the data (excluding the overall median).
3. Step 3: Use technology (e.g., a graphing calculator, spreadsheet software, or statistical software) to calculate the quartiles and create the box-and-whisker plot. Input the ordered data set into the software and use its built-in functions to compute Q1, Q2, Q3, and the interquartile range (IQR).
4. Step 4: Determine the whiskers of the plot. The lower whisker extends to the smallest data point that is not an outlier, and the upper whisker extends to the largest data point that is not an outlier. Outliers are typically defined as data points that fall below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR.
5. Step 5: Draw the box-and-whisker plot using the calculated quartiles and whiskers. The box spans from Q1 to Q3, with a line at Q2 (the median). The whiskers extend to the minimum and maximum non-outlier values, and any outliers are plotted as individual points.

Key Concepts

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 summarizing data distributions and identifying the spread and center of the data.

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 is the range between Q1 and Q3, while the 'whiskers' extend to the minimum and maximum values. This plot helps visualize the distribution, central tendency, and variability of the data.

Using Technology for Data Analysis

Using technology, such as statistical software or graphing calculators, allows for efficient computation and visualization of data. These tools can quickly calculate quartiles, create box-and-whisker plots, and perform other statistical analyses, making it easier to interpret complex data sets. Familiarity with these technologies enhances the ability to analyze and present data effectively.