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.


Vacation Days The number of vacation days used by a sample of 20 employees in a recent year
3 9 2 1 7 5 3 2 2 6
4 0 10 0 3 5 7 8 6 5

Verified step by step guidance

1. Step 1: Organize the data set in ascending order. This will make it easier to calculate the quartiles and create the box-and-whisker plot. The ordered data set is: 0, 0, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9, 10.
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: Draw the box-and-whisker plot. The box represents the interquartile range (from Q1 to Q3), with a line inside the box indicating the median (Q2). The whiskers extend from the minimum value to Q1 and from Q3 to the maximum value, unless there are outliers. Outliers are typically defined as values that are more than 1.5 × IQR below Q1 or above Q3.
5. Step 5: Label the box-and-whisker plot appropriately. Include a title, label the axes, and mark the quartiles, minimum, and maximum values on the plot. If there are outliers, indicate them with a separate marker (e.g., a dot or asterisk).

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, 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 represents the middle 50% of the data, while the 'whiskers' extend to the minimum and maximum values. This plot is useful for visualizing the distribution, central tendency, and variability of the data.

Technology in Statistics

Using technology in statistics involves employing software or tools to perform calculations, create visualizations, and analyze data efficiently. Programs like Excel, R, or Python libraries can automate the process of finding quartiles and generating box-and-whisker plots, making it easier to handle large data sets and perform complex statistical analyses without manual calculations.