Problem 4 - Algebra 2

Question

Make a box-and-whisker plot for each set of values. $$ \begin{array}{llllllll}{12} & {11} & {15} & {12} & {19} & {20} & {19}\end{array} $$

Step-by-Step Solution

Answer

The box-and-whisker plot will have a left whisker going from 12 to 11, a box going from 12 to 19 (with a line at 15 for the median), and a right whisker going from 19 to 20.

Step 1: Order the data

Arrange the numbers from smallest to largest. \[11, 12, 12, 15, 19, 19, 20]\]

Step 2: Find the median

Locate the middle value. In this case, with seven numbers, the median is the fourth number, which is 15.

Step 3: Determine the lower quartile

The lower quartile, Q1, is the median of the lower half of data (not including the median if the number of data points is odd). In this case, Q1 is the median of the numbers 11, 12, and 12, which is 12.

Step 4: Determine the upper quartile

The upper quartile, Q3, is the median of the upper half of data (not including the median for odd numbers of data points). Q3 in this case is the median of the numbers 19, 19, and 20, which is 19.

Step 5: Draw the box-and-whisker plot

The box part of the plot is drawn from Q1 to Q3, with a line in the box for the median. The whiskers extend from Q1 to the minimum value and from Q3 to the maximum value. So, the left whisker goes from 12 to 11, the box goes from 12 to 19, and the right whisker goes from 19 to 20.

Key Concepts

Median

The median is a critical concept when dealing with data sets, especially in box-and-whisker plots. It is essentially the middle value in an ordered list of numbers. This makes it a reliable measure of central tendency, as it is unaffected by extreme values or outliers.

In the context of our exercise, we first ordered the numbers from least to greatest. With seven numbers to consider, the median is the fourth value in the sequence, which is 15.

Finding the median in an odd number of data points is straightforward. Just pick the middle number. However, if the data set were even, you would average the two middle numbers to determine the median. Understanding the median is essential because it represents the center of the data in box-and-whisker plots.

Quartiles

Quartiles divide a data set into four equal parts, and they are vital for constructing box-and-whisker plots. The two main quartiles used in these plots are the lower quartile (Q1) and the upper quartile (Q3). These metrics help us understand the distribution of data around the median.

The lower quartile, Q1, is the median of the lower half of the data set. In our exercise, Q1 is calculated by finding the median of the numbers 11, 12, and 12. This gives us a value of 12 for Q1. This value indicates where the first 25% of the data lies.

Similarly, the upper quartile, Q3, represents the median of the upper half of the data set. For our set, the numbers are 19, 19, and 20. The median of these numbers, and thus Q3, is 19. It signifies where the top 25% of the data begins.

Together with the median, these quartiles provide insight into the spread and centering of the data set. This understanding is foundational for interpreting box-and-whisker plots.

Data Visualization

Data visualization transforms numerical data into graphical representations, allowing for easier interpretation and analysis. A box-and-whisker plot is a form of data visualization that visually presents data through a five-number summary.

• Minimum
• Lower Quartile (Q1)
• Median
• Upper Quartile (Q3)
• Maximum

The creation of a box-and-whisker plot involves a few simple steps. Once the quartiles and median have been calculated, the 'box' is drawn from Q1 to Q3, while a line inside the box marks the median. The 'whiskers' extend from the smallest value to Q1, and from Q3 to the largest value. In our exercise's context, this visualization method provides a compact summary of the data:
- The 'box' stretches from the lower quartile, 12, to the upper quartile, 19.
- The left whisker moves from 12 to the minimum, 11, and the right whisker extends from 19 to the maximum, 20.

Utilizing box-and-whisker plots for data visualization offers a clear view of the data's spread, highlighting median value as well as the variability and skewness within the dataset.