# 8.12 - Boxplots

8.12  The student will

a)  represent numerical data in boxplots;

b)  make observations and inferences about data represented in boxplots; and

c)  compare and analyze two data sets using boxplots.

### BIG IDEAS

• I can determine if there is a correlation between number of study hours and test scores, predict how much my cell phone bill will be, and decide if I need to worry about global warming.
• I will realize that real-life data often follows a linear pattern which provides easy predictions.

### UNDERSTANDING THE STANDARD

2016 VDOE Curriculum Framework - 8.12 Understanding

·  A boxplot (box-and-whisker plot) is a convenient and informative way to represent single-variable (univariate) data.

·  Boxplots are effective at giving an overall impression of the shape, center, and spread of the data. It does not show a distribution in as much detail as a stem and leaf plot or a histogram.

·  A boxplot will allow you to quickly analyze a set of data by identifying key statistical measures (median and range) and major concentrations of data.

·  A boxplot uses a rectangle to represent the middle half of a set of data and lines (whiskers) at both ends to represent the remainder of the data. The median is marked by a vertical line inside the rectangle.

·  The five critical points in a boxplot, commonly referred to as the five-number summary, are lower extreme (minimum), lower quartile, median, upper quartile, and upper extreme (maximum).
Each of these points represents the bounds for the four quartiles. In the example below, the lower extreme is 15, the lower quartile is 19, the median is 21.5, the upper quartile is 25, and the upper extreme is 29.

·  The range is the difference between the upper extreme and the lower extreme. The interquartile range (IQR) is the difference between the upper quartile and the lower quartile. Using the example above, the range is 14 or 29-15. The interquartile range is 6 or 25–19.

·  When there are an odd number of data values in a set of data, the median will not be considered when calculating the lower and upper quartiles.

-  Example: Calculate the median, lower quartile, and upper quartile for the following data values:

3  5  6  7  8  9  11  13  13

Median: 8;  Lower Quartile: 5.5;  Upper Quartile: 12

·  In the pulse rate example, shown below, many students incorrectly interpret that longer sections contain more data and shorter ones contain less. It is important to remember that roughly the same amount of data is in each section. The numbers in the left whisker (lowest of the data) are spread less widely than those in the right whisker.

·  Boxplots are useful when comparing information about two data sets. This example compares the test scores for a college class offered at two different times.

Using these boxplots, comparisons could be made about the two sets of data, such as comparing the median score of each class or the Interquartile Range (IQR) of each class.

### ESSENTIALS

• What are the inferences that can be drawn from sets of data points having a positive relationship, a negative relationship, and no relationship?
Sets of data points with positive relationships demonstrate that the values of the two variables are increasing. A negative relationship indicates that as the value of the independent variable increases, the value of the dependent variable decreases.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

·  8.12a1  Collect and display a numeric data set of no more than 20 items, using boxplots.

·  8.12b1  Make observations and inferences about data represented in a boxplot.

·  8.12b2  Given a data set represented in a boxplot, identify and describe the lower extreme (minimum), upper extreme (maximum), median, upper quartile, lower quartile, range, and interquartile range.

·  8.12c1  Compare and analyze two data sets represented in boxplots.

### KEY VOCABULARY

scatterplots, line of best fit, positive relationship, negative relationship, no relationship

Updated: Nov 20, 2018