162e. Range and Interquartile Range
Learning Intentions
- The range and interquartile range (IQR) are two measures of spread for numerical data
- The range of a set of data is affected by an outlier, but the IQR is not
- Calculate the range of a set of numerical data
- Calculate the IQR of a set of numerical data
Pre-requisite Summary
- Numerical data is data recorded Use numbers
- A measure of spread describes how spread out the data values are
- To calculate spread measures correctly, the data must first be arranged in order
- The range is found by subtracting the smallest value from the largest value
- Quartiles divide an ordered data set into four parts
- The lower quartile
is the median of the lower half of the data - The upper quartile
is the median of the upper half of the data - The interquartile range is found using
- An outlier is a value that is unusually large or unusually small compared with the rest of the data
Worked Examples
Worked Example 1
Solve the range of the data set:
Worked Example 2
Find the lower quartile, upper quartile and IQR of the data set:
Worked Example 3
Find the range and IQR of the data set:
Worked Example 4
The data set is
Find the range and IQR, and comment on the effect of the outlier.
Worked Example 5
The data set is
Write the data in order, then find the range and IQR.
Worked Example 6
A student says that the range and IQR are always equal.
Use the data set
Problems
Problem 1
Find the range of the data set:
Problem 2
Find the lower quartile, upper quartile and IQR of the data set:
Problem 3
Find the range and IQR of the data set:
Problem 4
The data set is
Find the range and IQR, and comment on the effect of the outlier.
Problem 5
The data set is
Write the data in order, then find the range and IQR.
Problem 6
Test the claim that the range and IQR are always equal using the data set
Exercises
Understanding and Fluency
Exercise 1.
Find the range of each data set.
a)
b)
c)
Exercise 2.
Write each data set in order, then find the range.
a)
b)
c)
Exercise 3.
Find
a)
b)
c)
Exercise 4.
Find
a)
b)
c)
Exercise 5.
Find the range and IQR of each data set.
a)
b)
c)
Exercise 6.
Find the range and IQR of each data set.
a)
b)
c)
Exercise 7.
Compare the effect of an outlier on the range and IQR.
a) Data set A:
b) Data set B:
Find the range of each set.
Find the IQR of each set.
Exercise 8.
Compare the effect of an outlier on the range and IQR.
a) Data set A:
b) Data set B:
Find the range of each set.
Find the IQR of each set.
Reasoning
Exercise 9.
Explain why the data should be arranged in order before finding the range or IQR.
Exercise 10.
A student says the range of
Exercise 11.
Explain why an outlier usually changes the range more than the IQR.
Exercise 12.
A student says the IQR uses the middle half of the data, not the extreme values. Explain what this means.
Exercise 13.
Explain why the range and IQR are both measures of spread rather than measures of centre.
Problem-solving
Exercise 14.
The weekly reading times in minutes for seven students are
Find the range and IQR, and comment on the effect of the outlier.
Exercise 15.
The masses of eight bags in kg are
Find the range and IQR.
Exercise 16.
The test scores of seven students are
Find the range and IQR, and explain which measure is more affected by the high score.
Exercise 17.
A gardener measures the heights of nine plants in cm:
Find the range and IQR.
Exercise 18.
Two data sets are shown below.
Set A:
Set B:
Find the range and IQR of each set, then Describe how the outlier affects the results.
Potential Misunderstandings
- Thinking the range is found by adding the largest and smallest values instead of subtracting
- Forgetting to arrange the data in order before finding quartiles
- Using the median of the whole data set instead of the medians of the lower and upper halves when finding quartiles
- Confusing the median with the quartiles
- Thinking the IQR uses all of the data values equally
- Believing that the IQR is always larger than the range
- Thinking the range is not affected by an outlier
- Believing that the IQR changes just as much as the range when an outlier is added
- Mixing up measures of spread with measures of centre
- Forgetting that the range uses only the smallest and largest values, while the IQR focuses on the middle half of the data