161. Mean, Median and Mode
Learning Intentions
- The mean (also called average) and median are different measures of centre for numerical data
- The mean of a set of data can be affected significantly by an outlier, whereas the median is not affected
- Calculate the mean, median and mode for a set of numerical data
Pre-requisite Summary
- Numerical data is data that is recorded Use numbers
- To Solve the mean, all data values are added and the total is divided by the number of values
- To find the median, the data must first be arranged in order
- The mode is the value that occurs most often
- An outlier is a value that is unusually large or unusually small compared with the rest of the data
- Different measures of centre can Describe the same data set in different ways
Worked Examples
Worked Example 1
Find the mean, median and mode of the data set:
Worked Example 2
Find the mean, median and mode of the data set:
Worked Example 3
Find the median and mode of the data set:
Worked Example 4
Find the mean, median and mode of the data set:
Worked Example 5
The data set is
A new value,
Find the new mean and median, and compare them with the original mean and median.
Worked Example 6
A student says that the mean and median of a data set are always equal.
Use the data set
Problems
Problem 1
Find the mean, median and mode of the data set:
Problem 2
Find the mean, median and mode of the data set:
Problem 3
Find the median and mode of the data set:
Problem 4
Find the mean, median and mode of the data set:
Problem 5
The data set is
A new value,
Find the new mean and median, and compare them with the original mean and median.
Problem 6
Test the claim that the mean and median are always equal using the data set
Exercises
Understanding and Fluency
Exercise 1.
Find the mean of each data set.
a)
b)
c)
Exercise 2.
Find the median of each data set.
a)
b)
c)
Exercise 3.
Find the mode of each data set.
a)
b)
c)
Exercise 4.
Find the mean, median and mode of each data set.
a)
b)
c)
Exercise 5.
Find the mean, median and mode of each data set.
a)
b)
c)
Exercise 6.
For each data set, first write the values in order, then find the median.
a)
b)
c)
Exercise 7.
Find the mean and median of each data set.
a)
b)
Compare the results.
Exercise 8.
Find the mean and median of each data set.
a)
b)
Compare the results.
Reasoning
Exercise 9.
Explain why the mean and median are both called measures of centre.
Exercise 10.
A student says the median of
Exercise 11.
Explain why an outlier usually changes the mean more than the median.
Exercise 12.
A data set has mean
Problem-solving
Exercise 13.
The number of books read by five students in a month is
Find the mean, median and mode.
Exercise 14.
The test scores of six students are
Find the mean, median and mode, and comment on the effect of the high score.
Exercise 15.
A basketball player scores
Find the mean, median and mode.
Exercise 16.
A gardener records the heights of six plants in cm:
Find the mean, median and mode, and State whether there appears to be an outlier.
Exercise 17.
A class records the number of minutes students read at home:
Find the mean, median and mode, and explain which measure of centre best represents the data.
Exercise 18.
Two data sets are shown below.
Set A:
Set B:
Find the mean and median of each set, then describe how the outlier affects the results.
Potential Misunderstandings
- Thinking the mean, median and mode always have the same value
- Forgetting to arrange the data in order before finding the median
- Choosing the middle position incorrectly when there is an even number of values
- Thinking the mode must be the largest value rather than the most frequent value
- Adding the values incorrectly or dividing by the wrong number when finding the mean
- Believing that an outlier affects the median just as much as the mean
- Ignoring repeated values when identifying the mode
- Thinking every data set must have exactly one mode
- Forgetting that the mean can be a value that is not actually in the data set