Mean, Median, Mode and Range

In this Data Analysis revision note, you will learn to find the mean, median, mode and range of a dataset.

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What is Mean, Median, Mode and Range

Mean, Median, Mode and Range refers to measurements within a single set of data. They are each a single typical value that provides insights into the central tendency and variability of a data set. Grasping these concepts is essential for effective data analysis and making informed decisions based on statistical data

Mean

In statistics, the mean stands as a central measure of a dataset, often referred to as the “average.” 

Calculating mean is the same as calculating average in primary school.

Mean = Sum of Data Value ÷ Number of Data

Mean Example 1

6 students took a test of total of 20 marks. Their scores are 0, 13, 13, 15, 18, and 19. 

(i) Find the sum of their scores.

(ii) Find the average of their scores.

Sum of total scores = 0 + 13 + 13 + 15 + 18 + 19 = 78

Average of their scores = 78 ÷ 6 = 13

Mean Example 2

The masses (in kg) of 30 people are shown below.

Use the table below to calculate an estimate of the mean mass.

Mean Mass = 1780 ÷ 30 = 59.3 kg

Median

In statistics, the “median” is a measure of central tendency that represents the middle number in a dataset when it is ordered from smallest to largest. To find the middle values in a dataset, you can:

1. arrange the data values in ascending order.
2. regardless of an even or odd number of data, determine the position of the median using:

[n + 1] ÷ 2 

3. state the corresponding value of the median [Apply average for even number of data where necessary. See Example 2 below].

Median Example 1 [odd number of data]

Look at these numbers: 3, 13, 7, 41, 21, 23, 39, 23, 42

If we put those numbers in order we have: 3, 7, 13, 21, 23, 23, 39, 41, 42

There are nine numbers. (9 + 1) ÷ 2 = 5.

The median is at the 5th position.

The median value of this set of numbers is 23.

Median Example 2 [even number of data]

Look at these numbers: 3, 13, 7, 41, 21, 23, 39, 23

If we put those numbers in order we have: 3, 7, 13, 21, 23, 23, 39, 41

There are now eight numbers. (8 + 1) ÷ 2 = 4.5 [4.5 lies between 4 and 5]

The median lies between the 4th and 5th numbers.

In this example, the middle numbers are 21 and 23.

To find the value half-way between them, add them together and divide by 2:

And, so, the median in this example is 22. 

Median Example 3

The number of hours spent online for a class of students for a particular Sunday is recorded in the table below.

Find the largest possible value of n if the median number of hours spent online is 2.

Observe that there is a group of 7 students who spent 2 hours online. For n to be largest, let the median position be at the last person in the group of 7 students.

The number of students on the left and right hand sides of the median student must be equal.

21 = n + 10

n = 21 – 10 = 11 (ans)

Note: If we let the median be at the 1st person in the group of 7 students, then we have the following situation.

Then

15 = n+16

n = -1 (not possible)

Mode

In statistics, the “mode” is a measure of central tendency that represents the most frequently observed values in a dataset. In other words, the mode is the value that appears with the highest frequency.

Consider the following dataset of numbers:

3, 3, 5, 7, 12, 12, 12, 12, 20, 23, 23, 23, 23, 29, 39

In this dataset, the value “23” appears most frequently (four times), so the mode is 23.

If a dataset has no numbers occurring more than once, then the dataset has no mode. A dataset can have more than one mode.

Mode Example 1

Find the mode of the following data set.

4, 3, 7, 1, 8, 7, 6, 4, 7

The mode is 7

Mode Example 2

The number of books read in a semester for a class of students is recorded in the table below.

If the modal number of books read in a semester is 2, find the largest possible value of x.

Since the given mode has a frequency of 8 counts, then all other data must have a frequency of less than 8 counts. ∴ largest x = 7.

Range

In statistics, the “range” refers to the difference between the largest and smallest values in a dataset. It is a measure of the spread or variability of a set of values and provides a quick way to assess the extent to which the values differ.

Here’s how to calculate it:

Step 1. Find the maximum value (largest number) in your dataset.

Step 2. Find the minimum value (smallest number) in your dataset.

Step 3. Subtract the minimum value from the maximum value.

For example, if your dataset is 

2, 5, 8, 3, 10

The range would be 10 (maximum) − 2 (minimum) = 8.

In general, the bigger the value of the range, the more “spread out” the data is.

Range Example 1

A dataset is given to be 

3, 5, 0, 3, 9, 1

State the range of the dataset.

Ans: 9 – 0 = 9

Range Example 2

The stem-and-leaf diagram below shows the test scores of 25 students for a particular Math test.

When the test score of another student was included in the stem-and-leaf diagram, the range of the test scores increased by 1.
What are the possible scores that the last student could have obtained?

Current range = 38 – 7 = 31

New range = 31 + 1 = 32

To increase the range, 

(i)the minimum score must be lower or

(ii)the maximum score must be higher.

Possible scores are 7 – 1 = 6 or 38 + 1 = 39.

Advantages and Limitations For Mean, Median, Mode

To analyse data using the mean, median, and mode, we usually use the most suitable measure of central tendency.