# Mean, Median, Mode and Range

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

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### Mean, Median, Mode

Mean, Median, and Mode are measures of central tendency that provide a single value representing the center or typical value of a dataset.

#### 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

 What happens if we have a large volume of data to calculate the mean? It is quite common that we need to group the data. Meaning the data will have to be tabulated in table form when we can be handling with large quantity of data or dealing with a wide range of numbers.

#### 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 value of a dataset when it is ordered from smallest to largest.
To find the median of a dataset,

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 occurring value 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.

#### 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.

#### Mean

Representativeness: The mean provides a single value that summarises the entire dataset, offering a representative measure of central tendency.

Ease of Computation: Calculating the mean is relatively straightforward and involves adding up all the values in a dataset and dividing by the number of values. This simplicity makes it a commonly used and easily understandable measure.

Sensitivity to Outliers: The mean is highly sensitive to extreme values (outliers) in the dataset. A single extremely high or low value can significantly affect the mean, leading to a potentially misleading representation of the data.

Skewed Distributions: In skewed distributions (when data tends to be collected on one side of the median), the mean may not accurately reflect the central tendency, as it tends to be pulled in the direction of the skewness (towards the direction with more data).

#### Median

Resilience to Outliers: Unlike the mean, the median is not strongly influenced by extreme values or outliers in the dataset. It is less affected by skewed data distributions.

Ease of Calculation: Finding the median involves ordering the dataset and identifying the middle value. This process is straightforward and does not require complex calculations, making it easily interpretable.

Does Not Utilise Full Information: In calculating the median, information about the values above and below the median is not fully utilised. Unlike the mean, which uses all data points, the median only considers the position of the middle value. This can overlook valuable information about the spread and distribution of the data, especially when dealing with outliers.

#### Mode

Identifying the most frequent value: It highlights the most common occurrence within a dataset, immediately revealing the most prevalent category or value.

Resilience to Outliers: The median is unaffected by outliers.

Lack of information about spread and distribution: Unlike mean and median, the mode only reveals the most frequent value but doesn’t tell us anything about the overall spread or distribution of the data. This can lead to limited understanding of data patterns.

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