Measure of Centre Revision Notes for Leaving Cert Mathematics

Measure of Centre

Overview

The measure of centre summarises a data set by identifying a single value that represents the typical or central point of the data. The most common measures of the centre are the mean, median, and mode.

Key Measures of Centre

Mean:

Also known as the average.
Calculated by summing all the data values and dividing by the total number of values:

\text{Mean} = \frac{\sum x}{n}

Suitable for numerical data but sensitive to extreme values (outliers).

Median:

The middle value when the data is ordered from smallest to largest.
If the data set has an even number of values, the median is the average of the two middle numbers:

\text{Median} = \frac{\text{middle value 1} + \text{middle value 2}}{2}

Less affected by outliers than the mean.

Mode:

The most frequently occurring value in the data set.
A data set can have one mode (unimodal), more than one mode (multimodal), or no mode if all values occur only once.

Worked Examples

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Example 1: Calculating the Mean

Problem: Find the mean of the data set: $5, 7, 8, 4, 6$

Solution:

Step 1: Add all the values:

$5 + 7 + 8 + 4 + 6 = :highlight[30]$

Step 2: Divide by the total number of values ( $n = :highlight[5]$ ):

\text{Mean} = \frac{30}{5} = :success[6]

Answer: The mean is $6$ .

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Example 2: Finding the Median

Problem: Determine the median of the data set: $9, 2, 5, 6, 7$

Solution:

Step 1: Arrange the values in ascending order: .

$2, 5, 6, 7, 9$

Step 2: Identify the middle value:

For $n = :highlight[5]$ (odd number of values), the middle value is $6$

Answer: The median is $6$ .

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Example 3: Determining the Mode

Problem: Identify the mode of the data set: $4, 5, 6, 4, 7, 6, 4$

Solution:

Step 1: Count the frequency of each value:

$4$ : 3 times.
$5$ : 1 time.
$6$ : 2 times.
$7$ : 1 time.

Answer: The mode is $4$

Summary

The mean is the average and is calculated as:

\text{Mean} = \frac{\sum x}{n}

The median is the middle value of the data when ordered, which is useful for avoiding the effects of outliers.
The mode is the most frequent value in the data set and can indicate patterns in categorical data.
Choose the appropriate measure based on the data type and presence of outliers:
- Use the mean for balanced data without outliers.
- Use the median for skewed data or when outliers are present.
- Use the mode for categorical data or to identify the most common value.

Measure of Centre (Leaving Cert Mathematics): Revision Notes