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.

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Worked Examples

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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.