9.3 Averages and Measures of Spread

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Calculate the mean, median, mode and range for individual data (in a list or frequency table) and explain when each is most useful.
  2. Calculate the mean, median, mode, quartiles, range and interquartile range for individual data and explain when each is most useful.
  3. Calculate an estimate of the mean for grouped data (grouped discrete or grouped continuous).
  4. Identify the modal class from a grouped frequency distribution.

Part 1 — The Four Averages and the Range

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What is an "average"?

An average is a single number that represents a whole set of data. It gives you a sense of what a "typical" value looks like. There are three types of average you need to know: the mean, the median, and the mode.

1.1 The Mode

The mode is the value that appears most often in a data set.

  • If the data is: 3, 5, 5, 7, 9 → the mode is 5 (it appears twice, more than any other value).
  • There can be more than one mode. For example: 2, 2, 4, 4, 6 has two modes: 2 and 4.
  • If every value appears the same number of times, there is no mode.

When to use the mode: Use it when the data is non-numerical (for example, favourite colours or types of pets), or when you want to know the most popular/common value. If there are two or more modes, the mode becomes less useful.

1.2 The Median

The median is the middle value when all values are arranged in order from smallest to largest.

Finding the median — step by step:

Step 1: Write all the values in order from smallest to largest. Step 2: Find the middle value.

  • If there is an odd number of values, the median is the exact middle one.
  • If there is an even number of values, the median is the mean of the two middle values (add them and divide by 2).

Finding the position of the median:

Median position=n+12th term\text{Median position} = \frac{n+1}{2} \text{th term}

where n is the total number of values.

Example (odd number): Data: 4, 7, 9, 11, 15 There are 5 values. Position = (5+1)/2 = 3rd term. The 3rd value is 9. So the median = 9.

Example (even number): Data: 3, 6, 8, 12 There are 4 values. Position = (4+1)/2 = 2.5th term → average of 2nd and 3rd values. (6 + 8) ÷ 2 = 7. So the median = 7.

When to use the median: The median is very useful when the data contains extreme values (very high or very low numbers that don't represent the rest of the data). Unlike the mean, the median is not affected by these extreme values.

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