How to Find the Mean, Median and Mode

Mean, median and mode are three ways to describe the "middle" or "typical" value of a data set. They come up in statistics questions from Year 7 onwards and are used constantly in everyday life, from sports averages to survey results. Each one tells you something slightly different, and knowing when to use which is just as important as knowing how to calculate them.

The mean (average)

The mean is what most people think of when they hear the word "average." You add up all the values and divide by how many there are.

Method:

1. Add all the values together.
2. Divide the total by the number of values.

Example: Find the mean of 4, 7, 9, 12, 8.

1. Total: 4 + 7 + 9 + 12 + 8 = 40
2. Number of values: 5
3. Mean: 40 / 5 = 8

The mean of this data set is 8.

Watch out for outliers

The mean is affected by extreme values. Consider the data set: 5, 6, 7, 8, 54. The mean is 80 / 5 = 16. But 16 does not really represent the "typical" value here because 54 is dragging the mean up. In cases like this, the median is often a better measure.

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The median (middle value)

The median is the middle value when the data is arranged in order from smallest to largest. It is not affected by extreme values, which makes it very useful.

Method:

1. Arrange the values in order from smallest to largest.
2. If there is an odd number of values, the median is the one in the middle.
3. If there is an even number of values, the median is the mean of the two middle values.

Odd number of values

Example: Find the median of 3, 7, 1, 9, 4.

1. In order: 1, 3, 4, 7, 9
2. There are 5 values, so the middle one is the 3rd value.
3. Median: 4

Even number of values

Example: Find the median of 2, 5, 8, 11.

1. Already in order: 2, 5, 8, 11
2. There are 4 values. The two middle values are 5 and 8.
3. Median: (5 + 8) / 2 = 6.5

A quick way to find the position of the median: if there are n values, the median is at position (n + 1) / 2. For 5 values, that is position 3. For 8 values, that is position 4.5, meaning you average the 4th and 5th values.

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The mode (most common value)

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

Example: Find the mode of 3, 5, 7, 5, 9, 5, 2.

The value 5 appears three times, more than any other. The mode is 5.

Special cases

• No mode: If every value appears the same number of times, there is no mode. For example, 2, 4, 6, 8 has no mode.
• Two modes (bimodal): 1, 2, 2, 3, 4, 4, 5 has two modes: 2 and 4.
• More than two modes (multimodal): This is possible but less common in school problems.

The mode is the only measure that works with non-numerical data. If you survey people's favourite colour and the most common answer is blue, then blue is the mode.

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Which one should you use?

Measure	Best used when	Weakness
Mean	Data is spread fairly evenly with no extreme values	Pulled up or down by outliers
Median	Data has outliers or is skewed	Ignores the actual values of most data points
Mode	You want the most common value, or data is categorical	May not exist, or there may be several

In many Year 7 and 8 questions, you will be asked to find all three and then explain which is the most appropriate for a given situation. The explanation is just as important as the calculation.

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A full worked example

Data set: 12, 15, 15, 18, 20, 22, 15, 19

Mean: (12 + 15 + 15 + 18 + 20 + 22 + 15 + 19) / 8 = 136 / 8 = 17

Median: Arrange in order: 12, 15, 15, 15, 18, 19, 20, 22. There are 8 values, so the median is the average of the 4th and 5th: (15 + 18) / 2 = 16.5

Mode: 15 appears three times. Mode = 15

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Common mistakes

• Forgetting to order the data before finding the median. This is the most common error and it gives you the wrong answer every time.
• Dividing by the wrong number for the mean. Count carefully. If there are 8 values, divide by 8, not 7 or 9.
• Confusing the three measures. If you cannot remember which is which: mean is the calculated average, median is the middle, mode is the most frequent. "Mode" and "most" both start with "mo."

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Practise these skills

Statistics questions are some of the most approachable in maths because the calculations are straightforward. The challenge is in understanding what the numbers tell you about the data. Practise with real data sets, such as sports scores, temperatures, or class test results, so the concepts feel concrete.

imSteyn covers mean, median and mode as part of the Year 7 statistics unit in the Australian Curriculum. It gives you data sets to work with and coaches you through the process, asking guiding questions rather than just showing the answer. Try it free to see how it works.