Strand 1 — Statistics & Probability
Summary Statistics: Mean, Median, Mode, and Range
1st Year · 2nd Year · 3rd Year (Junior Cert)
- ✓By the end of this lesson students will be able to calculate the mean, median, and mode of a set of data.
- ✓By the end of this lesson students will be able to determine the range of a set of data.
- ✓By the end of this lesson students will be able to interpret what the mean, median, mode, and range tell us about a data set.
- ✓By the end of this lesson students will be able to understand the concept of spread in data and how range relates to it.
Key concepts
The mean (or average) is calculated by adding up all the values in a data set and then dividing by the total number of values. It is a measure of the 'centre' of the data.
The median is the middle value in a data set when the data is arranged in order from smallest to largest. If there is an odd number of data points, the median is the single middle value. If there is an even number of data points, the median is the average of the two middle values.
The mode is the value that appears most frequently in a data set. A data set can have one mode (unimodal), more than one mode (bimodal, trimodal, etc.), or no mode if all values appear with the same frequency.
The range is a measure of the spread or dispersion of a data set. It is calculated by subtracting the smallest value from the largest value in the data set. A larger range indicates greater spread.
Spread refers to how 'spread out' or 'dispersed' the data values are. A small spread means the data points are clustered closely together, while a large spread means they are far apart. The range is a simple measure of spread, giving an idea of the total variability within the data. For example, a small range suggests consistency, while a large range suggests variability.
Key facts to remember
- 1The mean is the sum of all values divided by the number of values.
- 2The median is the middle value when the data is ordered from smallest to largest.
- 3The mode is the value that occurs most frequently in a data set.
- 4The range is the difference between the highest and lowest values in a data set.
- 5Always order the data before finding the median or range.
- 6The mean is affected by extreme values (outliers), while the median is less affected.
- 7The range is a simple measure of the spread or variability of data.
- 8A data set can have no mode, one mode, or multiple modes.
Worked examples
Example 1
For the following set of marks obtained by 7 students in a maths test: 12, 15, 10, 18, 15, 13, 17. Calculate the mean, median, mode, and range.
Answer
Mean = 14.3, Median = 15, Mode = 15, Range = 8
Always order the data first when finding the median or range.
Example 2
The number of goals scored by a football team in their last 6 matches were: 2, 0, 3, 1, 2, 4. Find the mean, median, mode, and range for the number of goals scored.
Answer
Mean = 2, Median = 2, Mode = 2, Range = 4
When there's an even number of data points, the median is the average of the two middle values.
Example 3
Two different classes, Class A and Class B, took the same maths test. Their scores (out of 20) are summarised below: Class A scores: 10, 12, 14, 15, 16, 18 Class B scores: 8, 10, 14, 16, 18, 20 Calculate the range for each class and comment on the spread of scores.
Answer
Class A Range = 8, Class B Range = 12. Class B's scores are more spread out than Class A's scores.
A larger range means the data is more spread out, while a smaller range means the data is more clustered.
Common mistakes
- ✗Not ordering the data before finding the median.
- ✗Confusing the mean with the median or mode.
- ✗Making calculation errors when summing values for the mean.
- ✗Forgetting to subtract the lowest value from the highest value when calculating the range.
- ✗Incorrectly identifying the mode, especially in data sets with multiple modes or no mode.
Exam tips
- ★Always show all your working steps clearly, even for simple calculations, as marks are often awarded for method.
- ★Read the question carefully to ensure you calculate all required statistics (mean, median, mode, range).
- ★Use your calculator efficiently for summing values and dividing, but double-check your input.
- ★If asked to comment on spread, use the range as evidence and explain what it means in context (e.g., 'more consistent' or 'more varied').
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