Range
In statistics, when we collect data we want to understand not just where the data is centred, but also how spread out the values are. A dataset where all values cluster tightly together tells a very different story from one where values are scattered widely. Measures of spread — also called measures of dispersion — quantify this variability, and the range is the simplest and most intuitive of them all. The range of a dataset is defined as the difference between the largest value (the maximum) and the smallest value (the minimum). For example, if the highest temperature recorded in a city during July was 38°C and the lowest was 19°C, the range of temperatures is 38 − 19 = 19°C. This single number immediately tells us that temperatures varied by up to 19 degrees over the month. The range is easy to compute and easy to interpret, which makes it a natural first step when exploring any new dataset. Despite its simplicity, the range has important limitations. Because it depends entirely on just two data points — the extreme values — it is highly sensitive to outliers. A single unusually large or small measurement can inflate the range dramatically, giving a misleading impression of how spread out the bulk of the data really is. For this reason, statisticians often pair the range with other measures such as the interquartile range, variance, and standard deviation to get a fuller picture of the data's distribution.
- List All Data Values
- Identify the Maximum and Minimum
- Subtract: Range = Maximum − Minimum
- Interpret the Result in Context
- Compare Ranges Across Datasets
List All Data Values
Write out every value in the dataset. It helps to organise them in order from smallest to largest, which makes the next steps straightforward and reduces the chance of error. For example: 4, 7, 9, 12, 15, 21.
Identify the Maximum and Minimum
The maximum is the largest value in the dataset and the minimum is the smallest. In our ordered list 4, 7, 9, 12, 15, 21, the maximum is 21 and the minimum is 4. When data is already sorted, these are simply the last and first values in the list.
Subtract: Range = Maximum − Minimum
Compute the range by subtracting the minimum from the maximum: Range = 21 − 4 = 17. This means the data spans 17 units from its lowest to its highest point. Always include the units of the original data in your answer — if the values were in kilograms, the range is 17 kg.
Interpret the Result in Context
A large range indicates the data is widely spread; a small range indicates the data clusters closely together. However, always ask: are there outliers driving the range? If a dataset of student test scores is 55, 60, 62, 65, 68, 70, 98, the range is 43 points, but 43 out of 45 students scored between 55 and 70 — the range here is dominated by a single high outlier and may be misleading about typical variability.
Compare Ranges Across Datasets
The range is especially useful when comparing the spread of two or more datasets with the same units and similar centres. If Class A has a range of 12 marks and Class B has a range of 35 marks on the same test, Class B's scores are considerably more varied. This comparison is instant and requires no complex calculation.
The Key Equation
- \text{Range} — Range (same as data): The measure of spread; how far the data extends from its lowest to its highest point.
- x_{\max} — Maximum value (same as data): The largest observed value in the dataset.
- x_{\min} — Minimum value (same as data): The smallest observed value in the dataset.
Worked Example
Problem. A sports coach records the times (in seconds) of eight athletes running 100 m: 11.2, 10.8, 12.5, 11.9, 10.5, 13.1, 11.4, 12.0. Find the range and comment on what it tells you about the spread of the athletes' performance.
Answer. Range = 2.6 seconds, indicating a moderate spread in the athletes' 100 m times.
Origins · 1 minKarl Pearson and how this was discovered
Karl Pearson was a pioneering British statistician who formalised many of the tools of modern descriptive and inferential statistics. He introduced the concept of the standard deviation as a rigorous measure of spread, established the correlation coefficient, and founded the journal Biometrika. His work in the late 19th and early 20th centuries transformed statistics from an informal collection of techniques into a rigorous mathematical discipline, laying the groundwork for all modern measures of spread including those that extend beyond the simple range. Pearson was so committed to his statistical work that he reportedly kept meticulous data records on nearly every aspect of his own life, treating his personal experiences as datasets to be analysed.
Related Concepts
- Interquartile Range (IQR) — The IQR is a more robust measure of spread that focuses on the middle 50% of the data, reducing the influence of outliers that can distort the simple range.
- Variance — Variance measures spread by averaging the squared deviations from the mean, using all data points rather than just the two extremes like the range.
- Standard Deviation — The standard deviation is the square root of variance and is the most widely used measure of spread, providing context that the range alone cannot give.
- Box-and-Whisker Plot — A box plot visually displays the range, IQR, median, and outliers simultaneously, making it one of the best tools for comparing spread across datasets.
- Outliers — Outliers are extreme values that disproportionately affect the range; understanding outliers is essential to correctly interpreting what the range does and does not tell you.