What the Range Measures
In Resistant Versus Non-Resistant Statistics, you learned that a statistic is non-resistant when a few extreme observations can affect it greatly. The range is another non-resistant statistic: it uses only the two endpoints of the data, the minimum and the maximum. That makes it quick to calculate, but also means that one unusually small or large value can determine the reported spread.
As introduced in Describing Spread in Context, the range is the maximum minus the minimum. The result measures the full numerical span of the observed values. It has the same units as the variable. For example, a range of 6 minutes means the largest observed time is 6 minutes greater than the smallest observed time. It does not mean that times are typically 6 minutes apart.
The range depends on the endpoints, not on how the observations between them are arranged. If one observation changes but remains between the same minimum and maximum, the range stays the same. If the unique maximum increases while the minimum stays fixed, the range increases by exactly the amount the maximum increased. A single extreme observation can therefore dominate this measure.
This endpoint focus is also the range’s main limitation. The range does not describe where most observations lie, how tightly they cluster, or whether there are gaps or several concentrations. It can give the same result for data sets with very different patterns. A dotplot, histogram, or other display can help reveal those features; as discussed in Choosing the Best Display for Quantitative Data, the best display depends on the question and the information you need.
A Careful Way to Calculate and Report the Range
To find the range, first identify the smallest and largest values in the data. Do not subtract the first value listed from the last unless the list is already ordered. Then subtract the minimum from the maximum, keeping the units, and interpret the result in context. The maximum and minimum must come from the same group and variable.
Name the group and quantitative variable you are summarizing.
Determine the minimum and maximum, checking all observations rather than relying on their order in the list.
Use maximum minus minimum, not the reverse, and include the variable’s units.
State how far apart the smallest and largest observations are, and do not describe the range as a typical distance or as a summary of the whole distribution.
Worked Examples: Calculating and Interpreting the Range
Worked Example: Wait Times at a Bike-Share Station
A fictional student group records how long, in minutes, seven riders wait for a bike to become available: 4, 5, 6, 7, 8, 9, and 10. Find and interpret the range.
Solution. The minimum wait is 4 minutes, and the maximum wait is 10 minutes. Subtract the minimum from the maximum:
The observed waits span 6 minutes, from 4 minutes to 10 minutes. This does not mean that riders typically waited 6 minutes, or that every pair of riders’ waits differed by 6 minutes. It only reports the difference between the longest and shortest observed waits.
The calculation also leaves out the five values between the endpoints. A range of 6 minutes alone cannot tell us whether those waits are evenly spread across the interval or concentrated near one end.
Worked Example: One Extreme Delivery Distance Changes the Range
A fictional delivery team records the distances, in kilometers, of seven local deliveries: 3, 4, 4, 5, 6, 6, and 7. A later delivery is included instead of the 7-kilometer delivery, giving distances of 3, 4, 4, 5, 6, 6, and 31 kilometers. Compare the ranges.
Solution—Find the original range. The minimum is 3 kilometers and the maximum is 7 kilometers:
Find the new range. After one delivery distance changes, the minimum is still 3 kilometers, but the maximum is now 31 kilometers:
The maximum increased by \(31-7=24\) kilometers, and the range increased by \(28-4=24\) kilometers. In this example, changing one unique maximum while leaving the minimum fixed makes the range larger by exactly the same amount. The new range is much larger even though the other six distances did not change.
Interpret in context. Before the change, the delivery distances spanned 4 kilometers; with the 31-kilometer delivery included, they spanned 28 kilometers. The new range reflects the difference between the shortest local delivery and the longest delivery. By itself, it does not show how the other six distances are distributed or establish whether the 31-kilometer trip is an error. As emphasized in What Outliers Mean in Context, an unusual observation should be investigated in context rather than automatically removed.
Worked Example: Changing an Interior Value Does Not Change the Range
A fictional greenhouse records the heights, in centimeters, of five seedlings: 12, 14, 15, 17, and 19. One interior height is later measured as 16 centimeters instead of 15. Compare the ranges before and after the correction.
Solution—Find the original range. The minimum is 12 centimeters and the maximum is 19 centimeters:
Find the new range. The revised data are 12, 14, 16, 17, and 19 centimeters. The minimum is still 12 and the maximum is still 19:
The range is unchanged because the observation that changed is still between the same endpoints. The revised measurement may matter for other summaries or for describing the data, but it does not affect the range as long as the minimum and maximum remain 12 and 19 centimeters. This example shows that the range is not sensitive to every data change—only changes that alter an endpoint can affect it.
Why the Range Is a Limited Measure of Spread
The range is useful when the question is specifically about the full span between the smallest and largest observed values. It is easy to compute and interpret, and it can quickly indicate whether the observed values cover a narrow or wide interval. But it compresses all the data into one difference and gives no separate information about the endpoints or the observations in between.
For example, consider these two sets of fictional processing times, in seconds:
| Set | Processing times, in seconds | Minimum | Maximum | Range |
|---|---|---|---|---|
| A | 10, 10, 10, 10, 16 | 10 | 16 | 6 seconds |
| B | 10, 11, 12, 14, 16 | 10 | 16 | 6 seconds |
Both sets have a range of 6 seconds. Yet in Set A, four of the five times equal 10 seconds, while in Set B, the observations are spread across several values. The range does not reveal that difference because the endpoints are identical. A display or additional summaries are needed to describe the pattern between the endpoints.
A range can also be strongly affected by one extreme value. If most observations are close together and one observation lies far away, that single value may stretch the full span substantially. In Resistant Versus Non-Resistant Statistics, you learned to call a statistic non-resistant when a few extreme observations can affect it greatly. The range is non-resistant because an extreme minimum or maximum can change it directly.
The range also depends on the sample’s observed extremes. A different sample from the same setting may include a more extreme value, changing the range even when most observations are similar. Thus, the range describes the span in the data actually observed; on its own, it is not a complete description of the distribution or a guarantee about values outside the sample.
For this reason, do not choose the range as though it were interchangeable with every other measure of spread. In Describing Spread in Context, you learned that the IQR describes the width of the middle 50% of the data, while standard deviation describes typical distance from the mean. Those measures answer different questions. The range answers only how far apart the minimum and maximum are.
Common Mistakes and AP Exam Tips
- Subtracting in the wrong order. The range is maximum minus minimum. Subtracting maximum from minimum produces a negative value, which is not the range.
- Using the first and last listed values without checking. A list may not be in order. Inspect all observations to identify the true minimum and maximum.
- Calling the range a typical difference. A range is determined by two endpoints. A full-credit interpretation says the largest observed value is a stated number of units greater than the smallest, in context.
- Assuming every data change affects the range. A change to an observation between the unchanged endpoints leaves the range alone. Check whether the minimum or maximum actually changes.
- Treating a large range as a complete description of spread. A single extreme value can create a large range even if many observations are close together. Mention that the range reports only the full span, and consult the display or other summaries to describe the rest of the distribution.
- Deleting an extreme value automatically. A large range does not prove an observation is invalid. Investigate the observation and explain any decision about it in context.
For full credit, show the endpoint subtraction and name the group, variable, and units in the interpretation. For example: “The delivery distances ranged from 3 to 31 kilometers, so their range was \(31-3=28\) kilometers. This is the full span between the shortest and longest observed deliveries; it does not describe how the other distances are distributed.”
Check Your Understanding
Answer each question using the endpoints and the context, and explain what the range does or does not tell you.
- A park records the lengths of five walks, in kilometers: 2, 3, 3, 5, and 8. What is the range, and how would you interpret it in context?
- The minimum in a data set stays fixed, while its unique maximum increases by 5 units. How does the range change?
- If an observation changes but remains between the same minimum and maximum, what happens to the range? Explain.
- Two data sets have the same minimum and maximum. What can you conclude about their ranges, and what can you not conclude about the patterns of their observations?
- Why is the range considered non-resistant? Describe how one extreme observation can affect it.