These questions ask you to calculate a range, recover an extreme value, or compare how spread out data sets are. They may include unordered lists, equal values, negative numbers or one unusually distant value.
Find both endpoints before subtracting
Scan the whole data set to identify the smallest and largest values. Then subtract smallest from largest. For example, the values 14, 4, 8 and 6 run from 4 to 14, so their range is 10. The numbers between the endpoints do not enter the subtraction. If every value is 7, both endpoints are 7 and the range is zero.
| Question pattern | Method or meaning |
|---|---|
| An unordered list | Inspect every value; the first and last entries may not be the extremes. |
| A negative minimum | Subtracting a negative moves upward across zero, so the distance stays non-negative. |
| The minimum and range are known | Largest equals smallest plus range. |
| Two data sets are compared | Calculate each pair of endpoints separately; matching total-based averages do not imply matching spread. |
| An endpoint is removed | Find the new endpoint before recalculating; the range may change sharply. |
A range answers how far apart the extremes are. It does not tell you what a usual observation looks like or how many observations there are. Rearranging the middle values cannot alter this measure while the minimum and maximum stay fixed.
The traps
| Tempting route | How to reject it |
|---|---|
| Report the largest value | The endpoint is not the distance. Subtract the smallest value from it. |
| Subtract the first entry from the last | List order is irrelevant. Search the entire set for the true extremes first. |
| Add the two endpoints | A sum measures a combined amount, while spread is the distance found by subtraction. |
| Read the range as an average or usual value | Range measures width only. Inspect the observations themselves to discuss what is typical. |
| Assume every value affects the range | Only the smallest and largest matter. Changing or removing an endpoint can have a large effect; changing a middle value may have none. |
| Produce a negative range | Reverse the subtraction. Largest minus smallest is always zero or positive. |
Try it
Four temperature readings are -3, 0, 2 and 6 degrees Celsius. What is the range of these readings? Enter the number of degrees Celsius.
Set A contains 4, 5 and 6, and Set B contains 1, 5 and 9. Both sets have a mean of 5. Which set is more spread out, and why?
- Set A, because its values are closer to the mean.
- Neither set, because the two sets have the same mean.
- Set A, because it has a range of 8 against Set B's range of 2.
- Set B, because it has a range of 8 against Set A's range of 2.
Set P contains 2, 3, 4 and 10, and Set Q contains 2, 9, 9 and 10. Which statements about the ranges of these two sets are true? Select all that apply.
- Set P has the smaller range, because its middle values are closer together.
- Set P has a range of 8.
- Set Q has the smaller range, because its values are clustered near 9.
- Set Q has a range of 8.
- The two sets have the same range.