Why spread matters
Two groups can have the same average but look quite different. Consider two sets of scores: 4, 4, 4, 4 and 1, 3, 5, 7. Both have a mean of 4, yet the first has no variation and the second spans several values. A measure of spread describes that difference. It adds context to a typical value and helps readers judge whether the values are tightly clustered or widely dispersed.
Range, interquartile range (IQR), and standard deviation all describe spread, but they answer different questions. Range looks at the endpoints. IQR looks at the middle half of the data. Standard deviation summarizes how far values tend to be from the mean. Understanding those distinctions makes it easier to choose a useful measure rather than reporting whichever number is available.
Range: the distance between the extremes
The range is the maximum value minus the minimum value. For the sorted values 2, 4, 4, 5, 7, 8, 10, the range is 10 minus 2, or 8. It has the same units as the original data: if these are hours, the range is 8 hours.
Range is quick to calculate and useful when the full observed span matters. If you are planning storage for objects of different lengths, for example, knowing the shortest and longest observed lengths may be more immediately useful than knowing how tightly most lengths cluster.
Its weakness is that it depends entirely on two values. If the largest value in the example were 30 instead of 10, the range would jump to 28 even though the other six values would be unchanged. A large range might reflect broad variation throughout a dataset, one unusual observation, or an error that needs checking. The range alone cannot tell you which.
Interquartile range: the width of the middle half
To find the IQR, first sort the values. Identify the first quartile, Q1, near the 25th percentile, and the third quartile, Q3, near the 75th percentile. Subtract Q1 from Q3. Using the seven values above and a method that excludes the median when splitting the data into halves, Q1 is 4 and Q3 is 8, so the IQR is 4.
The IQR describes the width of the middle 50% of the ordered data. Because it leaves the outer quarters out of the calculation, a change to the highest or lowest value may leave the IQR unchanged. That makes it helpful when a few extreme values would distract from the spread of more typical observations. It is often paired with the median, which also resists being pulled by extreme values.
Quartiles are not calculated identically by every textbook or software package, particularly for small datasets. If exact reproducibility matters, state which method you used. Also remember what the IQR omits: it does not describe how far the most extreme observations lie beyond the middle half.
Standard deviation: variation around the mean
Standard deviation describes how far values typically lie from their mean, using every observation in the calculation. Its calculation starts by finding each value’s difference from the mean, squaring those differences, and averaging them; the final step is to take a square root. For a sample, the usual formula divides the sum of squared differences by one less than the number of observations before taking that root. For a full population, it divides by the number of observations. The resulting standard deviation is in the original units.
Return to the two groups with a mean of 4. The four identical scores have a standard deviation of zero, while 1, 3, 5, 7 has a positive standard deviation because its values differ from the mean. Unlike the range, standard deviation uses information from every value. Unlike the IQR, it is sensitive to extreme values: squaring a large difference gives that observation substantial weight.
Standard deviation is useful alongside the mean when variation around the mean is the question of interest. It does not, by itself, describe the shape of a distribution or guarantee that a particular share of values falls within one standard deviation. Those interpretations need additional assumptions about the data.
Choosing and reporting a measure
Start with the question you want the data to answer. Use the range to show the observed endpoints, especially when those limits matter in practice. Use the IQR with the median to describe the middle of data that may contain unusual values or have a lopsided shape. Use standard deviation with the mean to describe variation around that mean. More than one measure can be useful: reporting both the IQR and the range shows the middle spread and the full span.
Before calculating anything, check that the values represent comparable observations and that missing entries or obvious recording mistakes have been handled deliberately. When presenting results, name the measure, include its units, and give enough context to interpret it. “IQR: 4 hours” says more than “spread: 4,” while a brief note about an extreme value can explain why the range and IQR tell different stories.
