Range

The range is maximum minus minimum: $R=x_{\max}-x_{\min}$.

What range measures

Range describes the total span covered by a numerical data set. For $4,7,7,10,15$, the minimum is $4$, maximum is $15$, and range is $15-4=11$.

The range has the same unit as the data. If temperatures are in degrees Celsius, the range is a temperature difference in degrees Celsius.

A reliable calculation

Sorting is helpful but not required if minimum and maximum are identified accurately.

Range versus endpoints

The interval from $28$ to $41$ contains the observed times, but its range is $13$, not “$28$ to $41$.” Endpoints describe location; their difference describes spread.

This distinction is similar to the distance between positions on a number line.

Sensitivity to extremes

Range uses only two observations, so one outlier can change it dramatically. Adding $120$ to the travel-time set raises the range from $13$ to $92$ minutes even though most values stay together.

This sensitivity can be useful when extremes matter, such as operating limits, but it can make range a poor description of typical variability.

Interquartile range

The interquartile range is

It measures the span of the middle $50\%$ of ordered data and resists extreme values. Reporting both range and IQR distinguishes total spread from central spread.

Comparing data sets

Two sets can have equal means but different ranges. The sets $8,9,10,11,12$ and $2,8,10,12,18$ both have mean $10$, but ranges $4$ and $16$. The second set is much more dispersed.

Fair comparisons should use the same units and similar sampling conditions. Range also tends to increase with sample size because larger samples have more chances to include extremes.

Transformations

Adding the same constant to every value shifts minimum and maximum equally, leaving range unchanged. Multiplying every value by positive factor $k$ multiplies range by $k$.

For a negative factor, order reverses, but the new range is multiplied by $|k|$. Unit conversions therefore scale range just as they scale individual differences.

Range in context

A classroom temperature range of $3^\circ$C may indicate stable conditions. A stock-price range does not show the path taken within the interval, and a test-score range does not reveal how scores cluster. Range is one summary, not the whole distribution.

Common mistakes

Adding maximum and minimum. Range is a difference.

Reporting the two endpoints instead of their distance. State both only when useful.

Ignoring an outlier’s effect. Investigate extremes and consider IQR.

Comparing ranges from incompatible units or sample sizes. Standardize the comparison first.

Quick self-check

• Have I identified the true minimum and maximum?
• Did I subtract minimum from maximum?
• Does an outlier dominate the result?
• Would IQR or standard deviation add a better picture?

Related topics

• Mean
• Median
• Box Plots
• Standard Deviation