Range

Range is the distance from the minimum to maximum and gives a quick, outlier-sensitive measure of total spread.

A box plot divides ordered data into four quarter groups. A number line carries a box-and-whisker plot with minimum, quartiles, median, and maximum labelled. Unequal segment lengths reveal spread without implying unequal counts per quartile.

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.

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.

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?