Range

Scientific Inquiry & Measurement

The range is the difference between the largest and smallest values in a set of trials, a quick measure of how spread out the results are.

Formula

\text{range} = \text{max} - \text{min}

Definition

The range tells you how spread out your trials are: subtract the smallest measurement from the largest, $\text{range} = \text{max} - \text{min}$. In a lab it is the simplest measure of precision, and reporting the mean with its range tells a reader how much the individual trials varied. It is the crudest measure of dispersion, using only the two extreme values, growing with sample size, and dominated by a single outlier, so it serves for a quick precision estimate with a handful of trials and for flagging inconsistent data, while standard deviation is preferred for larger sets and formal uncertainty analysis. Half the range is sometimes quoted as a rough uncertainty for small samples.

Example

Cart roll distances of $82$, $85$, and $84\ \text{cm}$ have a range of $85 - 82 = 3\ \text{cm}$, so the trials agree well. Dissolving times at $30^\circ\text{C}$ of $25$, $27$, and $26\ \text{s}$ have a range of $2\ \text{s}$, while times at $50^\circ\text{C}$ of $13$, $17$, and $12\ \text{s}$ have a range of $5\ \text{s}$, so the hotter trials were less consistent, perhaps because the water cooled during the runs. Five cart speeds from $2.35$ to $2.44\ \text{m/s}$ have a range of $0.09\ \text{m/s}$, a rough uncertainty of $\pm 0.045\ \text{m/s}$ on a mean of $2.40\ \text{m/s}$, and a speed of $2.55\ \text{m/s}$ at the next angle lies well outside that band.

Key Insight

A small range means your experiment is repeatable; a big range means something is changing between trials that you need to find. Compare the range to the difference between conditions: if two temperatures differ by $12\ \text{s}$ and the range within each is $2$ to $5\ \text{s}$, the difference is real, but if the range were $15\ \text{s}$ it would not be. Range measures precision, not accuracy; five trials can agree within $0.01\ \text{m/s}$ and all be wrong by $0.2\ \text{m/s}$ if the photogate spacing was measured incorrectly.