Rate Estimates From Rounded Measurements

Why Rounding Matters for an Average Rate

An average rate is calculated from changes between two endpoints. If those endpoints come from measurements, the recorded numbers may be rounded rather than exact. The arithmetic can be correct and still give a rate that looks more precise than the data justify.

For example, suppose a position is recorded at two times. The average velocity is the change in position divided by the change in time. If either position or time was rounded, the changes—and therefore the calculated rate—are uncertain too. A useful report should distinguish the rate calculated from the displayed values from the range of rates consistent with the measurement precision.

From Rounded Measurements to Possible Values

Rounding to the nearest \(q\) units means that, under the usual nearest-value convention, the actual value is at most \(q/2\) units from the recorded value. For example, a length recorded to the nearest \(0.1\) metre represents a value within \(0.05\) metre of that reading. At an exact halfway point, the rounding convention determines which reading is recorded; for bounds, we can use a closed interval that includes both endpoints.

Suppose two input measurements are recorded as \(\hat{x}_1\) and \(\hat{x}_2\), and the corresponding output measurements as \(\hat{y}_1\) and \(\hat{y}_2\). The average rate calculated from the recorded values is

This is the central estimate based on the displayed readings. It is a useful starting point, but it treats the readings as exact for the purpose of calculation. To assess the effect of rounding, first bound each change.

If the two output readings have uncertainties \(\delta_{y_1}\) and \(\delta_{y_2}\), then the actual output change \(y_2-y_1\) differs from the displayed change \(\hat{y}_2-\hat{y}_1\) by at most \(\delta_{y_1}+\delta_{y_2}\). The same reasoning applies to the input change. These bounds allow the two endpoints to vary independently across their rounding intervals; if the measurements are known to have linked errors, that additional information can sometimes make the bounds narrower.

Why the Change Uncertainty Adds

Write each actual measurement as its recorded value plus an error: \(x_i=\hat{x}_i+\varepsilon_{x_i}\) and \(y_i=\hat{y}_i+\varepsilon_{y_i}\). If measurement \(x_i\) was rounded to the nearest \(q_{x_i}\), then \(|\varepsilon_{x_i}|\leq q_{x_i}/2\). Subtracting the two actual input values gives

The triangle inequality gives \(|\varepsilon_{x_2}-\varepsilon_{x_1}|\leq|\varepsilon_{x_2}|+|\varepsilon_{x_1}|\). Therefore the actual input change is within \(\delta_{x_1}+\delta_{x_2}\) of its displayed change. Repeating the argument for the output gives the corresponding output-change bound.

This argument gives a complete bound when the errors can vary independently within their stated limits. For a calculated rate, there is one more condition: the possible input change must stay positive. If it could be zero, the rate may be undefined or extremely sensitive to tiny changes in the endpoints.

Bounding the Rate

Let the possible output changes be between \(N_-\) and \(N_+\), and the possible input changes be between \(D_-\) and \(D_+\). If \(D_->0\), every possible rate has the form \(N/D\), where \(N_-\leq N\leq N_+\) and \(D_-\leq D\leq D_+\).

The minimum and maximum rates can be found by evaluating \(N/D\) at the four pairs of endpoints. To see why, hold \(D\) fixed: because \(D>0\), increasing \(N\) increases \(N/D\). Then hold \(N\) fixed: as \(D\) changes, the quotient is either increasing or decreasing according to the sign of \(N\), or is constant if \(N=0\). Thus an extreme occurs at an endpoint of each interval, so checking the four corner combinations captures the minimum and maximum.

If the output-change interval is entirely positive, the smallest rate uses the smallest output change and the largest input change; the largest rate uses the largest output change and the smallest input change. If the output-change interval is entirely negative, the denominator choices reverse for the extremes. When the output-change interval includes zero, checking all four combinations avoids guessing which choice gives each extreme.

Worked Examples

A Common Mistake: Counting Rounding Error Only Once

In the first example, each position reading is uncertain by \(0.05\) metre. A common mistake is to assign the position difference an uncertainty of only \(0.05\) metre, as though subtracting two rounded readings cancels one of the endpoint errors. That is not guaranteed: the first error could be negative while the second is positive, increasing the difference by \(0.10\) metre.

For instance, using an incorrect \(0.05\)-metre uncertainty for the position change and \(0.05\)-second uncertainty for the time change would give an apparent rate range from \(19.35/3.05\approx6.344\) to \(19.45/2.95\approx6.593\text{ m/s}\). This fails to include possible rates such as \(19.3/3.1\approx6.226\text{ m/s}\). The correction is to add the two endpoint uncertainties for each difference: \(0.05+0.05=0.10\), then calculate the rate bounds from the resulting change intervals.

Reporting a Reasonable Estimate

A rate calculated from rounded data is not automatically wrong; it is an estimate based on the displayed readings. The range describes what the stated rounding precision allows, not a guarantee that every value in the range is equally likely. In particular, this range is a bound, not a statistical confidence interval.

Use the measurement units in the rate: output units divided by input units. Then report a central estimate at a sensible precision and, when useful, a range that communicates the effect of rounding. Avoid giving many decimal places for the central estimate when the underlying measurements support only a rough value. If the input-change interval includes zero, do not use the positive-denominator rule: the available measurements may not determine a finite, stable rate range at all.