Definition:Cumulative Rounding Error
Definition
Let $S$ be a set of continuous data which is to be rounded to the nearest $n$th power of $10$.
The cumulative rounding error of $S$ is defined as:
- $R = \ds \sum_{x \mathop \in S} \paren {x - \overline x}$
where $\overline x$ denotes the rounded value of a given $x$.
Examples
Illustration of Round to Even
Let $S$ be the following set of numbers reported to $2$ decimal places:
- $S = \set {4.35, 8.65, 2.95, 12.45, 6.65, 7.55, 9.75}$
The sum $\sum S$ of the elements of $S$ is:
- $\sum S = 4.35 + 8.65 + 2.95 + 12.45 + 6.65 + 7.55 + 9.75 = 52.35$
We desire to round the elements of $S$ to $1$ decimal place before adding them.
We need to decide which strategy to use for the treatment of the half:
- rounding up, so, for example, $4.35 \to 4.4$ and $12.45 \to 12.5$
- rounding down, so, for example, $4.35 \to 4.3$ and $12.45 \to 12.4$
- rounding to even, so, for example, $4.35 \to 4.4$ and $12.45 \to 12.4$
First, we use the strategy of rounding up.
Let $S_u$ be the set consisting of the elements of $S$ rounded up:
- $S_u = \set {4.4, 8.7, 3.0, 12.5, 6.7, 7.6, 9.8}$
- $\sum {S_u} = 4.4 + 8.7 + 3.0 + 12.5 + 6.7 + 7.6 + 9.8 = 52.7$
Next, we use the strategy of rounding down.
Let $S_d$ be the set consisting of the elements of $S$ rounded down:
- $S_d = \set {4.3, 8.6, 2.9, 12.4, 6.6, 7.5, 9.7}$
- $\sum {S_d} = 4.3 + 8.6 + 2.9 + 12.4 + 6.6 + 7.5 + 9.7 = 52.0$
Next, we use the strategy of rounding to even.
Let $S_e$ be the set consisting of the elements of $S$ rounded to even:
- $S_e = \set {4.4, 8.6, 3.0, 12.4, 6.6, 7.6, 9.8}$
- $\sum {S_e} = 4.4 + 8.6 + 3.0 + 12.4 + 6.6 + 7.6 + 9.8 = 52.4$
As can be seen, rounding to even gets us closest to the true value.
Also see
- Definition:Calculation Rounding Error, of which a cumulative rounding error is an instance
Sources
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Rounding of Data
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors