Definition:Cumulative Rounding Error

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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


Sources

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