Cumulative Rounding Error/Examples/Illustration of Round to Even

Example of Cumulative Rounding Error
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.