Rounding
Rounding is the process of reducing the number of significant digits in a number. The result of rounding is a "shorter" number having fewer non-zero digits yet similar in magnitude. The result is less precise but easier to use. There are several slightly different rules for rounding.
Example: 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80.
Rounding can be analyzed as a form of quantization.
Common method
This method is commonly used, for example in accounting.
- Decide which is the last digit to keep.
- Increase it by 1 if the next digit is 5 or more (this is called rounding up)
- Leave it the same if the next digit is 4 or less (this is called rounding down)
Example: 3.046 rounded to hundredths is 3.05 (because the next digit [6] is 5 or more).
Round-to-even method
This method is also known as statistician's rounding or as bankers' rounding. It is identical to the common method of rounding except when the digit(s) following to rounding digit start with a five and have no non-zero digits after it. The new algorithm is:
- Decide which is the last digit to keep.
- Increase it by 1 if the next digit is 6 or more, or a 5 followed by one or more non-zero digits.
- Leave it the same if the next digit is 4 or less
- Round up or down to the nearest even digit if the next digit is a five followed (if followed at all) only by zeroes. That is, increase the rounded digit if it is currently odd; leave it if it is already even.
With all rounding schemes there are two possible outcomes: increasing the rounding digit by one or leaving it alone. With traditional rounding, if the number has a value less than the half-way mark between the possible outcomes, it is rounded down; if the number has a value exactly half-way or greater than half-way between the possible outcomes, it is rounded up. The round-to-even method is the same except that numbers exactly half-way between the possible outcomes are sometimes rounded up—sometimes down.
Although it is customary to round the number 4.5 up to 5, in fact 4.5 is no nearer to 5 than it is to 4 (it is 0.5 away from either). When dealing with large sets of scientific or statistical data, where trends are important, traditional rounding on average biases the data upwards slightly. Over a large set of data, or when many subsequent rounding operations are performed as in digital signal processing, the round-to-even rule tends to reduce the total rounding error, with (on average) an equal portion of numbers rounding up as rounding down. This generally reduces the upwards skewing of the result.
Round-to-even is used rather than round-to-odd as the latter rule would prevent rounding to a result of zero.
Examples:
- 3.016 rounded to hundredths is 3.02 (because the next digit (6) is 6 or more)
- 3.013 rounded to hundredths is 3.01 (because the next digit (3) is 4 or less)
- 3.015 rounded to hundredths is 3.02 (because the next digit is 5, and the hundredths digit (1) is odd)
- 3.045 rounded to hundredths is 3.04 (because the next digit is 5, and the hundredths digit (4) is even)
- 3.04501 rounded to hundredths is 3.05 (because the next digit is 5, but it is followed by non-zero digits)
Other methods of rounding
Other methods of rounding include "round towards zero" (also known as truncation) and "round away from zero". These introduce more round-off error and therefore are rarely used in statistics and science; they are still used sometimes because they are slightly easier to do. Two specialized methods used in mathematics and computer science are the floor (always round down to the nearest integer) and ceiling (always round up to the nearest integer).
Stochastic rounding is a method that rounds to the nearest integer, but when the two integers are equidistant (e.g., 3.5), then it is rounded up with probability 0.5 and down with probability 0.5. This eliminates any drift, but adds randomness to the process.