Rounding
From Wikipedia, the free encyclopedia
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.
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[edit] Common method
This method is commonly used, for example in accounting. It is the one generally taught in basic mathematics classes. This method is also known as Symmetric Arithmetic Rounding or Round-Half-Up (Symmetric Implementation)
- 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)
Examples:
- 3.044 rounded to hundredths is 3.04 (because the next digit, 4, is less than 5).
- 3.046 rounded to hundredths is 3.05 (because the next digit, 6, is 5 or more).
- 3.0447 rounded to hundredths is 3.04 (because the next digit, 4, is less than 5).
For negative numbers the absolute value is rounded.
Examples:
- −2.1349 rounded to hundredths is −2.13
- −2.1350 rounded to hundredths is −2.14
[edit] Round-to-even method
This method is also known as unbiased rounding or as statistician's rounding or as bankers' rounding. It is identical to the common method of rounding except when the digit(s) following the 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
- Otherwise, all that follows the last digit is a 5 and possibly trailing zeroes; then change the last digit to the nearest even digit. 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)
[edit] History
The Round-to-even method has been the ASTM (E-29) standard since 1940. The origin of the terms unbiased rounding and statistician's rounding are fairly self-explanatory. In the 1906 4th edition of Probability and Theory of Errors [1] Robert Woodward called this "the computer's rule" indicating that it was then in common use by people ("computers") who were calculating mathematical tables. Churchill Eisenhart's 1947 paper "Effects of Rounding or Grouping Data" (in Selected Techniques of Statistical Analysis, McGrawHill, 1947, Eisenhart, Hastay, and Wallis, editors) indicated that the practice was already "well established" in data analysis.
The origin of the term bankers' rounding is more obscure. If this rounding method was ever a standard in banking, the evidence has proved extremely difficult to find. To the contrary, section 2 of the European Commission report The Introduction of the Euro and the Rounding of Currency Amounts [2] suggests that there had previously been no standard approach to rounding in banking.
[edit] Criticism
At first glance, it would seem certain that the round-to-even method would correct a statistical oversight in the common method; but this may not be the case because there is an inherent ambiguity in the way that any number is represented in decimal (see 0.999...). For example, the number 4.5000000... (with a never-ending series of zeros) could also be represented as 4.4999999... (with a never-ending series of nines); the common method would round the former representation up to 5, but the latter representation down to 4, even though they are mathematically identical. So the common method, arguably, already provides the balance sought by the round-to-even method, and the latter method could actually introduce a slight downward bias.
However, it should be noted that this criticism may be flawed. Nothing on earth can actually store an infinite series of anything; so although 4.5000000... may in principle be the same as 4.4999999..., in practice it is extremely rare to see a number visualized with an infinite series of trailing nines (or trailing zeros—which, unlike trailing nines, can be truncated without effect). In particular, the floating-point arithmetic seen in computers always assumes trailing zeros, so for computer calculations the round-to-even method may be more appropriate.
[edit] Other methods of rounding
Other methods of rounding exist, but use is mostly restricted to computers and calculators, statistics and science. In computers and calculators, these methods are used for one of two reasons: speed of computation or usefulness in certain computer algorithms. In statistics and science, the primary use of alternate rounding schemes is to reduce bias, rounding error and drift—these are similar to round-to-even rounding. They make a statistical or scientific calculation more accurate.
[edit] Ease of computation
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 in computer algorithms because they are slightly easier and faster to compute. 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).
[edit] Statistical accuracy
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 reduces any drift, but adds randomness to the process. Thus, if you perform a calculation with stochastic rounding twice, you may not end up with the same answer. The motivation is similar to statistician's rounding.
[edit] Round functions in programming languages
- PHP:
- round(-3.5) gives -4.
- round(8.7352,3) gives 8.735.
- round(4278.5,-2) gives 4300.
- JavaScript:
- Uses Asymmetric Arithmetic Rounding
- Math.round (-3.5) gives -3.
- Visual Basic for Applications:
- Uses Round-Half-Even (Banker's Rounding)
- ? Round(2.5, 0) gives 2.
- http://support.microsoft.com/kb/194983
- Microsoft SQL Server:
- Uses either Symmetric Arithmetic Rounding or Symmetric Round Down (Fix) depending on arguments
- SELECT Round(2.5, 0) gives 3.
- Microsoft Excel:
- Uses Symmetric Arithmetic Rounding
- = ROUND(2.5, 0) gives 3.
- = ROUND(3.5, 0) gives 4.
- = ROUND(-3.5, 0) gives -4
The Round() function is not implemented in a consistent fashion among different Microsoft products for historical reasons.
How To Implement Custom Rounding Procedures
[edit] See also
[edit] External links
- Rounding Numbers
- An introduction to different rounding algorithms that is accessible to a general audience but especially useful to those studying computer science and electronics.
