Significance arithmetic is a collection of rules-of-thumb which attempt to indicate the propagation of error in a scientific experiment or in statistics when perfect accuracy is not attainable or not required.
The rules are derived on an assumption that the number of significant figures in the operands to an operation is a useful guide to the error bounds of the number.
Multiplication and division using significance arithmetic
When multiplying and dividing numbers together, the product or quotient is rounded to the number of significant figures of that of the factor with the least. For instance, using significant figures rules:
- 8 × 8 = 60
- 8 × 8.0 = 60
- 8.0 × 8.0 = 64
- 8.02 × 8.02 = 64.3
In the above, all numbers are assumed to be measurements (therefore potentially inexact). For example: the answer yielded from 8 × 8 is actually 64, but because 8 is treated as a measurement, it only has one significant figure, and so the answer must be rounded to 60. If we are particularly unlucky in the measurement, this still might be incorrect; if each "8" is actually nearly 8.5, the result could be over 70.
Exact numbers are treated as having a limitless number of significant figures. A trivial example of such a number would be the quotient used in taking the mean, or a defined conversion factor.
When squaring or taking the square root of a value, the number of significant figures decreases by one using some systems of significant digits.
Addition and subtraction using significance arithmetic
When you add or subtract significant figures, limit to, and round your answer to the least number of decimal places in any of the numbers that make up the problem. Some examples using significant figures rules:
- 1 + 1.1 = 2
- 1 is significant up to the ones place, 1.1 is significant up to the tenths place. Of the two, the least accurate is the ones place. The answer cannot have any significant figures past the ones place.
- 1.0 + 1.1 = 2.1
- 1.0, 1.1 are significant up to the tenths place. So will the answer.
- 100 + 110 = 200
- 100 is significant up to the hundreds place, while 110 is up to the tenths place. Of the two, the least accurate is the hundreds place. The answer should not have significant digits past the hundreds place..
- 1.0×102 + 111 = 210
- 1.0×102 is significant up to the tens place while 111 has numbers up until the ones place. The answer will have no significant figures past the tens place.
- 123.25 + 46.0 + 86.26 = 255.5
- 123.25 and 86.26 are significant until the hundredths place while 46.0 is only significant until the tenths place. The answer will be significant up until the tenths place.
The even-odd rule
The even-odd rule, also known as "bankers' rounding", is a rounding rule concerned with situations when the rounding digit is 5 (such as rounding 4.5 to the nearest integer). Its main purpose is to prevent the skewing of data upwards, although it does not exactly prevent the problem so much as provide a counterbalance that, on average, balances out the rounding.
In common rounding rules, if the number directly to the right of the digit to be rounded to is less than five, the digit stays the same; if five or more, the digit is rounded up.
However, when dealing with large batches of data, like in statistics or scientific research, to always round up or to round down, if the rounding digit is equal to exactly five, would skew data upwards. Although we normally round the number 4.5 up to 5, in actuality 4.5 is not any nearer to 5 than it is to 4 (it is 0.5 away from either). Thus, by automatically rounding up when we have a rounding digit of 5, as is done in many everyday cases, this would skew data upwards towards the higher number, when in actuality the data should be in the exact middle.
Thus, when using the significant figures system and rounding in such a situation, the even-odd rule is used: round in whichever direction would make the last digit of the final product even. For example:
- If 3.5 had to be rounded to one significant figure, it would become 4.
- If 2.5 had to be rounded to one significant figure, it would become 2.
While this method is not exactly perfect, over a large set of data, the rounding should average out, with an approxamitely equal amount of numbers rounding up as rounding down. In this way, the even-odd rule avoids skewing data either upwards or downwards.
External links
References
- Daniel B. Delury. "Computation with Approximate Numbers". The Mathematics Teacher, v51, pp521-530. November 1958.