Significant figures
From Freepedia
The hypothetical idea of significant figures (sig figs or sf), also called significant digits (sig digs) is a method of expressing error in measurement.
Sometimes the term is used to describe some rules-of-thumb, known as significance arithmetic, which attempt to indicate the propagation of errors in a scientific experiment or in statistics when perfect accuracy is not attainable or not required. Scientific notation is often used when expressing the significant figures in a number.
The concept of significant figures originated from measuring a value and then estimating one degree below the limit of the reading; for example, if an object, measured with a ruler marked in millimeters, is known to be between six and seven millimeters and can be seen to be approximately 2/3 of the way between them, an acceptable measurement for it could be 6.6 mm or 6.7 mm, but not 6.666666 mm as a recurring decimal. This rule is based upon the principle of not implying more precision than can be justified when measurements are taken in this manner.
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Determining significant figures
Significant figures conventionally follow certain sets of rules. Such that:
All non-zero digits are significant: for example, 87.636 has five significant figures. In addition, any zeros that are between non-zeros are also considered significant; for example, 40.02 has four significant figures. Any zeros that follow immediately to the right of the decimal place in numbers smaller than one are not considered significant, e.g., 0.00057 has two sf. The situation regarding trailing zero digits that fall to the left of the decimal place in a number with no digits provided that fall to the right of the decimal place is less clear, but these are typically not considered significant unless the decimal point is placed at the end of the number to indicate otherwise (e.g., "2000." versus "2000"). However, any zeros that follow the last non-zero digit to the right of the decimal point are significant, e.g.: 0.002400 has four significant figures.
Conventionally, a number with value 0 is considered to have one significant figure.
In order to indicate exactly which digits are significant, values such as two thousand can be expressed in scientific notation, if necessary, using the correct number of significant figures. If only two digits — the '2' and the first '0' — are significant (i.e., the true value could be anywhere from 1950 to 2049), the conventional representation is 2.0 × 10³; if three are significant (the value is in the range 1995 to 2004) then it is 2.00 × 10³; if four are significant (from 1999.5 to 2000.4), then it could be either 2000 (two, zero, zero, zero) or 2.000 × 10³. (For clarity, the former form could be written 2000., with a decimal point; otherwise, some may read the number as having just one significant digit and three zeros for placement.) If five, it could be either 2000.0 or 2.0000 × 10³.
The same can be achieved by using another unit for the quantity expressed. A distance of 2000 m is supposed to have four significant digits, but 2 km has only one. More informally it can be done by using words to express numbers. The value 12 million has two significant digits, while officially 12,000,000 has 8. In practical situations it is wise to consider multiple trailing zeroes as insignificant.
Sometimes a bar over a trailing zero is used to indicate that it is significant. For example, <math>20 \bar{0} 0</math> appears to have four significant digits; the bar indicates that in fact the second zero is the last significant digit.
Measuring with significant figures
As illustrated in the above example involving the length measurement in millimeters, the significant figures method is that, when measuring using a non-electronic instrument, the observer should estimate within the nearest tenth of a division marked on the instrument. For example, if a graduated cylinder were marked off at every millilitre (ml), the observer should measure the amount of volume contained in the cylinder to the nearest tenth of a millilitre.
In order to express the degree of precision to which a value was measured, decimal numerals are used. When using significant figures rules, it should be assumed that the last significant digit of every measurement was estimated. Using the previous example, if the observer read the amount of liquid in the cylinder to be exactly at the 12 ml mark, the observer would write the value as 12.0 ml, which would indicate that the tenths place was the precision obtained, and the 0 was estimated. If the cylinder were marked off to every tenth of a ml, the observer would write the value as 12.00 ml.
Note that exact numbers obtained by counting discrete objects are not subject to the rules of significant figures and should be expressed as exact integers. Similarly, mathematical constants (such as π) do not have significant figures—they should be treated as having an infinite number of significant figures. The same is true of defined values, such as the speed of light or the atomic mass of carbon 12. Empirically-determined 'constants', however, do have error bounds; sometimes these bounds can be ignored because the value has been determined to much higher precision than other numbers in the expression. Likewise, some published values (perhaps inevitably) contain false accuracy. An example would be census estimates of a nation's population.
Rounding
There are two commonly-used rules for rounding off.
A good rule to avoid biased results is as follows: If the digit following the last reportable digit has a value of four (or less), then the last reportable digit stays as it was. If the digit following the last reportable digit is six or more, the last reportable digit increases by one. If the digit following the last reportable digit is a five followed by non-zero digits, the last reportable digit will be increased by one, but if the five is succeeded by nothing, odd last reportable digits are decreased by one, and even last reportable digits stay the same.
Alternatively, it is also common (e.g., in accounting) to use a rounding rule that does not depend on the value of the last reportable digit. Typically in such uses, a rounded digit equal to 5 with no following non-zero digits will cause the last reportable digit to be increased by one. In all other cases, the rule is the same as for the other rounding rule described above.
Other methods of rounding include round towards zero (AKA: truncate), round away from zero, floor (reduce non-round numbers to greatest lesser round number) and ceiling (increase non-round numbers to least greater round number).
Rounding can be analyzed as a form of quantization.
Problems with the concept of significant figures
Although the idea of significant digits attempts to deal with the real problem of expressing measurement and calculation error, the system itself leads to furthur (and unneccessary) error in expressing a measurement or calculation. The basis of significant figures is that of rounding, and rounding in itself reduces the accuracy of the measurement (this is because rounding is a technique that fundementally uses addition or subtraction - thus creating artificial error by the amount added or subtracted during rounding).
External links
- Significant Figures Calculator - Displays a number with the desired number of significant digits.
