Propagation of Errors
Significant Digits in Numerical Values
Fig. 1. Systematic errors in a linear instrument (full line).
Broken line shows response of an ideal instrument without error.
Examples of systematic errors caused by the wrong use of instruments are:
Examples of causes of random errors are:
Fig. 2. The Gaussian normal distribution. m = mean of measurements. s = standard deviation of measurements. 68% of the measurements lie in the interval m - s < x < m + s; 95% lie within m - 2s < x < m + 2s; and 99.7% lie within m - 3s < x < m + 3s.
The precision of a measurement is how close a number of measurements of the same quantity agree with each other. The precision is limited by the random errors. It may usually be determined by repeating the measurements.
A high resolution is necessary for high accuracy, but it does not give high accuracy when there are large systematic and random errors. We should choose instruments that have a sufficient resolution for the measurements to be made, but there is no need for a resolution greater than the systematic and random errors in the experiments.
In many calculations we may use two or three significant digits more than those justified by the accuracy of the original data. However, when the number of arithmetical operations is very large (which is possible when a computer is used), great care is needed to make sure that the errors do not accumulate and give inaccurate results.
The effects of small errors may often be studied by differentiation. The behavior of the errors is shown by the behavior of the differentials. For example, the error in a sum (or a difference) is the sum (or difference) of the errors in the separate terms:
d(x + y) = dx + dy, d(x - y) = dx - dy.
Since we do not know whether the errors are positive or negative, these equations are replaced by equations for the absolute errors as follows:
| d(x + y) | = | dx | + | dy |.
The fractional, or relative, error in a product (or quotent) is the sum (or difference) of the fractional, or relative, errors in the factors of the product (or quotient):
d(x.y)/(x.y) = dx/x + dy/y, d(x/y)/(x/y) = dx/x - dy/y.
As before, since we do not know whether the errors are positive or negative, we use equations for the absolute values of the relative errors, as follows:
| d(x.y)/(x.y) | = | dx/x | + | dy/y |, | d(x/y)/(x/y) | = | dx/x | + | dy/y |.
If a quantity u is calculated from several measurements x1, ... , xn by means of a function f as follows:
u = f(x1, ... , xn),
| du | = | u1dx1 | + ... + | undxn |,
where u1, ... , un are the values of the partial derivatives of f with respect to x1, ..., xn.
The accuracy of a numerical value may be indicated by the number of digits in the value given. We say that 15.6 has three significant digits, when the last decimal digit 6 may not be completely certain. The addition of more digits (for example, 15.620478) would be meaningless.
In multiplications and divisions, the number of significant digits in the final result should be the same as the number of significant digits in the least accurate factor.
(36.479×2.6)/14.85 = 6.3868956 = approximately 6.4. Although extra digits are kept in the intermediate steps of the calculation, the final result has only two significant digits, because the original factor 2.6 has only two significant digits.
In additions and subtractions, the number of digits after the decimal point in the final result should be the same as the smallest number of digits after the decimal point in the terms of the sums or differences.
17.524 + 2.4 - 3.56 = 16.364 = approximately 16.4. The final result has only one digit after the decimal point because the original term 2.4 has only one digit after the decimal point.
By R. H. B. Exell, 2001. King Mongkut's University of Technology Thonburi.