PRACTICAL MATHEMATICS

Error Analysis

Contents

Systematic Errors
Random Errors
Resolution
Propagation of Errors
Significant Digits in Numerical Values
Suggested Reading

Systematic Errors

Systematic errors in experimental observations usually come from the measuring instruments. They may occur because: Two types of systematic error can occur with instruments having a linear response:
  1. Offset or zero setting error in which the instrument does not read zero when the quantity to be measured is zero.
  2. Multiplier or scale factor error in which the instrument consistently reads changes in the quantity to be measured greater or less than the actual changes.
These errors are shown in Fig. 1. Systematic errors also occur with non-linear instruments when the calibration of the instrument is not known correctly.

Fig. 1.
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:

The accuracy of a measurement is how close the measurement is to the true value of the quantity being measured. The accuracy of measurements is often reduced by systematic errors, which are difficult to detect even for experienced research workers.

Random Errors

Random errors in experimental measurements are caused by unknown and unpredictable changes in the experiment. These changes may occur in the measuring instruments or in the environmental conditions.

Examples of causes of random errors are:

Random errors often have a Gaussian normal distribution (see Fig. 2). In such cases statistical methods may be used to analyze the data. The mean m of a number of measurements of the same quantity is the best estimate of that quantity, and the standard deviation s of the measurements shows the accuracy of the estimate. The standard error of the estimate m is s/sqrt(n), where n is the number of measurements.

Fig. 2.
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.

Resolution

The resolution of an instrument is the smallest possible difference between separate values of the output (in other words how fine the scale of the instrument is).

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.

Propagation of Errors

When measurements are used to calculate a result, the accuracy of the result is limited by the propagation of errors through the calculation. These errors arise from:
  1. errors in the original measurements,
  2. round-off errors in the calculation.

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(+  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),

then

| du | = | u1dx1 | + ... + | undxn |,

where u1, ... , un are the values of the partial derivatives of f with respect to x1, ..., xn.

Significant Digits in Numerical Values

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.

EXAMPLE

(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.

EXAMPLE

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.

EXERCISES

  1. Find the volume V of a circular cylinder with radius r = 1.26cm and height h = 7.3cm. (V = pi.r2.h)
  2. Find the sum: 0.056×102 + 11.8×10-1

Suggested Reading

  1. Topping, J. (1972). Errors of Observation and Their Treatment (4th Edition), London: Chapman and Hall. [KMUTT Library call number: QA275 TOP 1972]
  2. Acton, Forman S. (1959). Analysis of Straight-Line Data, New York: Dover. [KMUTT Library call number: QA276 ACT 1959]

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By R. H. B. Exell, 2001. King Mongkut's University of Technology Thonburi.