Systematic Errors

Random Errors

Resolution

Propagation of Errors

Significant Digits in Numerical Values

Suggested Reading

- there is something wrong with the instrument or its data handling system, or
- because the instrument is wrongly used by the experimenter.

**Offset**or**zero setting error**in which the instrument does not read zero when the quantity to be measured is zero.**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.

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:

- errors in measurements of temperature due to poor thermal contact between the thermometer and the substance whose temperature is to be found,
- errors in measurements of solar radiation because trees or buildings shade the radiometer.

Examples of causes of random errors are:

- electronic noise in the circuit of an electrical instrument,
- irregular changes in the heat loss rate from a solar collector due to changes in the wind.

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.

- errors in the original measurements,
- 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(x) | = | d+yx| + | 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 *x _{1}, ... , x_{n}* by means of a function

u=f(x),_{1}, ... , x_{n}

then

| du| = |ud_{1}x| + ... + |_{1}ud_{n}x|,_{n}

where *u _{1}, ... , u_{n}* are the values of the partial derivatives of

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.

## EXAMPLE17.524 + 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. |

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

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

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