**James Richard Fromm**

The difference between the actual value of a physical quantity and the measured value
of the quantity is the **error** of the measurement. **Quantitative
measurements are measurements with which error can be associated.**

The amount of error associated with a particular measurement may be considered from the
point of view of precision or the point of view of accuracy. The **precision**
of a measurement expresses the error, or deviation, of the measurement from the average of
a large number of measurements of the same quantity, while the **accuracy**
of a measured value expresses the deviation of the measurement from the true value of the
quantity. Error is considered from the point of view of accuracy when the true value is
known, but when the true value of a quantity is not known precision must be used in place
of accuracy. It is impossible to obtain accuracy if precision cannot be obtained, but
precision does not guarantee accuracy. Any significant **systematic error**
(an error which, for some systematic or determinate reason, influences the measurement in
a known or knowable way) may give results which are very precise--and highly inaccurate.

Scientists often obtain the precision of a measurement not by actually carrying out a large number of measurements but from knowledge of the limitations of the apparatus used to carry out the measurement procedure. The limitations of the apparatus available in an average basic chemistry laboratory used for undergraduate students are given in the Table below. These precisions can be obtained using proper measuring techniques and are a measure of the deviation expected in repetitive measurements.

Instrument Precision balance, platform +/- 0.5 g balance, triple-beam +/- 0.02 g balance, analytical 150 g +/- 0.0001 g graduated cylinder, 10 mL +/- 0.01 mL graduated cylinder, 50 mL +/- 0.5 mL buret, 50 mL +/- 0.1 mL pipet, 50 mL +/- 0.05 mL volumetric flask, 50 mL +/- 0.05 mL volumetric flask, 1000 mL +/- 0.30 mL barometer, mercury +/- 0.67 kPa thermometer, 110oC +/- 0.2 K meter stick +/- 0.001 m

Error can be expressed either as an absolute or a relative quantity; both forms of expression are commonly used. An absolute error has the same units as the measurement to which it relates.

Example. A mass which is actually 214.359206... g weighed on a triple-beam balance would have an error of 0.02 g. This measurement could be expressed as 214.36 +/- 0.02 g. In this expression sufficient figures must be reported for the quantity that the error is meaningful. The same mass weighed on a platform balance would be reported as 214.4 +/- 0.5 g, since no second place after the decimal point could be measured. On an analytical balance, the mass could be measured and reported as 214.3592 +/- 0.0001 g.

When scientific or engineering notation is used, both the error and the exponent can be
written as part of the number. A volume of 1.000 x 10^{+3} cm^{3} could be
measured in a 1.0 dm^{3} volumetric flask with a precision of +/-0.0003 dm^{3}.
(This precision might also be specified as +/-0.3 parts per thousand, abbreviated ppt, or
as +/-300 parts per million, abbreviated ppm.) Such a volume would be written using
scientific notation as 1.0000 +/- 0.0003 x 10^{+3} cm^{3} or as 1.0000(3)
x 10^{+3} cm^{3}. The error given in parentheses applies to the last
figure or figures given in the number. If the volume were measured using a flask
calibrated less precisely, the volume would be written as 1.0000 +/- 0.0020 x 10^{+3}
cm^{3} or as 1.0000(20) x 10^{+3} cm^{3}. Writing the volume as
1.000(20) x 10^{+3} cm^{3} would be in error, since this would mean 1.000
+/- 0.020 x 10^{+3} cm^{3}.

When error is expressed in a relative way, it has no net units because the error is
divided by the value of the measurement to which it refers. It is often convenient to
multiply this ratio, which is the **relative error**, by one hundred to
obtain the **percentage error**.

Example. The errors of the measurements given in the preceding paragraph would be
expressed as 0.009%, 0.2%, and 0.00005%. These errors are very small because measurements
of mass can be made very accurately. Measurements of other quantities are usually less
precise. A volume of 25.32 cm^{3} of water would be measured as 25.3 +/- 0.5 cm^{3}
using a graduated cylinder and as 25.3 +/- 0.1 cm^{3} using a buret. These errors,
expressed as a percentage error, are 2.0% and 0.4%.

When mathematical operations are performed upon numbers which are the result of quantitative measurements, it is often necessary (and always desirable) to know the error of the calculated result. The error in the calculated result will depend upon the errors in the measurements from which the result is calculated. Since it is not reasonable to believe that the error of a calculated result can be less than the error in the least accurate number used in the calculation, scientists assume that errors, like rabbits, always increase or propagate. The rules for propagation of error in arithmetic operations are simple.

**When two values are added or subtracted, the absolute error of the calculated result is taken to be the sum (never the difference) of the absolute errors of the values.****When two values are multiplied or divided, the relative error of the calculated result is taken to be the sum (never the difference) of the relative errors of the values.**

Relative errors must be used in multiplication and division because the results of these operations have different units from those of the original values.

Example. If a cube of iron has a dimension of 14.0 +/- 0.1 cm the relative error in the
dimension is 0.71%. If each dimension is measured separately, then the volume might be
(13.9)(14.0)(14.1) or 2744 cm^{3} with a relative error of 2.13% and would be
expressed as 2744 +/- 58 cm^{3} in absolute form. It would be unreasonable to
carry the decimals of 2743.86, since they cannot be significant if the relative error is
2.3. The mass of the cube, measured as 21.67 +/- 0.01 kg on a large-capacity scale, would
have an error of 0.05%. The density, which is the ratio of mass to volume, would then have
a relative error of 0.76% and could be reported as 7.90 +/- 0.06 g/cm^{3} in
absolute form.

The rules for propagation of error are useful in indicating how many figures should be
reported in an experimentally obtained quantity. A figure is significant only if it
conveys information which is not totally obscured by error. Thus the density in the above
example, which would appear on a calculator as 7.897230321 g/cm^{3}, should not be
reported with more figures than are significant. In this example any numbers beyond the
second decimal place are not significant.

As a rule of thumb, often elaborated into rules of significant figures, a calculated
result should not contain more significant figures than are present in the least precise
measurement from which the calculated result was obtained. **A zero whose only
function is to specify the location of a decimal point is not a significant figure.**

Example. In the value 0.004760 there are four significant figures because the first three zeroes serve only to locate the decimal point. The final zero in 0.004760 is just as significant as the nine in the value 0.004769.

Copyright 1997 James R. Fromm