Measurement

Abstract

Why measure? From a scientific perspective, measurement can provide a consistent yardstick for gauging differences. It thereby enables precise estimates of the degree of relationship between variables. I note in this connection that the more precise the measuring instrument and the more accurately the measurements are performed, the narrower is the difference between human perceptions of the physical world and the physical world itself. It is to these and other similar issues we now turn.

Appendices

Appendix A: Units of Measure

This appendix introduces the International System of Units, abbreviated SI. I do not, in general, elaborate on either why the various units of measure are defined as they are, or on how they are used in the various branches of science. For additional reading, see e.g. (http://physics.nist.gov/cuu/Units/), (www.unc.edu/~rowlett/units/index.html) and (Lee 2000; 147–158).

In SI there is only one unit for any physical quantity. Units are divided into two classes: (1) the seven base units, and (2) the derived units formed by combining base units.

• 1. SI base units/quantities

Quantity measured	Unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Thermodynamic temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

Their official definitions are:

Meter: The length of path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. Originally (in 1791) a metre was defined as 10⁻⁷ of the distance from the equator to the North Pole as measured along the meridian passing through Paris. In 1795 a provisional bar of brass was constructed to be used as the standard reference; in 1799 it was replaced by one of platinum; in 1889 an even more stable bar (alloy of platinum and iridium) was constructed to be used as the standard reference for the metre. Subsequent measurements have shown that this distance from the equator to the North Pole is closer to 10,002,290 m. Therefore, since 1960 a metal bar is no longer the standard reference, and the definition of a metre has been changed to that given here.

Kilogram: The unit of mass equal to the mass of the international prototype kilogram (still defined as the mass of water in a cube one-tenth of a metre on a side; a reference cube was made of platinum and iridium). This is the only basic unit still defined by a physical object. All other weight units, including those used in the UK and USA and which are not ordinarily expressed in terms of the metric system, are weighted against the standard kilogram.

Second: The duration of 9 192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of caesium-133 atom. A second was originally defined in terms of the Earth’s daily rotation; s = [1/(24 × 60 × 60)] × (the length of a day), but later it was shown that the speed of rotation of the earth is not constant. Then in 1956 a second was redefined in terms of the Earth’s complete rotation around the sun in a year. In 1967 it was decided that also this modified standard was not sufficiently stable/reliable compared to the current definition based on atomic physics. There is considerable activity at present to make certain that the three atomic watches that control each other in Paris are precise; this is to some extent due to the need of GPS-systems for more precise measurements.

Ampere: That constant current which, if maintained in two straight parallel conductors of infinite length, and of negligible circular cross section, and placed 1 m apart in a vacuum, would produce between these conductors a force equal to 2 × 10⁻⁷ newton per metre of length.

Kelvin: The unit of thermodynamic temperature; the zero point for Kelvin is absolute zero, or the lowest temperature theoretically possible. 0 °C = 273.15 K

Mole: The amount of substance of a system which contains as many elementary entities (atoms, electrons, ions, molecules, etc. or specified groups of such particles) as there are atoms in 0.012 kg of carbon 12. The number is not precisely known (it is roughly 6.02 × 10²³), though the mass of a “thing” relative to that of the carbon-12 atom can be determined.

Candela: The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 10¹² Hz and that has a radiant intensity in that direction of (1/673) watt per steradian. Originally, luminous intensity was defined in terms of amount of light given off by candles.

• 2. Derived units (created using the base units):

  The following is a list of some of the derived units. Some are named from the base units used to define them (e.g. metres/second) while others have been given special names (e.g. newton for force; pascal for pressure; note though that SI base units are used to determine all of them).

Derived quantity measured	Unit	Symbol
Area	Square metre	m2
Volume	Cubic metre	m3
Speed, velocity	Metre per second	m s−1
Acceleration	Metre/second squared	m s−2
Mass density	Kg/metre cubed	kg m−3
Specific volume	Metre cubed per kilogram	m3 kg−1
Plane angle	Radian	rad
Solid angle	Steradian	sr
Angular velocity	Radian per second	rad s−1
Frequency	Hertz	Hz
Force	Newton	N
Pressure, stress	Pascal	Pa
Energy, work	Joule	J
Power	Watt	W
Celsius temperature	Degree Celsius	°C
Current density	Ampere per metre squared	A/m2
Magnetic field strength	Ampere per metre	A/m
Luminance	Candela per metre squared	cd/m2

Prefixes are used when the sizes of the base units are not convenient for a given purpose, e.g. it is easier to write the distance from Shanghai to Copenhagen in kilometres than in metres, and similarly it is easier to measure distances at the level of the atom in terms of e.g. attometres (10⁻¹⁸), zeptometres (10⁻²¹), or yoctometres (10⁻²⁴) than in metres, centimetres or millimetres.

Note too that there are certain units which are not part of SI but which are accepted since they are used widely. For example, minute: 1 min = 60 s; degree: 1° = (Pi/180 rad); litre: 1 L = 10⁻³ m^3; hectare: 1 ha = 10⁴ m² …

Appendix B: Significant Digits/Figures and Rounding

The following guidelines are based on (Lee 2000; 159–164) and (Morgan 2014); many websites provide similar guidelines.

Significant Digits

All measurements have error. The digits that represent the accuracy of actual measurements are called significant digits or significant figures; their number indicates the accuracy of a measurement.

The following are some widely accepted simple rules for determining the number of significant digits in a measurement.

• 1. All numbers without zeroes are significant: 1234.56 kg has six significant digits.

• 2. Zeroes between non-zero digits are significant: 35.09 has four significant digits.

• 3. Zeroes to the left of the first non-zero digit are not significant (they only show the position of the decimal point): 0.00023 has two significant digits

• 4. Zeroes to the left of a decimal point and that are in a number greater than or equal to 10 are significant: 10.123 has 5 significant digits

• 5. Zeroes to the right of a decimal point and that are at the end of the number are significant: 5.60 has three significant digits.

These five rules should be clear. The next rule requires some comments.

• 6. When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 2090 can have three or four significant digits. If it means exactly 2090 it has four significant digits. If it does not mean exactly 2090, it could mean “about 2090”, which could mean closer to 2090 than to 2080 or 3000, i.e. 2090 plus or minus 5. In this case it would have three significant digits.

Possible ambiguities in this sixth rule can be avoided by writing numbers in exponential notation. In the example here, depending on whether the number of significant digits is three or four, we could write 2090 as: 2.09 × 10³ (three significant digits) or 2.090 × 10³ (four significant digits). In this way the number of significant digits is clearly indicated by the number of numerical figures in the ‘digit’ term.

An alternative way of avoiding ambiguities is as follows: If the number of significant digits is four (such that the measurement 2090 is in fact accurate to within ±0.5 due to the precision of the measuring instrument you are using), this could be written: 2090 (±0.5). Otherwise, in the absence of such a clarification, it would be natural to assume the smallest number of significant digits, here three, leading to the understanding that the measurement means 2090 s (±5).

When interpreting measurements provided by others, a good rule to follow is to assume the smallest number of significant digits unless there is information that the accuracy is greater. Similarly, when reporting your own measurements, you should make it clear what the number of significant digits is so that the reader does not have to guess/interpret the accuracy you are working with.

Measured quantities always have significant digits. Other numbers that are not the result of a measurement may have perfect accuracy; “Five people were interviewed” means exactly five people, not 4.5 to 5.5 people, and 1 m is by definition exactly 1000 mm.

Rounding

Quantities are rounded to bring them to the proper number of significant digits after either converting units (say from miles to kilometres) or using quantities in mathematical operations (say after multiplying two quantities).

It is improper to report quantities with greater accuracy than is justified by the accuracy of the original measurements; if you present data with more digits than are significant it misleads others into thinking that the measurements are better (more accurate) than they really are, and this affects how the data are interpreted.

When rounding, first determine the appropriate number of significant digits. Then the following general and broadly accepted rules can be followed:

• 1. If the first insignificant digit is greater than five, the last retained digit is increased by one: 8.472 rounded to two significant digits is 8.5.

• 2. If the first insignificant digit is less than five, the last retained digit is left unchanged: 13.37 rounded to two significant digits is 13.

• 3. If the first insignificant digit is 5 or 5 followed by anything other than zeroes, round up: 6.3753 with three significant digits rounds up to 6.38.

• 4. If the first insignificant digit is 5 or 5 followed only by zeroes, then round up if the last significant digit is odd, and round down if the last significant digit is even: 0.23500 rounded to two significant digits is 0.24 (since the last significant digit, 3, is odd) and 0.24500 rounds down to 0.24 (since the last significant digit is even). The rationale for this rule is to avoid bias in rounding: half of the time one rounds up, half the time one rounds down.

In general, to minimize error, it is best to round numbers after a calculation has been done. This can, for example, be important for many mathematical operations in statistics.

Converting units

When converting units a general rule is to round the resulting quantity so that the accuracy is approximately that of the original measurement. Converting 30 ft to metres on a calculator might give 30 ft × 0.3048 m/ft = 9.1440 m. This should be rounded to 9.1 m (two significant digits). The logic is as follows. The accuracy of the original measurement is one-half foot (6 in.) which is roughly comparable to 0.2 m: (6 in. × 0.0254 m/in. = 0.1524 m). So if one rounded down to 9 m (one significant digit, corresponding to ±0.5 m = ±19.69 in. or roughly ±20 in.), the result would be too imprecise compared to the accuracy of the original measurement. Similarly, if one rounded to 9.14 m (three significant digits, corresponding to ±0.005 m = 0.1969 in. or roughly 0.2 in.) the result would be too accurate compared to the accuracy of the original measurement.

Mathematical operations

When doing arithmetic with measurements, significant digits must be considered. The accuracy of a calculated result is limited by the least accurate measurement involved in the calculation.

Addition and subtraction: The result should be rounded so that there are no significant digits further right than the last significant digit of the least accurate measurement. For example:

• (a) 1585.236 + 234.76 + 1.2 = 1821.196 which is rounded to 1821.2 because the least accurate measurement, 1.2, has only one decimal place.

• (b) 1346.15 − 1218 = 128.15 which is rounded to 128 because the least accurate measurement, 1218, has no decimal places.

Multiplication and division: The result should have the same accuracy (same number of significant digits) as in the measurement with the least number of significant digits. For example:

• 13.32 (4 significant digits) × 1.08 (3 significant digits) × 2.0 (two significant digits) = 28.7712 which is rounded to 29 (2 significant digits).

Measurement ranges

When we have an estimate of a measurement error, for example when we know details about the precision of the measuring instrument we used, the best way to provide the result of a measurement is to give both the best estimate we can of the quantity being measured and of the range within which we can be confident that the quantity lies. In general, as a rule of thumb, one can say that the last significant digit in any reported result should be of the same order of magnitude (in the same decimal position) as the estimate of the uncertainty that is reported.

Thus a statement that the measured volume of a tooth of a fossil is 238,651.7924 ± 40 mm³ is misleading. The estimated uncertainty, 40, means that the digit 5 could be as large as 9 and as small as 1. Clearly the five digits to the right of the 5 have no significance and should be rounded. This leads to the more correct statement that the measured volume is 238,650 ± 40 mm³. If the estimate of the uncertainty is not 40 but 4, then the rounding should lead to 238,651 ± 4. And if the estimate of the uncertainty is 0.4, then the rounding rule of thumb here should lead to 238,651.8 ± 0.4.