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Handout 1 Errors, precision and significant figures

We express a measurement as a product of a number and a unit. The handout refers to rules in reporting

the number of the measurement. However, do not forget that all measurements or calculations must be

reported with their units.

All measurements are subject to error. A systematic error is an error that occurs regularly in the

measurement because of a built-in error in the measuring system. The systematic error is consistent. It

leads to a measurement reading that is either always too high or always too low. A random erroris an

error made by the observer due to limitations in his/her skill or ability to read a measurement. As its

name suggests, it is random leading to readings that fluctuate; sometimes too high and sometimes too low.

Precisionis the degree of reproducibility of a measured quantity. Accuracyis the agreement between the

measured value and the accepted or true value. Note that a measurement can be very precise yet

inaccurate or it can be accurate but with a poor precision. Obviously, we would prefer measurements that

are both very precise and accurate. However, since in many instances we do not know the true value, we

tend to prefer those measurements of high precision to those of poor precision.

When reporting measurements or results of calculations derived from experiments, one must use the

correct significant figures. The number of significant figuresin a measured quantity gives an indication

of the measuring device and the precision of the instrument. The last digit in the reported number is

considered to be uncertain. Unless otherwise stated, that uncertainty is considered to be 1 in the last

digit.

For a summary of rules, see the next page. You should also consult your textbook and the laboratory

manual for additional examples and practice.

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Rules for determining the number of significant figures in measurements

1. All non-zero digits are significant.

2. All zero digits between non-zero digits are significant.

3. All leading zeros, i.e. to the left of the first non-zero digit, are not significant.

4. Zeroes after non-zero digits are not significant if the number does not contain a decimal point.

Otherwise, they are significant.

Ex. 7500 2 significant figures

7500. 4 significant figures

7.050 4 significant figures

5. If necessary, revert to scientific notation to express a number with the correct significant figures.

Rules for determining the number of significant figures in calculations

1. The result of addition or subtraction must be expressed with the same number of digits beyond the

decimal point as the quantity carrying the smallest number of such digits.

Ex. 15.02 g + 9986.0 g + 3.518 g = 10004.5 g

2. The result of multiplication or division may contain only as many significant figures as the least

precisely known quantity in the calculation.

Ex. 14.79 cm x 12.11 cm x 5.05 cm = 904 cm3

3. Exact numbers are considered to have unlimited number of significant figures.

Exact numbers

There are 2 situations when a quantity appearing in a calculation may be exact

1. By definition (such as 3 ft = 1 yard)

2. Counting (such as 2 hydrogen atoms in a hydrogen molecule)

Note that numbers appearing in an equation are considered exact. For the circumference of a circle = 2r,

the number of significant figures will depend on the measured radius r and not on the numbers 2 or

Rounding off1. Determine the number of significant figures required.

2. Carry at least one more digit in your calculations (or as many as your calculator will carry).

3. Round off your final answer to the correct significant figures. If the dropped digit is 5, 6, 7, 8 or 9

increase the last remaining digit by one, otherwise leave it unchanged.

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SI units

The scientific system of measurement is called the Systeme Internationale d'Unitesand is abbreviated SI.

It is a modern version of the metric system which includes seven fundamental quantities from which all

other units of physical quantities can be derived. The seven fundamental units are listed below:

Physical Quantity Unit Abbreviation*

Length meter m

Mass kilogram kg

Time second s

Temperature kelvin K

Amount of substance mole mol

Electric current ampere A

Luminous intensity candela cd

Quantities differing from the base units by powers of ten are noted by the use of prefixes. Listed here are

some prefixes that you may encounter in the course and are expected therefore to know.

Multiple Prefix Name Prefix Symbol*

109 giga G

106 mega M

103 kilo k

102

centi c

103

milli m

106

micro

109

nano n

1012

pico p

1015

femto f

* It is important to use the correct case for units and prefixes. e.g. temperatures should be reported in K

not k and kilograms should be reported as kg not Kg.