Quantitative observations, i.e. those that require a numerical measurement, must communicate three things: a numerical value, proper units of measurement, and the precision with which the measurement is made. Some level of imprecision is inherent in all quantitative measurements. In the first place, the devices we use to measure can usually be read only to a limited number of decimal places. For example, if the smallest division on a ruler is one tenth of a centimeter you can probably read between the lines to estimate the length of an object to the nearest .05 cm but you can not read the measurement beyond the hundredths place with any reliability. If somebody else were to read the same length on the same ruler, they would easily agree with your reading to the nearest full tenth of a centimeter but might well disagree with the number of hundredths of a centimeter in the measurement (because reading the hundredths place requires some estimation and can be subjective). The decimal place in a numerical measurement about which there is some uncertainty in its value is called the least significant digit.

Although successive measurements of the same property should always yield the same value, in practice they do not because of small, uncontrollable changes that can occur from measurement to measurement. For example, the temperature might be a little different or your hands may not be quite in the same position. These random errors will make some of the measurements a little bit larger than they should be while making others a little bit smaller than they should be. If several successive measurements of the same property are made, however, we expect that these random errors will cancel out when an average is taken. Thus we will define precision as the level of agreement between successive measurements of the same property. The more precise our measurements, the more confident we are that the average of those measurements represents the true value of the property.

Precision is commonly confused with accuracy. Accuracy, however, is defined as the level of agreement between a measured value and the true value. Inaccurate measurements do not arise from random errors, but from systematic errors. Systematic errors commonly arise from faulty equipment, through poor design, maintenance, or calibration, or improperly followed procedures. Severe systematic errors will cause measured values to always be lower or always be higher than the true value. Thus no amount of averaging will remove systematic errors. Systematic error can only be removed from a measurement by fixing equipment or properly following procedures. We are, of course, interested in making measurements that are both precise and accurate.

There are many ways of communicating the level of precision (and, by extension, error) in our measurements and their calculated results. Perhaps the simplest method is the method of significant figures. In this method we will only report our measurements and the calculated results out to the least significant digit, and no further.

To determine how many significant figures are in a measurement, follow this simple rule: Count the number of digits in a measurement starting with the first non-zero digit as you read the number from right to left. Zeroes to the left of the first non-zero digit are placeholder zeroes that are there simply to place the decimal point. Placeholder zeroes will disappear if the units of measurement are changed to a smaller size, hence they do not reflect the precision of the measurement in any way. Trailing zeroes to the right of a measurement are assumed to be significant unless we are told otherwise. The best way to avoid confusion over which zeroes are significant and which aren't is to write our measurements and calculated answers in scientific (or exponential) notation. Measurements with more significant figures are more precise.

Three rules are observed in determining correct number of significant figures in a calculation. (1) If an addition or subtraction operation occurs, than the answer must have the same number of decimal places (not significant figures) as the smallest number of decimal places in the original calculation. (2) If a multiplication or division operation occurs, the answer must have the same number of significant figures as the smallest number of significant figures in the original calculation. (3) Exact numbers have an infinite number of significant figures, although they may not all be written, and never limit the number of significant figures in an answer. Exact numbers are counting numbers, such as the number of trials in an average or the number of atoms of an element in a molecule, or mathematically defined numbers, such as p or relationships between different sizes of units (1 kg = 1000 g, for example).

In a multi-step operation it is best to carry the answers to the intermediate steps out at least one decimal point beyond the least significant digit. This is done in order to avoid round-off error. Meanwhile the correct number of significant figures is noted for each of the intermediate answers so that the final answer can be written to the correct number of significant figures. In this week's experiment the final answers for which you will be expected to write the calculated values to the correct number of significant figures are: the average density of water at the end of Part I of the lab, and the specific gravities of your metal slugs at the end of part II of the lab.