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3 Numbers; counting and measuring, precision, and accuracy

      When we obtain data in our experiments, we either count the occurrence of an event or we make measurements of a specific parameter.

      A count can only be a whole number (i.e. an integer), while a measurement may have any number of decimal places depending on either the accuracy of the instrumentation of the equipment used to make the measurement or any subsequent calculation that uses such measurements. So, if we use a typical home room thermometer to measure temperature, then we should be able to observe whole degree Celsius differences in temperature. If we use a laboratory thermometer then, hopefully, we should achieve an accuracy of half a degree Celsius, and accuracy may be further improved to 0.1 of a degree Celsius by using a reasonably priced digital thermometer. However, if we have access to a high‐quality industrial thermometer for use with thermocouples or resistance thermometer probes to measure temperature, then the accuracy may be improved even further. However, no number of decimal places will yield the true temperature as we will always be unsure of what number lies beyond the last digit. Perhaps the best we can hope for, no matter what we are measuring, is to achieve data with the highest level of precision and accuracy available to us.

Precision and accuracy

      The term precision reflects the consistency of a series of measurements and is therefore the ability to obtain the same value on multiple occasions. So, the measure of precision is related to the spread of the data; the higher the precision of the data, then the less the spread of the values. The measurement of precision is provided by the coefficient of variation while accuracy reflects the nearness of the measurement to the true value.

      Consider the following example.

      Example 3.1

      Two groups of students were asked to make 10 measurements of the pH of a solution (with a known pH of 8.0). The resulting data is shown in the table below.

The respective coefficients of variation show that Group 2 was more precise in their pH measurements compared with the data obtained by Group 1; however, their estimation of the solution's pH was quite poor with a % accuracy of 107.4%. In contrast, Group 1 was 100% accurate in their estimation of the pH, but their measurements were highly variable. Consequently, while Group 2 was very precise, in contrast Group 1 was highly accurate but did not know it due to the inherent variation in their data (see Figure 3.1)!

Table 3.1 Estimation of pH.