The types of numbers we work with…
In science we work with two types of numbers. The first kind of numbers that we will discuss are the exact numbers, and as the name implies these values are known exactly. Most of these numbers are defined. For example, there are exactly 12 units in a dozen, 1000 milligrams in a gram, or 2.54 centimeters in an inch. Conversion factors within systems (e.g. metric to metric conversions) are exact. The “1” in any conversion factor is always exact even if the conversion factor itself is not exact (e.g. 1 lb. = 453.6 g). Exact numbers can also be the result of counting objects. For example, if we were to count apples in a basket, we might observe and record that there were 15 apples which would be an exact value.
The second kind of numbers that we will encounter are inexact numbers. These numbers are obtained by making measurements (which we constantly perform in chemistry!). Every instrument or tool used in chemistry for making measurements has inherent limitations (equipment errors). Additionally, the person operating the instrument or using the tool will see or do things slightly differently than another person. In other words, there will be differences in how each person makes the same measurement (human errors). For example, suppose a group of 5 students are all tasked with measuring the mass of a gold ingot. They each take the same sample to their respective balances and record the value displayed. The value recorded by each student is similar, but varies from measurement to measurement. This could be due to the calibration of the balance or in the way each student reads the mass provided by the balance. The take home message is that uncertainty exists in any measured value! So we will have to pay attention to this fact, and make sure that we take this into consideration when we report values in chemistry that are from measurements or use them in calculations.
Accuracy and precision…
When talking about uncertainty of measured values, often the terms accuracy and precision arise. Accuracy refers to how well the measured value agrees with the correct or actual value. Precision refers to how close the measured values are to one another (are they tightly grouped or all very different?). As an example, consider using a balance to perform several trials collecting the mass of the sample of gold mentioned in the previous section. We often perform multiple trials of experiments and report the result as an average. This allows us to examine the precision of the measurements which we express as a standard deviation (not the topic of this discussion, but worth examining on your own time for future courses in science). Simply put, this is the way that each measurement deviates from the average value reported in our experiment. Ideally, you would like the standard deviation to be small (i.e. for the masses obtained for each trial to be very similar), which indicates higher precision. However, we cannot assume that just because the precision is high that the measurement was accurate. The balance used for the experiment could have been poorly calibrated leading to high precision, but it was not close to the correct value and therefore inaccurate.
Let’s consider the following question:
Which of the three images below represents high accuracy and precision (assume the center of the target is the actual or correct value)?
Examining the images from left to right, we can see that in the first image the green X’s are grouped tightly which represents high precision. The X’s are not at the center of the target so the accuracy is low. In the middle image, the X’s are scattered around the target and not near the center. This would be an example of low accuracy and low precision. Lastly, the target to the right shows high accuracy and precision as all of the X’s are grouped tightly in the center.
To continue with our discussion of numbers in chemistry please see the next post which will cover measurements and significant figures!
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