Accuracy and Precision

Differences between accuracy and precision

When we make measurements and analyze data in chemistry, there are two things we always need to focus on: precision and accuracy.

Here's a diagram to visually explain accuracy and precision:

Fig. 1: Accuracy and precision as shown by darts. The goal is to get as close as possible to the smallest circle in the centre, so accurate shots will be either in or very close to that centric circle. For a player to have high precision, they will have to always shoot to the same point. Obviously, the best player is the one who can land the most darts close together in the centric circle.

The diagram above shows the results of four dart games, each with their own level of accuracy and precision. We are going to assume that the "bulls-eye" is our target/accepted value (since it gives you the most points). Let's break each down:

1. The game is precise since all the darts landed near each other, but it has low accuracy since the darts aren't near the bulls-eye

2. The game is accurate since the darts are around/close to the bulls-eye, but it has low precision since they aren't near each other.

3. This game is neither accurate nor precise since the darts aren't near each other or the bulls-eye 4. This game is both accurate and precise since the darts are near each other and the bulls-eye

The main difference between accuracy and precision is what value we are trying to get close to in each case: other data points we made (precision) or a target/accepted value (accuracy).

Examples of accuracy in everyday life

Before we look at precision and accuracy in chemistry, let's look at some examples in everyday life.

Accuracy:

• Making a basket in basketball.
• Getting the correct answer on a test.
• Hitting the correct notes while singing.

Precision:

• Consistently hitting the rim in basketball.
• Redoing a problem and getting the same answer each time.
• Holding a note steady while singing.

Precision and accuracy in chemistry

Precision and accuracy are helpful in determining how effective a method is, and what errors might be occurring. Ideally, we would want high accuracy and precision, but that isn't always possible. Having low accuracy and/or precision can actually be helpful when evaluating certain methods.

Here's an example:

So what happened? Since your data was precise but inaccurate, systematic errors occurred.

When running the experiment again, you notice that the scale wasn't properly calibrated, so it subtracted 3 g from all of your data points. But what about experiments that are low precision, but high accuracy? When this happens, it is often due to one of two types of errors: random error or gross error.

When measuring accuracy, we typically use the average of our data, so if you have one gross outlier or measurements below and above the desired value, the average may end up close to our desired value. While it may seem accurate, this accuracy is based on luck more than anything else. High accuracy but low precision experiments aren't desirable in a lab setting, since the "accuracy" is more so due to chance than proper measurement.

How to calculate accuracy and precision

While there are several methods to calculate accuracy and precision, the most common (and simple) are percent error (for accuracy) and standard deviation (for precision).

Percent error and standard deviation formulas

Both percent error and standard deviation have formulas that can help us calculate how inaccurate or imprecise our experiments were:

Percent error formula:

\[\delta = \big| \frac{v_A - v_E}{v_E} \big | \cdot 100\%\]

Where,

• \(\delta\) is the percent error,
• \(v_A\) the actual value that was observed and
• \(v_E\) the expected or target value.

Standard deviation formula

\[\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}}\]

Where:

• \(\sigma\) is the standard deviation of your measures,
• \(N\) is the number of samples or measurements made,
• \(x_i\) is each value from your measurements and
• \(\mu\) is the mean of your measurements.

No matter what, there will always be some error, but we should always aim to minimize them. There isn't a standard "good" percent error, as it depends on how difficult the measurements you make are. The main goal is to keep the value as close to 0 as possible.

How to calculate accuracy and precision - exercise

Let's work on an example problem:

This percent error is very low. Depending on the type of experiment you are4 running, the percent error will vary. Essentially, the more steps you add, the more that can go wrong, and the higher a (good) percent error can be.

Like with percent error, a "good" standard deviation is dependent on the experiment you are running and your measurements. The example above is a good standard deviation, since we are only off by 1.49% (\(\frac{s}{\bar{x}}*100\%)\) of the average.

Importance of accuracy and precision

As mentioned previously, accuracy and precision can help us see if our experimental methods are useful for what we are trying to accomplish, and clue us into what error may be occurring, but that isn't the only reason why they are important.

Let's say you are a chemist manufacturing a life-saving drug. If your methods aren't precise, each pill could contain a different amount of the active ingredient, which could be very harmful. If your methods aren't accurate, you could have way too much or way too little, and that also could be very harmful!

As scientists, we always want our data to be as reliable and accurate as possible. Even if you are doing something as simple as weighing a sample, you always want the best data possible.