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# Accuracy and Precision

In 1853, William Henry Perkin attempted so synthesize quinine, a substance used to treat malaria. However, that isn't what he ended up making. What he actually made was a crude mixture that, when extracted with alcohol, produced a bright light purple color. This mixture was then mass-produced as a dye, which we know today as mauve. While he didn't make the treatment he wanted, he did create a discovery that made society much more vibrant.

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Perkin's experiment is a good example of something with low accuracy, but high precision. In this article, we will be defining these terms and learn why they are so important.

• First, we will define these two terms and learn about the differences between them.
• Next, we will look at some examples of accuracy and precision in everyday life.
• Then, we will see how they apply to chemistry and how to calculate them.
• Lastly, we will explain why accuracy and precision are so important.

## 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.

Precision is a measure of how close measurements are to each other.

Accuracy is a measure of how close measurements are to a target or accepted value.

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:

• 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:

Let's say you are trying to measure how much salt is in a glass of salt water. You do this by evaporating the water and weighing the salt crystals left behind. After three trials, you get this data: 2.5 g, 2.7 g, 2.5 g. Your teacher tells you that there was actually 5.6 g of salt in the solution. This means that, while precise, your data was inaccurate.

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

A systematic error is caused by consistent deviations due to:

• Personal error (such as putting the decimal in the wrong place).
• Methodological error (using the wrong method, such as using the wrong solvent).
• Instrumental error (such as the machines being calibrated wrong).

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.

Random error is caused by uncontrollable fluctuations during experimentation. Unlike systematic error, random error can fluctuate from being higher to lower than expected and can also not be replicated. An example would be copying down a measurement wrong.

Gross error is an error that is significantly off due to personal error or negligence. For example, spilling a portion of your sample before measuring.

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.

If the error wasn't due to personal mistakes, a high accuracy, low precision experiment can indicate that the method you used wasn't the best for what you are trying to measure.

## 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 is a measure of how accurate a measurement/average of measurements is. The formula is:

$$\text{Percent error}=\frac{|\text{experimental value-accepted/expected value}|}{\text{accepted/expected value}}*100\%$$

The vertical bars represent absolute value (i.e. the value is always positive)

### 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:

A student measures a sample of magnesium using a mass balance. After three measurements they get these results: 0.625 g, 0.619 g, 0.623 g. The actual mass of the sample is 0.618 g. What is the student's percent error?

Since we are given three measurements, we need to first calculate the average. An average is just the sum of the data points divided by the number of data points.

$$\text{average}=\frac{0.625\,g+0.619\,g+0.623\,g}{3}$$

$$\text{average}=0.622\,g$$

Now we can plug our values into the percent error formula

$$\text{Percent error}=\frac{|\text{experimental value-accepted/expected value}}{\text{accepted/expected value}|}*100\%$$

$$\text{Percent error}=\frac{|0.622\,g-0.618\,g|}{0.618\,g}*100\%$$

$$\text{Percent error}=\frac{|0.004\,g|}{0.618\,g}*100\%$$

$$\text{Percent error}=0.647%$$

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.

Now let's talk about standard deviation.

Standard deviation measures how close a set of data points are to each other. The formula is:

$$s=\sqrt{\frac{\sum({x_i-\bar{x}})^2}{n-1}}$$

Where: $$x_i\,\text{is a data point}\,,\bar{x}\,\text{is the average of the data points, and}\, n\, \text{is the number of data points}$$

The standard deviation is written as: $$\bar{x} \pm s$$

The standard deviation tells us how our deviated above or below (shown by plus or minus symbol) the average. The higher the standard deviation, the less precise the data.Let's work on an example.

A student is measuring the initial temperature of a 500 mL beaker of water. The measurements they obtain are: 20.0 °C, 20.4 °C, 19.8 °C, and 20.3 °C. What is the standard deviation?

Our first step is to calculate the average.

$$\bar{x}=\frac{20.0^\circ C+20.4^\circ C+19.8^\circ C+20.3^\circ C}{4}$$$$\bar{x}=\frac{80.5^\circ C}{4}=20.1^\circ C$$

Next we need to subtract the average from each data point, square that, then add these values together.

$$\sum{(x_i-\bar{x})^2}=(20.0^\circ C-20.1^\circ C)^2+(20.4^\circ C-20.1^\circ C)^2+(19.8^\circ C-20.1^\circ C)^2+(20.3^\circ C-20.1^\circ C)^2$$

$$\sum{(x_i-\bar{x})^2}=(0.1^\circ C)^2 +(0.3^\circ C)^2+(-0.3^\circ C)^2 +(0.2^\circ C)^2$$

$$\sum{(x_i-\bar{x})^2}=0.01^\circ C +0.09^\circ C +0.09^\circ C +0.04^\circ C$$

$$\sum{(x_i-\bar{x})^2}=0.23^\circ C$$

Now we need to divide this value by the number of data points minus 1:

$$\frac{\sum{(x_i-\bar{x})^2}}{n-1}=\frac{0.23^\circ C}{n-1}$$

$$\frac{\sum{(x_i-\bar{x})^2}}{n-1}=\frac{0.23^\circ C}{3}$$

$$\frac{\sum{(x_i-\bar{x})^2}}{n-1}=0.0767^\circ C$$

Next, we need to take the square root:

$$\sqrt{\frac{\sum({x_i-\bar{x}})^2}{n-1}}=\sqrt{0.0767^\circ C}$$$$=\sqrt{\frac{\sum({x_i-\bar{x}})^2}{n-1}}=0.3^\circ C$$

Lastly, we write our expression for standard deviation:

$$\bar{x} \pm s$$

$$20.1^\circ C \pm 0.3^\circ C$$

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.

## Accuracy and Precision - Key takeaways

• Precision is a measure of how close measurements are to each other.
• Accuracy is a measure of how close measurements are to a target or accepted value.

• Systematic error is caused by consistent deviations due to: -Personal error (such as putting the decimal in the wrong place).

-Methodological error (using wrong method, such as using the solvent).

-Instrumental error (such as being calibrated wrong).

• Random error is caused by uncontrollable fluctuations during experimentation. Unlike systematic error, random error can fluctuate from being higher to lower than expected and can also not be replicated. An example would be copying down a measurement wrong.

• Gross error is error that is significantly off due to personal error or negligence. For example, spilling a portion of your sample before measuring.

• Percent error is a measure of how accurate a measurement/average of measurements is. The formula is:

$$\text{Percent error}=\frac{|\text{experimental value-accepted/expected value}|}{\text{accepted/expected value}}*100\%$$

• Standard deviation measures how close a set of data points are to each other. The formula is:

$$s=\sqrt{\frac{\sum({x_i-\bar{x}})^2}{n-1}}$$

#### Flashcards in Accuracy and Precision 13

###### Learn with 13 Accuracy and Precision flashcards in the free StudySmarter app

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What is the difference between precision and accuracy?

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).

What is accuracy?

Accuracy is a measure of how close measurements are to a target or accepted value.

What is precision?

Precision is a measure of how close measurements are to each other.

What is an example of being precise but not accurate?

An easy example of being precise but not accurate is solving a problem multiple times to get the same incorrect answer. You are getting always the same result, so you are being precise, but the answer is incorrect, so you are not being accurate.

Why is accurate measurement important?

Accurate measurements are important, because if measures are incorrect, errors will occur during the process you are trying to accomplish. For example, you might measure the wrong amount of a substance that you need to create a medicine, making it useless or even toxic, or you might measure the wrong amount of concrete to build a structure that can then fall or crumble, putting people in danger.

In the best-case scenario, an inaccurate measurement makes your experiment unreliable or not usable for the target question.

## Test your knowledge with multiple choice flashcards

Which of the following is an example of accuracy?

Which of the following is an example of precision?

High precision, but low accuracy, is often due to what?

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