# How to Accurately Perform Basic Error Analysis

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

In this lesson, learn about the different types of error that can affect a measurement and how to correct for them. In addition, learn how to quantify the amount of error in a measurement, a process called error analysis.

## What Is Error Analysis?

As part of a science experiment, you need to know the temperature of a beaker full of water. How are going to do this? Perhaps you might put a thermometer in the beaker, look at it, and record the temperature. This is a good idea, but is it enough?

Should you measure it more than once? How can you tell if your measurement is 'good' or not? To begin to answer these questions, you need to understand error analysis, which is the process of determining how accurate and precise a measurement is.

## Accuracy and Precision

Error analysis helps you determine the accuracy and precision of a measurement. In science, these words have very specific meanings.

• Accuracy is how close a measurement is to the true or accepted value.
• Precision measures how repeatable a measurement is.

If you measured the length of a pencil 3 times and got 10.3 cm every time, your measurements would be very precise.

Of course, you want all your measurements to be as accurate and precise as possible, but precision and accuracy are NOT the same thing, and it is possible to have one without the other.

For example, say you repeat a measurement several times and got very different values each time. Your results would be imprecise, but you then find that the average value is very close to the true value, so your measurement would be accurate.

A measurement could also be precise but not accurate if you got the same value over and over, but it was not close to the true value.

## Sources of Error

When you make any measurement, there is always the possibility of error. Experimental error falls into two general categories, systemic error and random error.

Systemic error is error that is caused by a problem with the measurement equipment. This includes instruments with hard-to-read scales, improperly calibrated instruments, and instruments that are not being used correctly. Systemic errors typically result in measurements that are precise but not accurate, because the equipment may be giving you the same result every time, even if that result is not accurate.

To reduce systemic errors, make sure that you use the equipment in the correct way and that it is calibrated and functioning correctly. For example, when trying to read the temperature from the thermometer, you want to be sure to put your eyes level with the top of the liquid so that you can make an accurate measurement. If you look at it from above, you are likely to make a type of error called a parallax error, which is due to observing a measuring device at an angle.

In contrast to systemic error, random error is due to unpredictable and uncontrollable factors that can affect the experiment. Random error is unavoidable, but you can reduce the effects of random error by making several measurements and taking an average. That way, even if random error causes your measurements to have low precision, they can still be accurate if you average several values.

## How to Calculate Percent Error

Now that we know the different types of error and some ways that you can eliminate or compensate for it, let's look at how to quantify exactly HOW accurate your measurements are. If you know the true value of what you are trying to measure, you can calculate the percent error to get a number that tells you just how accurate you were.

To find the percent error, first average all your measurements. Then, find the difference between your average and the true value. Finally, divide this difference by the true value and multiply by 100 to make it a percent.

Did you see the symbols that look like this ( | ) on each side of the numerator? These tell you to take the absolute value of the numerator. This means that it should always be positive. You will NEVER get a negative value for percent error.

## Measuring the Temperature of the Water

Let's go back to the problem of measuring the water temperature and see if we can conduct an error analysis on your measurements. You measure the temperature three times and get these values (in degrees Celsius): 34.5, 34.6, 34.4.

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