Error Calculation

Few things in physics are as fundamental to the experimental framework as error calculations. Error calculation is used throughout every physics topic to find how large or small the error for a given result might be. This can then be used to understand the level of uncertainty in the results of an experiment. As such, we need to go over the different ways of representing errors and how to calculate these error values.

Meaning of Error Calculation

Before we can go any further, we need to understand what error calculations are. When gathering any data in physics, whether measuring the length of a piece of string using a ruler or reading the temperature of an object from a thermometer, we can introduce errors to our results. Generally speaking, the errors are not an issue as long as we can explain why they have occurred and understand the uncertainty they add to the experiment results. This is where error calculation comes in. We use error calculation to help us understand how accurate our results are and talk about why they have occurred.

Types of Errors

There are two main types of errors that you will need to know about when it comes to physics: systematic errors and random errors. Systematic errors are In contrast, random errors are errors that are just that! Random! There’s no reason for an unexpected error to occur; they just happen occasionally. Both of these kinds of errors can often be addressed by taking an average, or by identifying them as anomalies.

Systematic Errors

A systematic error is an error created by a mistake in the way the experimental procedure is carried out and can be caused by the instruments or equipment being used, a change in the environment, or errors in how the experiment is carried out.

Instrument error

An instrument error is perhaps the most obvious source of error in an experiment - they occur when the reading on an instrument is different from the true value being measured. This can be caused by the instrument being calibrated incorrectly. For example, if the scales in the image below read \(6\;\mathrm{g}\) when there is nothing on them, then this will introduce an error of \(6\;\mathrm{g}\) into any readings made with them. In this case, the true mass of the strawberries would be \(140\;\mathrm{g}\).

Fig. 1 - Some strawberries being weighed on a digital scale.

When an instrument introduces a consistent error into results through poor calibration this is often described as instrument bias. The good news is that if the bias is identified, it is usually easy to correct by recalibrating the instrument and readings. Instruments with poor precision can also introduce random errors in the results, which are much harder to correct.

Procedural error

Procedural errors are introduced when the experimental procedure is followed inconsistently, resulting in variation in how the final results are arrived at. An example could be how results are rounded - if a value is rounded up in one reading, and down in the next, this would introduce procedural errors into the data.

Environmental error

Errors can also be introduced by variations in how the experiment behaves due to changes in environmental conditions. For example, if an experiment required a very precise measurement to be made of the length of a specimen, variation in the temperature could cause the specimen to expand or contract slightly - introducing a new source of error. Other variable environmental conditions such as humidity, noise levels, or even the amount of wind could also introduce potential sources of error into the results.

Human error

Humans may be the most common cause of error in your high school physics lab! Even in more professional settings, humans are still liable to introduce errors to results. The most common sources of human error are a lack of accuracy when reading a measurement (such as parallax error), or recording the measured value incorrectly (known as a transcriptional error).

Random Errors

As random errors are by their nature, random, they can be harder to control when carrying out an experiment. There will inevitably be inconsistencies when taking repeated measurements, due to variations in the environment, a change in the part of the sample or specimen being measured, or even the resolution of the instrument causing the true value to be rounded up or down.

In order to reduce the potential impacts of random errors in results, typically experiments will take several repeat measurements. As random errors are expected to be randomly distributed, rather than biased in a certain direction, taking an average of multiple readings should give a result closest to the true value. The difference between the average value and each reading can be used to identify anomalies, which may be excluded from the final results.

Importance of Error Calculation

It's always important to analyze the errors you may have in a set of experimental results in order to understand how to correct or deal with them. Another important reason to carry out this kind of analysis is the fact that many scientific studies are carried out using results or data from previous investigations. In this case, it's important that results are presented with a level of uncertainty, as this allows errors to be considered throughout the subsequent analysis and prevents error propagation from leading to unknown errors.

Precision vs Accuracy

Another essential thing to remember when doing error analysis in physics is the difference between precision and accuracy. For example, you can have a set of scales that are extremely precise but make a measurement that is wildly inaccurate because the scales were not calibrated correctly. Or alternatively, the scales could be highly accurate (having an average reading very close to the true value), but imprecise, resulting in a high amount of variation in the readings. The illustration below demonstrates the difference between accuracy and precision.

Error Calculation Formula

The different representations of errors each have a calculation that you need to be able to use. Check out the equations below to see how we calculate each of them using the measured value \(x_m\) and the actual value \(x_a\):

\[ \text{Absolute error}\; D_a = \text{Actual value} - \text{Measured value} \]

\[D_a=x_a-x_m\]

\[ \text{Relative error} \; D_r= \dfrac{\text{Absolute error}}{\text{Actual value}} \]

\[D_r=\frac{(x_a-x_m)}{x_a}\]

\[ \text{Percentage error} \; D_\%= \text{Relative error}\times 100\%\]

\[D_\%=\left|\frac{(x_a-x_m)}{x_a}\times100\%\right|\]

In each of these equations, the \(\text{Actual value}, x_a \) can be considered the mean average of multiple readings when the true value is unknown.

These formulae are simple to remember, and you should use them both sequentially to complete thorough error analysis of your completed experiment. The best way to do this is by using a spreadsheet to record your results, which can be set up to automatically calculate these three values as each reading is entered.

Error Analysis Examples