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.

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    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.

    Error calculation is the process used to find the significance of errors in a given dataset or set of results.

    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.

    An anomaly is a result that unexpectedly deviates from the normal value due to random errors.

    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 \(140\;\mathrm{g}\).

    Error Calculation Photo of strawberries on a scale StudySmarterFig. 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).

    Parallax errors are easily encountered when reading a measurement from a scale, such as on a thermometer or ruler. They occur when your eye is not directly above the measurement marker, resulting in an incorrect reading being taken due to the 'skew' view. An example of this effect is shown in the animation below - notice how the relative positions of the rows of houses seem to change as they move from the left to the right of the viewer.

    Error Calculation Animation of parallax effect StudySmarterFig. 2 - Animation showing the parallax effect while passing in front of buildings.

    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.

    Precision describes how repeatable, or tightly grouped, the readings from an instrument are. A precise instrument will have low levels of random error.

    Accuracy describes how close the average readings from an instrument are to the true value. An accurate instrument must have low levels of systematic error.

    Uncertainty in Results

    Unavoidable random errors in an experiment will always result in readings from an instrument having a level of uncertainty. This defines a range around the measured value that the true value is expected to fall into. Typically, the uncertainty of a measurement will be significantly smaller than the measurement itself. There are different techniques to calculate the amount of uncertainty, but a common rule of thumb for the amount of error to assign readings taken by eye from an instrument such as a ruler is half of the increment value.

    For example, if you read a measurement of \(194\;\mathrm{mm}\) from a ruler with \(1\;\mathrm{mm}\) increments, you would record your reading as: \((194\pm0.5)\;\mathrm{mm}\).

    This means that the true value is between \(193.5\;\mathrm{mm}\) and \(194.5\;\mathrm{mm}\).

    Error Propagation

    When analyzing results, if a calculation is performed it is important that the effect of error propagation is accounted for. The uncertainties present for variables within a function will affect the uncertainty of the function result. This can get complicated when performing complex analyses, but we can understand the effect using a simple example.

    Imagine that in the previous example, the specimen you measured was a \((194\pm0.5)\;\mathrm{mm}\) long piece of string. You then measure an additional specimen, and record this length as \((420\pm0.5)\;\mathrm{mm}\). If you want to calculate the combined length of both specimens, we also need to combine the uncertainties - as both strings could be either at the shortest or longest limits of their stated length.

    $$(194\pm0.5)\;\mathrm{mm}+(420\pm0.5)\;\mathrm{mm}=(614\pm1)\;\mathrm{mm}$$

    This is also why is it important to state final results with an uncertainty level - as any future work using your results will know the range that the true value is expected to fall within.

    Methods of error calculation

    Errors in experimental measurements can be expressed in several different ways; the most common are absolute error \(D_a\), relative error \(D_r\) and percentage error \(D_\%\).

    Absolute error

    Absolute error is an expression of how far a measurement is from its actual or expected value. It is reported using the same units as the original measurement. As the true value may not be known, the average of multiple repeated measurements can be used in place of the true value.

    Relative error

    Relative error (sometimes called proportional error) expresses how large the absolute error is as a portion of the total value of the measurement.

    Percentage error

    When the relative error is expressed as a percentage, it is called a percentage error.

    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

    You have a summer job at a chicken farm, and one of the hens has just laid a potentially record-breaking egg. The farmer has asked you to perform an accurate measurement of the giant egg to determine if the hen is potentially prize-winning poultry. Luckily you know that in order to correctly state your measurements of the egg, you'll have to perform some error analysis!

    Error calculation Photo of a chicken with eggs StudySmarterFig. 3 - Clearly, the chicken must have been there before the eggs.

    You take 5 measurements the mass of the egg, and record your results in the table below.

    No.Mass (g)Absolute error \(D_a\)Relative error \(D_r\)Percentage error \(D_\%\)
    1\(71.04\)
    2\(70.98\)
    3\(71.06\)
    4\(71.00\)
    5\(70.97\)
    Average \(x_a\)

    Having calculated the mean average of the set of measurements, you can then use this as the \(\mathrm{actual}\;\mathrm{value},x_a,\) in order to calculate the error values using the formulas given earlier.

    No.Mass (g)Absolute error \(D_a\)Relative error \(D_r\)Percentage error \(D_\%\)
    1\(71.04\)\(-0.57\)\(-0.008\)\(0.8\%\)
    2\(70.98\)\(-0.63\)\(-0.009\)\(0.9\%\)
    3\(71.06\)\(-0.55\)\(-0.008\)\(0.8\%\)
    4\(74.03\)\(2.42\)\(0.034\)\(3.4\%\)
    5\(70.97\)\(-0.64\)\(-0.009\)\(0.9\%\)
    Average \(x_a\)\(71.61\)Average\(1.36\%\)

    By analyzing the error values, we can see that measurement number 4 has a significantly larger error than the other readings, and that the average percentage error values for all the measurements is reasonably large. This indicates that measurement 4 may have been an anomaly due to some environmental factor, and as such we decide to remove it from the dataset and recalculate the errors in the table below.

    No.Mass (g)Absolute error \(D_a\)Relative error \(D_r\)Percentage error \(D_\%\)
    1\(71.04\)\(0.03\)\(0.0004\)\(.04\%\)
    2\(70.98\)\(-0.03\)\(-0.0004\)\(.04\%\)
    3\(71.06\)\(0.05\)\(0.0007\)\(.07\%\)
    474.03N/AN/AN/A
    5\(70.97\)\(-0.04\)\(-0.0006\)\(.06\%\)
    Average \(x_a\)\(71.01\)\(.05\%\)

    After recalculating the error values, we can see that the average percentage error is now much lower. This gives us a greater degree of confidence in our average measurement of \(71.01\;\mathrm{g}\) approximating the true mass of the egg.

    In order to present our final value scientifically, we need to include an uncertainty. While the rule-of-thumb presented earlier in the article is suitable when using an instrument such as a ruler, we can clearly see that our results vary by more than half of the smallest increment on our scale. Instead, we should look at the values of absolute error in order to define a level of uncertainty that encompasses all of our readings.

    We can see that the largest absolute error in our readings is \(0.05\), therefore we can state our final measurement as:

    \[\mathrm{Egg}\;\mathrm{mass}=71.01\pm0.05\;\mathrm{g}\]

    Error Calculation - Key takeaways

      • Error calculation is the process used to find how significant an error is from a given dataset or set of results.
      • There are two main types of errors that you will need to know about when it comes to physics experiments: systematic errors and random errors.
      • Absolute error \(D_a\) is an expression of how far a measurement is from its actual value.
      • Relative \(D_r\) and percentage error \(D_\%\) both express how large the absolute error is compared with the total size of the object being measured.
      • By performing error calculation and analysis, we can more easily identify anomalies in our datasets. Error calculation also helps us assign an appropriate level of uncertainty to our results, as no measurement can ever be perfectly accurate.

    References

    1. Fig 1: My first ever digital kitchen scale (https://www.flickr.com/photos/jamieanne/4522268275) by jamieanne licensed by CC-BY-ND 2.0 (https://creativecommons.org/licenses/by-nd/2.0/)
    Frequently Asked Questions about Error Calculation

    What is error calculation?

    Error calculation is the process used to find how significant an error is from a given dataset or set of results.

    What is the formula for error calculation?

    Both absolute and relative errors each have a calculation that you need to be able to use. Check out the word equations below to see how we calculate each of them:


    Absolute error = Actual value - Measured value


    Relative error = Absolute error/Known value


    These formulae are extremely simple to remember, and you should use them both one after the other to complete a thorough error analysis of your completed experiment.


    What is an example of error calculation?

    For example, if you just completed an experiment where you calculated acceleration due to gravity, you would have to compare your result to the known result of gravitational acceleration and then explain why your result differs from the known result. This difference in results arise due to several factors and such analysis of factors is error calculation. 

    How are error rates calculated?

    Error rate or percent error is calculated as follows:

    ( Actual value - Measured value/Known value )*100%

    How do you calculate systematic error and random error?

    The best thing you can do when noticing a systematic error is to restart your experiment, making sure that you have fixed the issue that was causing the systematic error in the first place. Random errors are random, and they do not come about because of our experimental procedure. Instead, we can make their impact less by performing the exact measurement multiple times. A percentage error is used to determine how close a measured value is to an actual value.


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    Test your knowledge with multiple choice flashcards

    An error analysis is a part of the data analysis that you should complete as a part of any scientific experiment.

    Systematic errors are random. 

    Percentage error is an expression of how far a measurement is from the actual value being measured.

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