experimental error definition biology

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• J Cell Biol
• v.177(1); 2007 Apr 9

Error bars in experimental biology

Geoff cumming.

1 School of Psychological Science and 2 Department of Biochemistry, La Trobe University, Melbourne, Victoria, Australia 3086

Fiona Fidler

David l. vaux.

Error bars commonly appear in figures in publications, but experimental biologists are often unsure how they should be used and interpreted. In this article we illustrate some basic features of error bars and explain how they can help communicate data and assist correct interpretation. Error bars may show confidence intervals, standard errors, standard deviations, or other quantities. Different types of error bars give quite different information, and so figure legends must make clear what error bars represent. We suggest eight simple rules to assist with effective use and interpretation of error bars.

Journals that publish science—knowledge gained through repeated observation or experiment—don't just present new conclusions, they also present evidence so readers can verify that the authors' reasoning is correct. Figures with error bars can, if used properly ( 1 – 6 ), give information describing the data (descriptive statistics), or information about what conclusions, or inferences, are justified (inferential statistics). These two basic categories of error bars are depicted in exactly the same way, but are actually fundamentally different. Our aim is to illustrate basic properties of figures with any of the common error bars, as summarized in Table I , and to explain how they should be used.

Common error bars

What do error bars tell you?

Descriptive error bars..

Range and standard deviation (SD) are used for descriptive error bars because they show how the data are spread ( Fig. 1 ). Range error bars encompass the lowest and highest values. SD is calculated by the formula

where X refers to the individual data points, M is the mean, and Σ (sigma) means add to find the sum, for all the n data points. SD is, roughly, the average or typical difference between the data points and their mean, M . About two thirds of the data points will lie within the region of mean ± 1 SD, and ∼95% of the data points will be within 2 SD of the mean.

It is highly desirable to use larger n , to achieve narrower inferential error bars and more precise estimates of true population values.

Descriptive error bars. Means with error bars for three cases: n = 3, n = 10, and n = 30. The small black dots are data points, and the column denotes the data mean M . The bars on the left of each column show range, and the bars on the right show standard deviation (SD). M and SD are the same for every case, but notice how much the range increases with n . Note also that although the range error bars encompass all of the experimental results, they do not necessarily cover all the results that could possibly occur. SD error bars include about two thirds of the sample, and 2 x SD error bars would encompass roughly 95% of the sample.

Descriptive error bars can also be used to see whether a single result fits within the normal range. For example, if you wished to see if a red blood cell count was normal, you could see whether it was within 2 SD of the mean of the population as a whole. Less than 5% of all red blood cell counts are more than 2 SD from the mean, so if the count in question is more than 2 SD from the mean, you might consider it to be abnormal.

As you increase the size of your sample, or repeat the experiment more times, the mean of your results ( M ) will tend to get closer and closer to the true mean, or the mean of the whole population, μ. We can use M as our best estimate of the unknown μ. Similarly, as you repeat an experiment more and more times, the SD of your results will tend to more and more closely approximate the true standard deviation (σ) that you would get if the experiment was performed an infinite number of times, or on the whole population. However, the SD of the experimental results will approximate to σ, whether n is large or small. Like M , SD does not change systematically as n changes, and we can use SD as our best estimate of the unknown σ, whatever the value of n .

Inferential error bars.

In experimental biology it is more common to be interested in comparing samples from two groups, to see if they are different. For example, you might be comparing wild-type mice with mutant mice, or drug with placebo, or experimental results with controls. To make inferences from the data (i.e., to make a judgment whether the groups are significantly different, or whether the differences might just be due to random fluctuation or chance), a different type of error bar can be used. These are standard error (SE) bars and confidence intervals (CIs). The mean of the data, M , with SE or CI error bars, gives an indication of the region where you can expect the mean of the whole possible set of results, or the whole population, μ, to lie ( Fig. 2 ). The interval defines the values that are most plausible for μ.

Confidence intervals. Means and 95% CIs for 20 independent sets of results, each of size n = 10, from a population with mean μ = 40 (marked by the dotted line). In the long run we expect 95% of such CIs to capture μ; here 18 do so (large black dots) and 2 do not (open circles). Successive CIs vary considerably, not only in position relative to μ, but also in length. The variation from CI to CI would be less for larger sets of results, for example n = 30 or more, but variation in position and in CI length would be even greater for smaller samples, for example n = 3.

Because error bars can be descriptive or inferential, and could be any of the bars listed in Table I or even something else, they are meaningless, or misleading, if the figure legend does not state what kind they are. This leads to the first rule. Rule 1: when showing error bars, always describe in the figure legends what they are.

If you carry out a statistical significance test, the result is a P value, where P is the probability that, if there really is no difference, you would get, by chance, a difference as large as the one you observed, or even larger. Other things (e.g., sample size, variation) being equal, a larger difference in results gives a lower P value, which makes you suspect there is a true difference. By convention, if P < 0.05 you say the result is statistically significant, and if P < 0.01 you say the result is highly significant and you can be more confident you have found a true effect. As always with statistical inference, you may be wrong! Perhaps there really is no effect, and you had the bad luck to get one of the 5% (if P < 0.05) or 1% (if P < 0.01) of sets of results that suggests a difference where there is none. Of course, even if results are statistically highly significant, it does not mean they are necessarily biologically important. It is also essential to note that if P > 0.05, and you therefore cannot conclude there is a statistically significant effect, you may not conclude that the effect is zero. There may be a real effect, but it is small, or you may not have repeated your experiment often enough to reveal it. It is a common and serious error to conclude “no effect exists” just because P is greater than 0.05. If you measured the heights of three male and three female Biddelonian basketball players, and did not see a significant difference, you could not conclude that sex has no relationship with height, as a larger sample size might reveal one. A big advantage of inferential error bars is that their length gives a graphic signal of how much uncertainty there is in the data: The true value of the mean μ we are estimating could plausibly be anywhere in the 95% CI. Wide inferential bars indicate large error; short inferential bars indicate high precision.

Science typically copes with the wide variation that occurs in nature by measuring a number ( n ) of independently sampled individuals, independently conducted experiments, or independent observations.

Rule 2: the value of n (i.e., the sample size, or the number of independently performed experiments) must be stated in the figure legend.

It is essential that n (the number of independent results) is carefully distinguished from the number of replicates, which refers to repetition of measurement on one individual in a single condition, or multiple measurements of the same or identical samples. Consider trying to determine whether deletion of a gene in mice affects tail length. We could choose one mutant mouse and one wild type, and perform 20 replicate measurements of each of their tails. We could calculate the means, SDs, and SEs of the replicate measurements, but these would not permit us to answer the central question of whether gene deletion affects tail length, because n would equal 1 for each genotype, no matter how often each tail was measured. To address the question successfully we must distinguish the possible effect of gene deletion from natural animal-to-animal variation, and to do this we need to measure the tail lengths of a number of mice, including several mutants and several wild types, with n > 1 for each type.

Similarly, a number of replicate cell cultures can be made by pipetting the same volume of cells from the same stock culture into adjacent wells of a tissue culture plate, and subsequently treating them identically. Although it would be possible to assay the plate and determine the means and errors of the replicate wells, the errors would reflect the accuracy of pipetting, not the reproduciblity of the differences between the experimental cells and the control cells. For replicates, n = 1, and it is therefore inappropriate to show error bars or statistics.

If an experiment involves triplicate cultures, and is repeated four independent times, then n = 4, not 3 or 12. The variation within each set of triplicates is related to the fidelity with which the replicates were created, and is irrelevant to the hypothesis being tested.

To identify the appropriate value for n , think of what entire population is being sampled, or what the entire set of experiments would be if all possible ones of that type were performed. Conclusions can be drawn only about that population, so make sure it is appropriate to the question the research is intended to answer.

In the example of replicate cultures from the one stock of cells, the population being sampled is the stock cell culture. For n to be greater than 1, the experiment would have to be performed using separate stock cultures, or separate cell clones of the same type. Again, consider the population you wish to make inferences about—it is unlikely to be just a single stock culture. Whenever you see a figure with very small error bars (such as Fig. 3 ), you should ask yourself whether the very small variation implied by the error bars is due to analysis of replicates rather than independent samples. If so, the bars are useless for making the inference you are considering.

Inappropriate use of error bars. Enzyme activity for MEFs showing mean + SD from duplicate samples from one of three representative experiments. Values for wild-type vs. −/− MEFs were significant for enzyme activity at the 3-h timepoint (P < 0.0005). This figure and its legend are typical, but illustrate inappropriate and misleading use of statistics because n = 1. The very low variation of the duplicate samples implies consistency of pipetting, but says nothing about whether the differences between the wild-type and −/− MEFs are reproducible. In this case, the means and errors of the three experiments should have been shown.

Sometimes a figure shows only the data for a representative experiment, implying that several other similar experiments were also conducted. If a representative experiment is shown, then n = 1, and no error bars or P values should be shown. Instead, the means and errors of all the independent experiments should be given, where n is the number of experiments performed.

Rule 3: error bars and statistics should only be shown for independently repeated experiments, and never for replicates. If a “representative” experiment is shown, it should not have error bars or P values, because in such an experiment, n = 1 ( Fig. 3 shows what not to do).

Rule 4: because experimental biologists are usually trying to compare experimental results with controls, it is usually appropriate to show inferential error bars, such as SE or CI, rather than SD. However, if n is very small (for example n = 3), rather than showing error bars and statistics, it is better to simply plot the individual data points.

Standard error (SE).

Suppose three experiments gave measurements of 28.7, 38.7, and 52.6, which are the data points in the n = 3 case at the left in Fig. 1 . The mean of the data is M = 40.0, and the SD = 12.0, which is the length of each arm of the SD bars. M (in this case 40.0) is the best estimate of the true mean μ that we would like to know. But how accurate an estimate is it? This can be shown by inferential error bars such as standard error (SE, sometimes referred to as the standard error of the mean, SEM) or a confidence interval (CI). SE is defined as SE = SD/√ n . In Fig. 4 , the large dots mark the means of the same three samples as in Fig. 1 . For the n = 3 case, SE = 12.0/√3 = 6.93, and this is the length of each arm of the SE bars shown.

Inferential error bars. Means with SE and 95% CI error bars for three cases, ranging in size from n = 3 to n = 30, with descriptive SD bars shown for comparison. The small black dots are data points, and the large dots indicate the data mean M . For each case the error bars on the left show SD, those in the middle show 95% CI, and those on the right show SE. Note that SD does not change, whereas the SE bars and CI both decrease as n gets larger. The ratio of CI to SE is the t statistic for that n , and changes with n . Values of t are shown at the bottom. For each case, we can be 95% confident that the 95% CI includes μ, the true mean. The likelihood that the SE bars capture μ varies depending on n , and is lower for n = 3 (for such low values of n , it is better to simply plot the data points rather than showing error bars, as we have done here for illustrative purposes).

The SE varies inversely with the square root of n , so the more often an experiment is repeated, or the more samples are measured, the smaller the SE becomes ( Fig. 4 ). This allows more and more accurate estimates of the true mean, μ, by the mean of the experimental results, M .

We illustrate and give rules for n = 3 not because we recommend using such a small n , but because researchers currently often use such small n values and it is necessary to be able to interpret their papers. It is highly desirable to use larger n , to achieve narrower inferential error bars and more precise estimates of true population values.

Confidence interval (CI).

Fig. 2 illustrates what happens if, hypothetically, 20 different labs performed the same experiments, with n = 10 in each case. The 95% CI error bars are approximately M ± 2xSE, and they vary in position because of course M varies from lab to lab, and they also vary in width because SE varies. Such error bars capture the true mean μ on ∼95% of occasions—in Fig. 2 , the results from 18 out of the 20 labs happen to include μ. The trouble is in real life we don't know μ, and we never know if our error bar interval is in the 95% majority and includes μ, or by bad luck is one of the 5% of cases that just misses μ.

The error bars in Fig. 2 are only approximately M ± 2xSE. They are in fact 95% CIs, which are designed by statisticians so in the long run exactly 95% will capture μ. To achieve this, the interval needs to be M ± t ( n –1) ×SE, where t ( n –1) is a critical value from tables of the t statistic. This critical value varies with n . For n = 10 or more it is ∼2, but for small n it increases, and for n = 3 it is ∼4. Therefore M ± 2xSE intervals are quite good approximations to 95% CIs when n is 10 or more, but not for small n . CIs can be thought of as SE bars that have been adjusted by a factor ( t ) so they can be interpreted the same way, regardless of n .

This relation means you can easily swap in your mind's eye between SE bars and 95% CIs. If a figure shows SE bars you can mentally double them in width, to get approximate 95% CIs, as long as n is 10 or more. However, if n = 3, you need to multiply the SE bars by 4.

Rule 5: 95% CIs capture μ on 95% of occasions, so you can be 95% confident your interval includes μ. SE bars can be doubled in width to get the approximate 95% CI, provided n is 10 or more. If n = 3, SE bars must be multiplied by 4 to get the approximate 95% CI.

Determining CIs requires slightly more calculating by the authors of a paper, but for people reading it, CIs make things easier to understand, as they mean the same thing regardless of n . For this reason, in medicine, CIs have been recommended for more than 20 years, and are required by many journals ( 7 ).

Fig. 4 illustrates the relation between SD, SE, and 95% CI. The data points are shown as dots to emphasize the different values of n (from 3 to 30). The leftmost error bars show SD, the same in each case. The middle error bars show 95% CIs, and the bars on the right show SE bars—both these types of bars vary greatly with n , and are especially wide for small n . The ratio of CI/SE bar width is t ( n –1) ; the values are shown at the bottom of the figure. Note also that, whatever error bars are shown, it can be helpful to the reader to show the individual data points, especially for small n , as in Figs. 1 and ​ and4, 4 , and rule 4.

When comparing two sets of results, e.g., from n knock-out mice and n wild-type mice, you can compare the SE bars or the 95% CIs on the two means ( 6 ). The smaller the overlap of bars, or the larger the gap between bars, the smaller the P value and the stronger the evidence for a true difference. As well as noting whether the figure shows SE bars or 95% CIs, it is vital to note n , because the rules giving approximate P are different for n = 3 and for n ≥ 10.

Fig. 5 illustrates the rules for SE bars. The panels on the right show what is needed when n ≥ 10: a gap equal to SE indicates P ≈ 0.05 and a gap of 2SE indicates P ≈ 0.01. To assess the gap, use the average SE for the two groups, meaning the average of one arm of the group C bars and one arm of the E bars. However, if n = 3 (the number beloved of joke tellers, Snark hunters ( 8 ), and experimental biologists), the P value has to be estimated differently. In this case, P ≈ 0.05 if double the SE bars just touch, meaning a gap of 2 SE.

Estimating statistical significance using the overlap rule for SE bars. Here, SE bars are shown on two separate means, for control results C and experimental results E, when n is 3 (left) or n is 10 or more (right). “Gap” refers to the number of error bar arms that would fit between the bottom of the error bars on the controls and the top of the bars on the experimental results; i.e., a gap of 2 means the distance between the C and E error bars is equal to twice the average of the SEs for the two samples. When n = 3, and double the length of the SE error bars just touch (i.e., the gap is 2 SEs), P is ∼0.05 (we don't recommend using error bars where n = 3 or some other very small value, but we include rules to help the reader interpret such figures, which are common in experimental biology).

Rule 6: when n = 3, and double the SE bars don't overlap, P < 0.05, and if double the SE bars just touch, P is close to 0.05 ( Fig. 5 , leftmost panel). If n is 10 or more, a gap of SE indicates P ≈ 0.05 and a gap of 2 SE indicates P ≈ 0.01 ( Fig. 5 , right panels).

Rule 5 states how SE bars relate to 95% CIs. Combining that relation with rule 6 for SE bars gives the rules for 95% CIs, which are illustrated in Fig. 6 . When n ≥ 10 (right panels), overlap of half of one arm indicates P ≈ 0.05, and just touching means P ≈ 0.01. To assess overlap, use the average of one arm of the group C interval and one arm of the E interval. If n = 3 (left panels), P ≈ 0.05 when two arms entirely overlap so each mean is about lined up with the end of the other CI. If the overlap is 0.5, P ≈ 0.01.

Estimating statistical significance using the overlap rule for 95% CI bars. Here, 95% CI bars are shown on two separate means, for control results C and experimental results E, when n is 3 (left) or n is 10 or more (right). “Overlap” refers to the fraction of the average CI error bar arm, i.e., the average of the control (C) and experimental (E) arms. When n ≥ 10, if CI error bars overlap by half the average arm length, P ≈ 0.05. If the tips of the error bars just touch, P ≈ 0.01.

Rule 7: with 95% CIs and n = 3, overlap of one full arm indicates P ≈ 0.05, and overlap of half an arm indicates P ≈ 0.01 ( Fig. 6 , left panels).

The rules illustrated in Figs. 5 and ​ and6 6 apply when the means are independent. If two measurements are correlated, as for example with tests at different times on the same group of animals, or kinetic measurements of the same cultures or reactions, the CIs (or SEs) do not give the information needed to assess the significance of the differences between means of the same group at different times because they are not sensitive to correlations within the group. Consider the example in Fig. 7 , in which groups of independent experimental and control cell cultures are each measured at four times. Error bars can only be used to compare the experimental to control groups at any one time point. Whether the error bars are 95% CIs or SE bars, they can only be used to assess between group differences (e.g., E1 vs. C1, E3 vs. C3), and may not be used to assess within group differences, such as E1 vs. E2.

Inferences between and within groups. Means and SE bars are shown for an experiment where the number of cells in three independent clonal experimental cell cultures (E) and three independent clonal control cell cultures (C) was measured over time. Error bars can be used to assess differences between groups at the same time point, for example by using an overlap rule to estimate P for E1 vs. C1, or E3 vs. C3; but the error bars shown here cannot be used to assess within group comparisons, for example the change from E1 to E2.

Assessing a within group difference, for example E1 vs. E2, requires an analysis that takes account of the within group correlation, for example a Wilcoxon or paired t analysis. A graphical approach would require finding the E1 vs. E2 difference for each culture (or animal) in the group, then graphing the single mean of those differences, with error bars that are the SE or 95% CI calculated from those differences. If that 95% CI does not include 0, there is a statistically significant difference (P < 0.05) between E1 and E2.

Rule 8: in the case of repeated measurements on the same group (e.g., of animals, individuals, cultures, or reactions), CIs or SE bars are irrelevant to comparisons within the same group ( Fig. 7 ).

Error bars can be valuable for understanding results in a journal article and deciding whether the authors' conclusions are justified by the data. However, there are pitfalls. When first seeing a figure with error bars, ask yourself, “What is n ? Are they independent experiments, or just replicates?” and, “What kind of error bars are they?” If the figure legend gives you satisfactory answers to these questions, you can interpret the data, but remember that error bars and other statistics can only be a guide: you also need to use your biological understanding to appreciate the meaning of the numbers shown in any figure.

Acknowledgments

This research was supported by the Australian Research Council.

David L. Vaux: [email protected]

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• Knowledge Base

Methodology

• Random vs. Systematic Error | Definition & Examples

Random vs. Systematic Error | Definition & Examples

Published on May 7, 2021 by Pritha Bhandari . Revised on June 22, 2023.

In scientific research, measurement error is the difference between an observed value and the true value of something. It’s also called observation error or experimental error.

There are two main types of measurement error:

Random error is a chance difference between the observed and true values of something (e.g., a researcher misreading a weighing scale records an incorrect measurement).

• Systematic error is a consistent or proportional difference between the observed and true values of something (e.g., a miscalibrated scale consistently registers weights as higher than they actually are).

By recognizing the sources of error, you can reduce their impacts and record accurate and precise measurements. Gone unnoticed, these errors can lead to research biases like omitted variable bias or information bias .

Table of contents

Are random or systematic errors worse, random error, reducing random error, systematic error, reducing systematic error, other interesting articles, frequently asked questions about random and systematic error.

In research, systematic errors are generally a bigger problem than random errors.

Random error isn’t necessarily a mistake, but rather a natural part of measurement. There is always some variability in measurements, even when you measure the same thing repeatedly, because of fluctuations in the environment, the instrument, or your own interpretations.

But variability can be a problem when it affects your ability to draw valid conclusions about relationships between variables . This is more likely to occur as a result of systematic error.

Precision vs accuracy

Random error mainly affects precision , which is how reproducible the same measurement is under equivalent circumstances. In contrast, systematic error affects the accuracy of a measurement, or how close the observed value is to the true value.

Taking measurements is similar to hitting a central target on a dartboard. For accurate measurements, you aim to get your dart (your observations) as close to the target (the true values) as you possibly can. For precise measurements, you aim to get repeated observations as close to each other as possible.

Random error introduces variability between different measurements of the same thing, while systematic error skews your measurement away from the true value in a specific direction.

When you only have random error, if you measure the same thing multiple times, your measurements will tend to cluster or vary around the true value. Some values will be higher than the true score, while others will be lower. When you average out these measurements, you’ll get very close to the true score.

For this reason, random error isn’t considered a big problem when you’re collecting data from a large sample—the errors in different directions will cancel each other out when you calculate descriptive statistics . But it could affect the precision of your dataset when you have a small sample.

Systematic errors are much more problematic than random errors because they can skew your data to lead you to false conclusions. If you have systematic error, your measurements will be biased away from the true values. Ultimately, you might make a false positive or a false negative conclusion (a Type I or II error ) about the relationship between the variables you’re studying.

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Random error affects your measurements in unpredictable ways: your measurements are equally likely to be higher or lower than the true values.

In the graph below, the black line represents a perfect match between the true scores and observed scores of a scale. In an ideal world, all of your data would fall on exactly that line. The green dots represent the actual observed scores for each measurement with random error added.

Random error is referred to as “noise”, because it blurs the true value (or the “signal”) of what’s being measured. Keeping random error low helps you collect precise data.

Sources of random errors

Some common sources of random error include:

• natural variations in real world or experimental contexts.
• imprecise or unreliable measurement instruments.
• individual differences between participants or units.
• poorly controlled experimental procedures.

Random error is almost always present in research, even in highly controlled settings. While you can’t eradicate it completely, you can reduce random error using the following methods.

Take repeated measurements

A simple way to increase precision is by taking repeated measurements and using their average. For example, you might measure the wrist circumference of a participant three times and get slightly different lengths each time. Taking the mean of the three measurements, instead of using just one, brings you much closer to the true value.

Increase your sample size

Large samples have less random error than small samples. That’s because the errors in different directions cancel each other out more efficiently when you have more data points. Collecting data from a large sample increases precision and statistical power .

Control variables

In controlled experiments , you should carefully control any extraneous variables that could impact your measurements. These should be controlled for all participants so that you remove key sources of random error across the board.

Systematic error means that your measurements of the same thing will vary in predictable ways: every measurement will differ from the true measurement in the same direction, and even by the same amount in some cases.

Systematic error is also referred to as bias because your data is skewed in standardized ways that hide the true values. This may lead to inaccurate conclusions.

Types of systematic errors

Offset errors and scale factor errors are two quantifiable types of systematic error.

An offset error occurs when a scale isn’t calibrated to a correct zero point. It’s also called an additive error or a zero-setting error.

A scale factor error is when measurements consistently differ from the true value proportionally (e.g., by 10%). It’s also referred to as a correlational systematic error or a multiplier error.

You can plot offset errors and scale factor errors in graphs to identify their differences. In the graphs below, the black line shows when your observed value is the exact true value, and there is no random error.

The blue line is an offset error: it shifts all of your observed values upwards or downwards by a fixed amount (here, it’s one additional unit).

The purple line is a scale factor error: all of your observed values are multiplied by a factor—all values are shifted in the same direction by the same proportion, but by different absolute amounts.

Sources of systematic errors

The sources of systematic error can range from your research materials to your data collection procedures and to your analysis techniques. This isn’t an exhaustive list of systematic error sources, because they can come from all aspects of research.

Response bias occurs when your research materials (e.g., questionnaires ) prompt participants to answer or act in inauthentic ways through leading questions . For example, social desirability bias can lead participants try to conform to societal norms, even if that’s not how they truly feel.

Your question states: “Experts believe that only systematic actions can reduce the effects of climate change. Do you agree that individual actions are pointless?”

Experimenter drift occurs when observers become fatigued, bored, or less motivated after long periods of data collection or coding, and they slowly depart from using standardized procedures in identifiable ways.

Initially, you code all subtle and obvious behaviors that fit your criteria as cooperative. But after spending days on this task, you only code extremely obviously helpful actions as cooperative.

Sampling bias occurs when some members of a population are more likely to be included in your study than others. It reduces the generalizability of your findings, because your sample isn’t representative of the whole population.

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You can reduce systematic errors by implementing these methods in your study.

Triangulation

Triangulation means using multiple techniques to record observations so that you’re not relying on only one instrument or method.

For example, if you’re measuring stress levels, you can use survey responses, physiological recordings, and reaction times as indicators. You can check whether all three of these measurements converge or overlap to make sure that your results don’t depend on the exact instrument used.

Regular calibration

Calibrating an instrument means comparing what the instrument records with the true value of a known, standard quantity. Regularly calibrating your instrument with an accurate reference helps reduce the likelihood of systematic errors affecting your study.

You can also calibrate observers or researchers in terms of how they code or record data. Use standard protocols and routine checks to avoid experimenter drift.

Randomization

Probability sampling methods help ensure that your sample doesn’t systematically differ from the population.

In addition, if you’re doing an experiment, use random assignment to place participants into different treatment conditions. This helps counter bias by balancing participant characteristics across groups.

Wherever possible, you should hide the condition assignment from participants and researchers through masking (blinding) .

Participants’ behaviors or responses can be influenced by experimenter expectancies and demand characteristics in the environment, so controlling these will help you reduce systematic bias.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Degrees of freedom
• Null hypothesis
• Discourse analysis
• Control groups
• Mixed methods research
• Non-probability sampling
• Quantitative research
• Ecological validity

Research bias

• Rosenthal effect
• Implicit bias
• Cognitive bias
• Selection bias
• Negativity bias
• Status quo bias

Random and systematic error are two types of measurement error.

Systematic error is a consistent or proportional difference between the observed and true values of something (e.g., a miscalibrated scale consistently records weights as higher than they actually are).

Systematic error is generally a bigger problem in research.

With random error, multiple measurements will tend to cluster around the true value. When you’re collecting data from a large sample , the errors in different directions will cancel each other out.

Systematic errors are much more problematic because they can skew your data away from the true value. This can lead you to false conclusions ( Type I and II errors ) about the relationship between the variables you’re studying.

Random error  is almost always present in scientific studies, even in highly controlled settings. While you can’t eradicate it completely, you can reduce random error by taking repeated measurements, using a large sample, and controlling extraneous variables .

You can avoid systematic error through careful design of your sampling , data collection , and analysis procedures. For example, use triangulation to measure your variables using multiple methods; regularly calibrate instruments or procedures; use random sampling and random assignment ; and apply masking (blinding) where possible.

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3.5 – Statistics of error

• Introduction
• Statistics of error

Error and the observer

Confidence in estimates, confidence interval to describe accuracy, significant figures, when to round, chapter 3 continues.

In this section, following the discussion about error in statistics, you’ll find a justification for use of confidence intervals, how to calculate confidence intervals, both as an approximation and with an example of exact calculation, use of confidence interval to quantify accuracy, and conclude with a brief discussion of rounding and significant figures.

An error in statistics means there was a difference between the measured value and the actual value for an object. Classical statistical approach developed a large body of calculated statistics, e.g., standard error the mean , which allows the user to quantify how large the error of measurement is given assumptions about the distribution of the errors. Thus, classical statistics requires user to make assumptions about the error distribution, the subject of our Chapter 6 . A critical issue to understand is that these methods assume large sample sizes are available; they are called asymptotic statistics or large-sample statistics ; the properties of the statistical estimates are evaluated as sample size approaches infinity.

Jackknife sampling and bootstrap sampling are permutation approaches to working with data when the Central Limit Theorem — as sample size increases, the distribution of sample means will tend to a normal distribution (see Chapter 6.7 ) — is unlikely to apply or, rather, we don’t wish to make that assumption ( Chapter 19 ). The jackknife is a sampling method involving repeatedly sampling from the original data set, but each time leaving one value out. The estimator, for example, the sample mean, is calculated for each sample. The repeated estimates from the jackknife approach yield many estimates which, collected, are used to calculate the sample variance. Jackknife estimators tend to be less biased than those from classical asymptotic statistics.

Bootstrapping, and not jackknife resampling, may now be the preferred permutation approach (e.g., Google Scholar search “bootstrap statistics” 36K hits; “jackknife statistics” 17K hits), but which method is best depends on qualities of the data set. Bootstrapping involves large numbers of permutations of the original data, which, in short, means we repeatedly take many samples of our data and recalculate our statistics on these sets of sampled data. We obtain statistical significance by comparing our result from the original data against how often results from our permutations on the resampled data sets exceed the originally observed results. By permutation methods, the goal is to avoid the assumptions made by large-sample statistical inference , i.e., reaching conclusions about the population based on samples from the population. Since its introduction, “bootstrapping” has been shown to be superior in many cases for statistics of error compared to the standard, classical approach (add citations).

There are many advocates for the permutation approaches, and, because we have computers now instead of the hand calculators our statistics ancestors used, permutation methods may be the approach you will take in your own work. However, the classical approach has its strengths — when the conditions, that is, when the assumptions of asymptotic statistics are met by the data, then the classical approaches tend to be less conservative than the permutation methods. By conservative, statisticians mean that a test performs at the level we expect it to. Thus, if the assumptions of classical statistics are met they return the correct answer more often than do the permutation tests.

Individual researchers make observations, therefore, we can talk about observer variation as a kind of error measurement. For repeated measures of the same object by an individual, we would expect the individual to return the same results. To the extent repeated measures differ, this is intraobserver error . In contrast, measures of the same object from different individuals is interobserver error . For a new instrument or measurement system, one would need to establish the reliability of the measure: confronted with the same object do researchers get the same measurement? Accounting for interobserver error applies in many fields, e.g., histopathology of putative carcinoma slides (Franc et al 2003), liver biopsies for cirrhosis (Rousselet et al 2005), blood cell counts (Bacus 1973).

A really useful concept in statistics is the idea that you can assign how confident you are to an estimate. This is another way to speak of the accuracy of an estimate. Clearly, we have more confidence in a sample estimate for a population parameter if many observations are made. Another factor in our ability to estimate is the magnitude of observation differences. In general, the larger the differences among values from repeated trials, the less confident we will be in out estimate, unless, again, we make our estimates from a large collection of observations. These two quantities,  sample size  and  variability,  along with our level of confidence, e.g., 95%, are incorporated into a statistic called the  confidence interval .

We will use this concept a lot throughout the course; for now, a simple but  approximate confidence interval  is to use the 2 x SEM rule (as long as sample size large): twice the standard error of the mean. Take your estimate of the mean, then add (upper limit) or subtract (lower limit) twice the value of the standard error of the mean (if you recall, that’s the standard deviation divided by the square-root of the sample size).

Example . Consider five magnetic darts thrown at a dart board (28 cm diameter, height of 1.68m from the floor) from a distance of 3.15 meters.

Figure 2. Magnetic dart board with 5 darts. Click image to view full sized image.

The distance in centimeters (cm) between where each of the five darts landed on the board compared to the bullseye is reported in (Table 1),

Table 1. Results of five darts thrown at a target (Figure 2)*

* distance in centimeters from the center. 

Note: Use of the coordinate plane, and including the angle measurement in addition to distance (the vector) from center, would be a better analysis. In the context of darts, determining accuracy of a thrower is an Aim-Point targeting problem and part of your calculation would be to get MOA (minute of angle). For the record, the angles (degrees) were

measured using imageJ . Because there seems to be an R package for just about every data analysis scenario, unsurprisingly, there’s an R package called shotGroups to analyze shooting data.

How precise was the dart thrower? We’ll use the coefficient of variation as a measure of precision. Second, how accurate were the throws? Use R to calculate

Note that the true value would be a distance of zero — all bullseyes. We need to calculate the standard error of the mean (SEM), then, we calculate the confidence interval around the sample mean.

The mean was 4.3 cm, therefore to get the lower limit of the interval subtract 2.65 (2*SEM = 2.645298) from the mean; for the upper limit add 2.65 to the mean. Thus, we report our approximate confidence interval as (1.7, 7.0), and we read this as saying we are about 95% confident the population value is between these two limits. Five is a very small sample number†, so we shouldn’t be surprised to learn that our approximate confidence interval would be less than adequate. In statistical terms, we would use the  t -distribution, and not the normal distribution, to make our confidence interval in cases like this.

†Note: As a rule, implied by Central Limit theory and use of asymptotic statistical estimation , a sample size of 30 or more is safer, but probably unrealistic for many experiments. This is sometimes called as the rule of thirty . (For example, a 96-well PCR array costs about $500; with n = 30, that’s $15,000 US Dollars for one group!). So, what about this rule? This type of thinking should be avoided as “a relic of the pre-computer era,” ( Hesterberg, T. (2008). It’s Time To Retire the” n>= 30″ rule. ). We can improve on asymptotic statistics by applying bootstrap principles ( Chapter 19 ). 

We made a quick calculation of the confidence interval; we can get make this calculation by hand by incorporating the t  distribution. We need to know the  degrees of freedom , which in this case is 4 ( n  – 1, where  n  = 5). We look up critical value of  t  at 5% (to get our 95% confidence interval),  t  = 2.78. Subtract for lower limit  t *SEM and add for upper limit  t *SEM to the sample mean for the 95% confidence interval. We can get help from R, by using the one-sample t-test with a test against the hypothesis that the true mean is equal to zero

t.test uses the function qt() , which provides the quantile function. To recreate the 95% CI without the additional baggage output from the t.test , we would simply write

where sd(darts)/sqrt(5) is the standard error of the mean.

Or, alternatively, download and take advantage of a small package called Rmisc (not to be confused with the RcmdrMisc package) and use the function CI

The advantage of using the CI() command from the package Rmisc is pretty clear; I don’t have to specify the degrees of freedom or the standard error of the mean. By default, CI reports the 95% confidence interval. we can specify any interval simply by adding to the command. For example, 

reports upper and lower limits for the 90% confidence interval.

And finally, we should respect significant figures , the number of digits which have meaning. Our data were measured to the nearest tenth of a centimeter, or two significant figures. Therefore, if we report the confidence interval as (0.6477381, 7.9922619), then we imply a false level of precision , unless we also report our random sampling error of measurement.

R has a number of ways to manage output. One option would be to set number of figures globally with the options() function — all values reported by R would hold for the entire session. For example, options(digits=3) would report all numbers to three significant figures. Instead, I prefer to use signif() function, which allows us to report just the values we wish and does not change reporting behavior for the entire session.

Note: The options() function allows the R user to set a number of  settings for an R session. After gaining familiarity with R, the advanced user recognizes that many settings can be changed to make the session work to report in ways more convenient to the user. If curious, submit options() at the R prompt and available settings will be displayed.

The R function signif() applies rounding rules. We apply rounding rules when required to report estimates to appropriate levels of precision. Rounding procedures are used to replace a number with a simplified approximation. Wikipedia provides a comprehensive list of rounding rules. Notable rules include 

• directed rounding to an integer, e.g., rounding up or down
• rounding to nearest integer, e.g., round half up (down) if the number ends with 5
• randomly rounding to an integer, e.g., stochastic rounding.

With the exception of stochastic rounding, all rounding methods impose biases on the sets of numbers. For example, the round half up method applied for numbers above 5, round down for numbers below 5 will increase the variance of the sample. In R, use round() for most of your work. If you need one of the other approaches, for example, to round up, the command is ceiling() ; to round down we use floor() .

No doubt your previous math classes have cautioned you about the problems of rounding error and their influence on calculation. So, as a reminder, if reporting calls for rounding, then always round after you’ve completed your calculations, never during the calculations themselves. 

A final note about significant figures and rounding. While the recommendations about reporting statistics are easy to come by (and often very proscriptive, e.g., Table 1, Cole 2015), there are other concerns. Meta-analysis , which are done by collecting information from multiple studies, would benefit if more and not fewer numbers are reported, for the very same reason that we don’t round during calculations.

• by one alternative method in R, demonstrated with examples in this page
• How many significant figures should be used for the volumetric pipettor p1000? The p200? The p20 (data at end of this page)?
• Another function,  round() , can be used. Try
• and report the results: vary the significant figures from 1 to 10 ( signif() will take digits up to 22).
• Note any output differences between  signif()  and  round() ? Don’t forget to take advantage of R help pages (e.g., enter  ?round at the R prompt) and see Wikipedia .
• Compare rounding by signif() and round() for the number 0.12345. Can you tell which rounding method the two functions use? 
• Calculate the coefficient of variation (CV) for each of the three volumetric pipettors from the data at end of this page. Rank the CV from smallest to largest. Which pipettor had the smallest CV and would therefore be judged the most precise? 
• (data provided at end of this page, click here ).
• Standards distinguish between within run precision and between run precision of a measurement instrument. The data in Table 1 were all recorded within 15 minutes by one operator. What kind pf precision was measured?
• Calculate the standard error of the means for each of the three pipettors from the data
• Repeat, but this time apply your preferred R method for obtaining confidence intervals.
• Compare approximate and R method confidence intervals. How well did the approximate method work?

Pipette calibration

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Random Error vs. Systematic Error

Two Types of Experimental Error

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No matter how careful you are, there is always some error in a measurement. Error is not a "mistake"—it's part of the measuring process. In science, measurement error is called experimental error or observational error.

There are two broad classes of observational errors: random error and systematic error. Random error varies unpredictably from one measurement to another, while systematic error has the same value or proportion for every measurement. Random errors are unavoidable but cluster around the true value. Systematic error can often be avoided by calibrating equipment, but if left uncorrected, it can lead to measurements far from the true value.

Key Takeaways

• The two main types of measurement error are random error and systematic error.
• Random error causes one measurement to differ slightly from the next. It comes from unpredictable changes during an experiment.
• Systematic error always affects measurements by the same amount or proportion, provided that a reading is taken the same way each time. It is predictable.
• Random errors cannot be eliminated from an experiment, but most systematic errors may be reduced.

Systematic Error Examples and Causes

Systematic error is predictable and either constant or proportional to the measurement. Systematic errors primarily influence a measurement's accuracy .

What Causes Systematic Error?

Typical causes of systematic error include observational error, imperfect instrument calibration, and environmental interference. For example:

• Forgetting to tare or zero a balance produces mass measurements that are always "off" by the same amount. An error caused by not setting an instrument to zero prior to its use is called an offset error.
• Not reading the meniscus at eye level for a volume measurement will always result in an inaccurate reading. The value will be consistently low or high, depending on whether the reading is taken from above or below the mark.
• Measuring length with a metal ruler will give a different result at a cold temperature than at a hot temperature, due to thermal expansion of the material.
• An improperly calibrated thermometer may give accurate readings within a certain temperature range, but become inaccurate at higher or lower temperatures.
• Measured distance is different using a new cloth measuring tape versus an older, stretched one. Proportional errors of this type are called scale factor errors.
• Drift occurs when successive readings become consistently lower or higher over time. Electronic equipment tends to be susceptible to drift. Many other instruments are affected by (usually positive) drift, as the device warms up.

How Can You Avoid Systematic Error?

Once its cause is identified, systematic error may be reduced to an extent. Systematic error can be minimized by routinely calibrating equipment, using controls in experiments, warming up instruments before taking readings, and comparing values against standards .

While random errors can be minimized by increasing sample size and averaging data, it's harder to compensate for systematic error. The best way to avoid systematic error is to be familiar with the limitations of instruments and experienced with their correct use.

Random Error Examples and Causes

If you take multiple measurements , the values cluster around the true value. Thus, random error primarily affects precision . Typically, random error affects the last significant digit of a measurement.

What Causes Random Error?

The main reasons for random error are limitations of instruments, environmental factors, and slight variations in procedure. For example:

• When weighing yourself on a scale, you position yourself slightly differently each time.
• When taking a volume reading in a flask, you may read the value from a different angle each time.
• Measuring the mass of a sample on an analytical balance may produce different values as air currents affect the balance or as water enters and leaves the specimen.
• Measuring your height is affected by minor posture changes.
• Measuring wind velocity depends on the height and time at which a measurement is taken. Multiple readings must be taken and averaged because gusts and changes in direction affect the value.
• Readings must be estimated when they fall between marks on a scale or when the thickness of a measurement marking is taken into account.

How Can You Avoid (or Minimize) Random Error?

Because random error always occurs and cannot be predicted , it's important to take multiple data points and average them to get a sense of the amount of variation and estimate the true value. Statistical techniques such as standard deviation can further shed light on the extent of variability within a dataset.

Cochran, W. G. (1968). "Errors of Measurement in Statistics". Technometrics. Taylor & Francis, Ltd. on behalf of American Statistical Association and American Society for Quality. 10: 637–666. doi:10.2307/1267450

Bland, J. Martin, and Douglas G. Altman (1996). "Statistics Notes: Measurement Error." BMJ 313.7059: 744.

Taylor, J. R. (1999). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books. p. 94. ISBN 0-935702-75-X.

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What is: Experimental Error

What is experimental error.

Experimental error refers to the difference between the measured value and the true value of a quantity in scientific experiments. It is an inherent aspect of any experimental process, arising from various sources such as measurement inaccuracies, environmental factors, and limitations in the experimental design. Understanding experimental error is crucial for data analysis and interpretation in fields like statistics, data science, and research.

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Types of Experimental Error

There are two primary types of experimental error: systematic error and random error. Systematic errors are consistent and repeatable inaccuracies that occur due to flaws in the measurement system or experimental setup. In contrast, random errors are unpredictable fluctuations that can arise from various sources, including human error, environmental changes, or limitations in measurement tools. Both types of errors can significantly impact the reliability of experimental results.

Systematic Error Explained

Systematic error can lead to biased results, as it consistently skews measurements in a particular direction. This type of error can often be identified and corrected through calibration of instruments or adjustments in the experimental procedure. For instance, if a scale consistently reads 0.5 grams too high, all measurements taken with that scale will be systematically biased. Recognizing and mitigating systematic errors is essential for achieving accurate and reliable data.

Random Error Explained

Random error, on the other hand, is characterized by its unpredictable nature. It can arise from various factors, such as fluctuations in environmental conditions, variations in the measurement process, or even human error during data collection. Unlike systematic errors, random errors can be reduced by increasing the number of observations or measurements, as the average of a large number of trials tends to converge on the true value. Understanding random error is vital for statistical analysis and hypothesis testing.

Impact of Experimental Error on Data Analysis

Experimental error can significantly affect the outcomes of data analysis and the conclusions drawn from experimental results. When errors are not accounted for, they can lead to incorrect interpretations and potentially flawed decisions based on the data. Researchers must employ statistical methods to quantify and minimize the impact of experimental error, ensuring that their findings are robust and reliable.

Quantifying Experimental Error

Quantifying experimental error involves calculating the uncertainty associated with measurements. This can be done using various statistical techniques, such as calculating the standard deviation, confidence intervals, and error propagation. These methods help researchers understand the degree of uncertainty in their measurements and provide a framework for making informed decisions based on the data collected.

Reducing Experimental Error

To enhance the accuracy of experimental results, researchers can implement several strategies to reduce experimental error. These include improving measurement techniques, using high-quality instruments, standardizing procedures, and conducting repeated trials. By systematically addressing potential sources of error, researchers can improve the reliability of their findings and contribute to the overall integrity of scientific research.

Role of Experimental Error in Scientific Research

Experimental error plays a critical role in scientific research, as it influences the validity and reliability of experimental findings. Acknowledging and addressing experimental error is essential for maintaining the integrity of scientific inquiry. Researchers must be transparent about the limitations of their studies and the potential sources of error, allowing for a more accurate interpretation of results and fostering trust in the scientific community.

Conclusion on Experimental Error

In summary, understanding experimental error is fundamental for anyone involved in statistics, data analysis, and data science. By recognizing the types of errors, quantifying their impact, and implementing strategies to minimize them, researchers can enhance the accuracy and reliability of their experimental results. This knowledge is crucial for making informed decisions based on data and advancing scientific knowledge.

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Error bars in experimental biology

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Geoff Cumming , Fiona Fidler , David L. Vaux; Error bars in experimental biology . J Cell Biol 9 April 2007; 177 (1): 7–11. doi: https://doi.org/10.1083/jcb.200611141

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Error bars commonly appear in figures in publications, but experimental biologists are often unsure how they should be used and interpreted. In this article we illustrate some basic features of error bars and explain how they can help communicate data and assist correct interpretation. Error bars may show confidence intervals, standard errors, standard deviations, or other quantities. Different types of error bars give quite different information, and so figure legends must make clear what error bars represent. We suggest eight simple rules to assist with effective use and interpretation of error bars.

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Common sources of error in biology lab experiments

We look at what causes errors in biology lab experiments and how lab automation can help reduce them.

Errors are an unfortunate staple of laboratory life, including in biology lab experiments. Conducting experiments on biological materials can be unpredictable. Every researcher will grapple with errors to some extent, whether that be simply accounting for their presence in day to day experiments, or resenting their invalidation of a set of hard-won results.

These errors, whatever form they take, can prove to have significant consequences. It is estimated, for instance, that between 24% and 30% of laboratory errors influence patient care, and more seriously that patient harm occurs in between 3% and 12% of cases . There is also a financial dynamic, with the lost productivity and wasted resources amounting to an approximate cost of $180,000 per year in the pre and post analytical stages for laboratories in the US .

What causes errors in the lab?

In general, errors in a laboratory environment can be divided into two main groups. The first of these is systematic errors, referring to faults or flaws in the experiment design or procedure, which shift all measurements and thereby reduces the overall accuracy of an experiment. Examples here could include faulty measurement equipment, inadequate sensitivity of instruments, or calibration errors, which have the result of meaning an experiment becomes ‘biased’.

The second of these groups are random errors, which are caused by unknown and unpredictable changes in a measurement. Errors of this type impact the precision of an experiment, which in turn reduces the reproducibility of a result. There are a wide array of sources of random errors, with some examples including an experiment’s environment changing as a result of measurement, experimenter fatigue or inexperience, and even intrinsic variability.

How can you reduce the impact of errors in the lab?

Fortunately, steps can be taken to reduce the impact and occurrence of errors. When it comes to systematic errors, the main means of doing this is ensuring that experiments are carefully designed, and steps are followed attentively. Laboratories could also reduce these errors by maintaining equipment to a high standard, and ensuring staff are trained properly in its use.

Random errors are slightly harder to reduce due to their unpredictable nature, but steps can be made to reduce their impact. Most important here is ensuring the use of a large sample size and taking multiple measurements, which will ensure that random errors which occur for any reason will have only a limited bearing on the overall experiment results. On the human side, proper training is again relevant here, as is ensuring that staff take sufficient breaks.

However, using these traditional means of error reduction can only do so much, and errors continue to frustrate researchers. Given the raft of technological innovation in the laboratory space, this is a shame, particularly when it comes to such basic issues as incorrect equipment reading.

What is the solution?

One potential solution is the use of total lab automation . In practice, this would mean linking together several workstations and automating entire workflows. Automata Labs for instance links together a series of modular pods which house a robotic arm along with tried-and-tested lab equipment, and links seamlessly into broader laboratory processes.

The uses of such automation are broad, and clinical laboratory automation for instance can help produce faster and more reliable results in areas such as diagnostic and drug discovery protocols, amongst a raft of other uses.

In terms of error reduction, automation allows for taking over the bulk of many manual activities, including specimens sorting, loading, centrifugation, decapping, aliquoting, and sealing, massively reducing the risks of error in the manually-intensive pre-analytical phase. Since studies suggest that up to 70% of errors occur during this stage, this offers a hugely significant positive impact, a fact confirmed by recent polling conducted by Automata which found that 85% of lab managers associate automation with error reduction.

In the longer-term too, reducing the need for scientists to perform such manual tasks can offer a raft of positive benefits. For instance, on the health side, issues such as eye strain from tasks such as cell counting and even carpal tunnel syndrome from repeated manipulations are mitigated to a major extent.

What do lab managers think about automation’s potential to reduce human error ?

Looking more broadly, scientists working in laboratories are also simply given more time. Freeing them from mundane and manual tasks allows scientists to focus on much more high-value and intellectually demanding tasks, which permit greater creativity.

Automata recently conducted research looking at the benefits of Automation in the lab. 78% of lab managers in Automata’s recent polling emphasised the importance of this creativity angle to them, with automation correspondingly offering opportunities to improve laboratory staff productivity and morale.

But, crucially, Lab managers feel their sector is operating under significant strain. Many citied time pressures and the ability to meet current demand as key issues. Error not only compounds these problems, but human error was also specifically identified by 65% of respondents as having an impact on their work.

However, our research also identified automation is a solution. The majority of lab managers feel that full-workflow automation would have a positive impact on their day-to-day work, for example in improving capacity (93%) and turnaround time (90%). Indeed, full workflow automation was also felt to have a positive impact on human error (85%), lab safety (80%) and patient outcomes (68%).

What is laboratory workflow automation?

Laboratory workflow automation solutions link multiple processes and workflows in order to automate entire assays in diagnostic testing. In the clinical lab, laboratory workflow automation normally encompasses both lab automation hardware and software to form an integrated system that charts the end-to-end progress of a sample. This solution will automate the processes of pre-analytics, analytics and post-analytics.

A laboratory workflow automation solution in diagnostic testing may process, test and store specimens autonomously, with minimal human intervention.

With the laboratory workflow automation of diagnostics processes, human error is removed and the quality of lab results is significantly improved. You can find out more about our laboratory workflow automation solution here .

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Error bars in experimental biology

Affiliation.

• 1 School of Psychological Science and 2Department of Biochemistry, La Trobe University, Melbourne, Victoria, Australia 3086. [email protected]
• PMID: 17420288
• PMCID: PMC2064100
• DOI: 10.1083/jcb.200611141

Error bars commonly appear in figures in publications, but experimental biologists are often unsure how they should be used and interpreted. In this article we illustrate some basic features of error bars and explain how they can help communicate data and assist correct interpretation. Error bars may show confidence intervals, standard errors, standard deviations, or other quantities. Different types of error bars give quite different information, and so figure legends must make clear what error bars represent. We suggest eight simple rules to assist with effective use and interpretation of error bars.

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Biological Mistakes: What They Are and What They Mean for the Experimental Biologist

Organisms and other biological entities are mistake-prone: they get things wrong. The entities of pure physics, such as atoms and inorganic molecules, do not make mistakes: they do what they do according to physical law, with no room for error except on the part of the physicist or their theory. We set out a novel framework for understanding biology and its demarcation from physics – that of mistake-making. We distinguish biological mistakes from mere failures. We then propose a rigorous definition of mistakes that, although invoking the concept of function, is compatible with various views about what functions are. The definition of mistake-making is agential, since mistakes do not just happen ­– at least in the sense analysed here – but are made . This requires, then, a notion of biological agency which we set out as a definition of the Minimal Biological Agent. The paper then considers a series of objections to the theory presented here, along with our replies. Two key features of our theory of mistakes are, first, that it is a supplement to, not a replacement for, existing general frameworks within which biology is understood and practised. Secondly, it is designed to be experimentally productive . Hence we end with a series of case studies where mistake theory can be shown to be useful in the potential generation of novel hypotheses of interest to the working biologist.

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Calculate Percent Error 5

Percent Error Definition

Percent error, sometimes referred to as percentage error, is an expression of the difference between a measured value and the known or accepted value . It is often used in science to report the difference between experimental values and expected values.

The formula for calculating percent error is:

Note: occasionally, it is useful to know if the error is positive or negative. If you need to know the positive or negative error, this is done by dropping the absolute value brackets in the formula. In most cases, absolute error is fine. For example, in experiments involving yields in chemical reactions, it is unlikely you will obtain more product than theoretically possible.

Steps to Calculate the Percent Error

• Subtract the accepted value from the experimental value.
• Take the absolute value of step 1
• Divide that answer by the accepted value.
• Multiply that answer by 100 and add the % symbol to express the answer as a percentage .

Example Calculation

Now let’s try an example problem.

You are given a cube of pure copper. You measure the sides of the cube to find the volume and weigh it to find its mass. When you calculate the density using your measurements, you get 8.78 grams/cm 3 . Copper’s accepted density is 8.96 g/cm 3 . What is your percent error?

Solution: experimental value = 8.78 g/cm 3 accepted value = 8.96 g/cm 3

Step 1: Subtract the accepted value from the experimental value.

8.78 g/cm 3 – 8.96 g/cm 3 = -0.18 g/cm 3

Step 2: Take the absolute value of step 1

|-0.18 g/cm 3 | = 0.18 g/cm 3

Step 3: Divide that answer by the accepted value.

Step 4: Multiply that answer by 100 and add the % symbol to express the answer as a percentage.

0.02 x 100 = 2 2%

The percent error of your density calculation is 2%.

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5 thoughts on “ calculate percent error ”.

Percent error is always represented as a positive value. The difference between the actual and experimental value is always the absolute value of the difference. |Experimental-Actual|/Actualx100 so it doesn’t matter how you subtract. The result of the difference is positive and therefore the percent error is positive.

Percent error is always positive, but step one still contains the error initially flagged by Mark. The answer in that step should be negative:

experimental-accepted=error 8.78 – 8.96 = -0.18

In the article, the answer was edited to be correct (negative), but the values on the left are still not in the right order and don’t yield a negative answer as presented.

Mark is not correct. Percent error is always positive regardless of the values of the experimental and actual values. Please see my post to him.

Say if you wanted to find acceleration caused by gravity, the accepted value would be the acceleration caused by gravity on earth (9.81…), and the experimental value would be what you calculated gravity as :)

If you don’t have an accepted value, the way you express error depends on how you are making the measurement. If it’s a calculated value, like, based on a known about of carbon dioxide dissolved in water, then you have a theoretical value to use instead of the accepted value. If you are performing a chemical reaction to quantify the amount of carbonic acid, the accepted value is the theoretical value if the reaction goes to completion. If you are measuring the value using an instrument, you have uncertainty of the instrument (e.g., a pH meter that measures to the nearest 0.1 units). But, if you are taking measurements, most of the time, measure the concentration more than once, take the average value of your measurements, and use the average (mean) as your accepted value. Error gets complicated, since it also depends on instrument calibration and other factors.

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