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How to Calculate Experimental Error in Chemistry

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• Ph.D., Biomedical Sciences, University of Tennessee at Knoxville
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Error is a measure of accuracy of the values in your experiment. It is important to be able to calculate experimental error, but there is more than one way to calculate and express it. Here are the most common ways to calculate experimental error:

Error Formula

In general, error is the difference between an accepted or theoretical value and an experimental value.

Error = Experimental Value - Known Value

Relative Error Formula

Relative Error = Error / Known Value

Percent Error Formula

% Error = Relative Error x 100%

Example Error Calculations

Let's say a researcher measures the mass of a sample to be 5.51 grams. The actual mass of the sample is known to be 5.80 grams. Calculate the error of the measurement.

Experimental Value = 5.51 grams Known Value = 5.80 grams

Error = Experimental Value - Known Value Error = 5.51 g - 5.80 grams Error = - 0.29 grams

Relative Error = Error / Known Value Relative Error = - 0.29 g / 5.80 grams Relative Error = - 0.050

% Error = Relative Error x 100% % Error = - 0.050 x 100% % Error = - 5.0%

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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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Absolute and Relative Error and How to Calculate Them

Absolute, relative, and percent error are the most common experimental error calculations in science. Grouped together, they are types of approximation error. Basically, the premise is that no matter how carefully you measure something, you’ll always be off a bit due to the limitations of the measuring instrument. For example, you may be only able to measure to the nearest millimeter on a ruler or the nearest milliliter on a graduated cylinder. Here are the definitions, equations, and examples of how to use these types of error calculations.

Absolute Error

Absolute error is the magnitude (size) of the difference between a measured value and a true or exact value.

Absolute Error = |True Value – Measured Value|

Absolute Error Example: A measurement is 24.54 mm and the true or known value is 26.00 mm. Find the absolute error. Absolute Error = |26.00 mm – 25.54 mm|= 0.46 mm Note absolute error retains its units of measurement.

The vertical bars indicate absolute value . In other words, you drop any negative sign you may get. For this reason, it doesn’t actually matter whether you subtract the measured value from the true value or the other way around. You’ll see the formula written both ways in textbooks and both forms are correct.

What matters is that you interpret the error correctly. If you graph error bars, half of the error is higher than the measured value and half is lower. For example, if your error is 0.2 cm, it is the same as saying ±0.1 cm.

The absolute error tells you how big a difference there is between the measured and true values, but this information isn’t very helpful when you want to know if the measured value is close to the real value or not. For example, an absolute error of 0.1 grams is more significant if the true value is 1.4 grams than if the true value is 114 kilograms! This is where relative error and percent error help.

Relative Error

Relative error puts absolute error into perspective because it compares the size of absolute error to the size of the true value. Note that the units drop off in this calculation, so relative error is dimensionless (unitless).

Relative Error = |True Value – Measured Value| / True Value Relative Error = Absolute Error / True Value

Relative Error Example: A measurement is 53 and the true or known value is 55. Find the relative error. Relative Error = |55 – 53| / 55 = 0.034 Note this value maintains two significant digits.

Note: Relative error is undefined when the true value is zero . Also, relative error only makes sense when a measurement scale starts at a true zero. So, it makes sense for the Kelvin temperature scale, but not for Fahrenheit or Celsius!

Percent Error

Percent error is just relative error multiplied by 100%. It tells what percent of a measurement is questionable.

Percent Error = |True Value – Measured Value| / True Value x 100% Percent Error = Absolute Error / True Value x 100% Percent Error = Relative Error x 100%

Percent Error Example: A speedometer says a car is going 70 mph but its real speed is 72 mph. Find the percent error. Percent Error = |72 – 70| / 72 x 100% = 2.8%

Mean Absolute Error

Absolute error is fine if you’re only taking one measurement, but what about when you collect more data? Then, mean absolute error is useful. Mean absolute error or MAE is the sum of all the absolute errors divided by the number of errors (data points). In other words, it’s the average of the errors. Mean absolute error, like absolute error, retains its units.

Mean Absolute Error Example: You weigh yourself three times and get values of 126 lbs, 129 lbs, 127 lbs. Your true weight is 127 lbs. What is the mean absolute error of the measurements. Mean Absolute Error = [|126-127 lbs|+|129-127 lbs|+|127-127 lbs|]/3 = 1 lb

• Hazewinkel, Michiel, ed. (2001). “Theory of Errors.”  Encyclopedia of Mathematics . Springer Science+Business Media B.V. / Kluwer Academic Publishers. ISBN 978-1-55608-010-4.
• Helfrick, Albert D. (2005). Modern Electronic Instrumentation and Measurement Techniques . ISBN 81-297-0731-4.
• Steel, Robert G. D.; Torrie, James H. (1960).  Principles and Procedures of Statistics, With Special Reference to Biological Sciences . McGraw-Hill. 

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• Published: 11 November 2022

Error, reproducibility and uncertainty in experiments for electrochemical energy technologies

• Graham Smith   ORCID: orcid.org/0000-0003-0713-2893 1 &
• Edmund J. F. Dickinson   ORCID: orcid.org/0000-0003-2137-3327 1  

Nature Communications volume  13 , Article number:  6832 ( 2022 ) Cite this article

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• Electrocatalysis
• Electrochemistry
• Materials for energy and catalysis

The authors provide a metrology-led perspective on best practice for the electrochemical characterisation of materials for electrochemical energy technologies. Such electrochemical experiments are highly sensitive, and their results are, in practice, often of uncertain quality and challenging to reproduce quantitatively.

A critical aspect of research on electrochemical energy devices, such as batteries, fuel cells and electrolysers, is the evaluation of new materials, components, or processes in electrochemical cells, either ex situ, in situ or in operation. For such experiments, rigorous experimental control and standardised methods are required to achieve reproducibility, even on standard or idealised systems such as single crystal platinum 1 . Data reported for novel materials often exhibit high (or unstated) uncertainty and often prove challenging to reproduce quantitatively. This situation is exacerbated by a lack of formally standardised methods, and practitioners with less formal training in electrochemistry being unaware of best practices. This limits trust in published metrics, with discussions on novel electrochemical systems frequently focusing on a single series of experiments performed by one researcher in one laboratory, comparing the relative performance of the novel material against a claimed state-of-the-art.

Much has been written about the broader reproducibility/replication crisis 2 and those reading the electrochemical literature will be familiar with weakly underpinned claims of “outstanding” performance, while being aware that comparisons may be invalidated by measurement errors introduced by experimental procedures which violate best practice; such issues frequently mar otherwise exciting science in this area. The degree of concern over the quality of reported results is evidenced by the recent decision of several journals to publish explicit experimental best practices 3 , 4 , 5 , reporting guidelines or checklists 6 , 7 , 8 , 9 , 10 and commentary 11 , 12 , 13 aiming to improve the situation, including for parallel theoretical work 14 .

We write as two electrochemists who, working in a national metrology institute, have enjoyed recent exposure to metrology: the science of measurement. Metrology provides the vocabulary 15 and mathematical tools 16 to express confidence in measurements and the comparisons made between them. Metrological systems and frameworks for quantification underpin consistency and assurance in all measurement fields and formal metrology is an everyday consideration for practical and academic work in fields where accurate measurements are crucial; we have found it a useful framework within which to evaluate our own electrochemical work. Here, rather than pen another best practice guide, we aim, with focus on three-electrode electrochemical measurements for energy material characterisation, to summarise some advice that we hope helps those performing electrochemical experiments to:

avoid mistakes and minimise error

report in a manner that facilitates reproducibility

consider and quantify uncertainty

Minimising mistakes and error

Metrology dispenses with nebulous concepts such as performance and instead requires scientists to define a specific measurand (“the quantity intended to be measured”) along with a measurement model ( ”the mathematical relation among all quantities known to be involved in a measurement”), which converts the experimental indicators into the measurand 15 . Error is the difference between the reported value of this measurand and its unknowable true value. (Note this is not the formal definition, and the formal concepts of error and true value are not fully compatible with measurement concepts discussed in this article, but we retain it here—as is common in metrology tuition delivered by national metrology institutes—for pedagogical purposes 15 ).

Mistakes (or gross errors) are those things which prevent measurements from working as intended. In electrochemistry the primary experimental indicator is often current or voltage, while the measurand might be something simple, like device voltage for a given current density, or more complex, like a catalyst’s turnover frequency. Both of these are examples of ‘method-defined measurands’, where the results need to be defined in reference to the method of measurement 17 , 18 (for example, to take account of operating conditions). Robust experimental design and execution are vital to understand, quantify and minimise sources of error, and to prevent mistakes.

Contemporary electrochemical instrumentation can routinely offer a current resolution and accuracy on the order of femtoamps; however, one electron looks much like another to a potentiostat. Consequently, the practical limit on measurements of current is the scientist’s ability to unambiguously determine what causes the observed current. Crucially, they must exclude interfering processes such as modified/poisoned catalyst sites or competing reactions due to impurities.

As electrolytes are conventionally in enormous excess compared to the active heterogeneous interface, electrolyte purity requirements are very high. Note, for example, that a perfectly smooth 1 cm 2 polycrystalline platinum electrode has on the order of 2 nmol of atoms exposed to the electrolyte, so that irreversibly adsorbing impurities present at the part per billion level (nmol mol −1 ) in the electrolyte may substantially alter the surface of the electrode. Sources of impurities at such low concentration are innumerable and must be carefully considered for each experiment; impurity origins for kinetic studies in aqueous solution have been considered broadly in the historical literature, alongside a review of standard mitigation methods 19 . Most commercial electrolytes contain impurities and the specific ‘grade’ chosen may have a large effect; for example, one study showed a three-fold decrease in the specific activity of oxygen reduction catalysts when preparing electrolytes with American Chemical Society (ACS) grade acid rather than a higher purity grade 20 . Likewise, even 99.999% pure hydrogen gas, frequently used for sparging, may contain more than the 0.2 μmol mol −1 of carbon monoxide permitted for fuel cell use 21 .

The most insidious impurities are those generated in situ. The use of reference electrodes with chloride-containing filling solutions should be avoided where chloride may poison catalysts 22 or accelerate dissolution. Similarly, reactions at the counter electrode, including dissolution of the electrode itself, may result in impurities. This is sometimes overlooked when platinum counter electrodes are used to assess ‘platinum-free’ electrocatalysts, accidentally resulting in performance-enhancing contamination 23 , 24 ; a critical discussion on this topic has recently been published 25 . Other trace impurity sources include plasticisers present in cells and gaskets, or silicates from the inappropriate use of glass when working with alkaline electrolytes 26 . To mitigate sensitivity to impurities from the environment, cleaning protocols for cells and components must be robust 27 . The use of piranha solution or similarly oxidising solution followed by boiling in Type 1 water is typical when performing aqueous electrochemistry 20 . Cleaned glassware and electrodes are also routinely stored underwater to prevent recontamination from airborne impurities.

The behaviour of electronic hardware used for electrochemical experiments should be understood and considered carefully in interpreting data 28 , recognising that the built-in complexity of commercially available digital potentiostats (otherwise advantageous!) is capable of introducing measurement artefacts or ambiguity 29 , 30 . While contemporary electrochemical instrumentation may have a voltage resolution of ~1 μV, its voltage measurement uncertainty is limited by other factors, and is typically on the order of 1 mV. As passing current through an electrode changes its potential, a dedicated reference electrode is often incorporated into both ex situ and, increasingly, in situ experiments to provide a stable well defined reference. Reference electrodes are typically selected from a range of well-known standardised electrode–electrolyte interfaces at which a characteristic and kinetically rapid reversible faradaic process occurs. The choice of reference electrode should be made carefully in consideration of chemical compatibility with the measurement environment 31 , 32 , 33 , 34 . In combination with an electronic blocking resistance, the potential of the electrode should be stable and reproducible. Unfortunately, deviation from the ideal behaviour frequently occurs. While this can often be overlooked when comparing results from identical cells, more attention is required when reporting values for comparison.

In all cases where conversion between different electrolyte–reference electrode systems is required, junction potentials should be considered. These arise whenever there are different chemical conditions in the electrolyte at the working electrode and reference electrode interfaces. Outside highly dilute solutions, or where there are large activity differences for a reactant/product of the electrode reaction (e.g. pH for hydrogen reactions), liquid junction potentials for conventional aqueous ions have been estimated in the range <50 mV 33 . Such a deviation may nonetheless be significant when onset potentials or activities at specific potentials are being reported. The measured potential difference between the working and reference electrode also depends strongly on the geometry of the cell, so cell design is critical. Fig.  1 shows the influence of cell design on potential profiles. Ideally the reference electrode should therefore be placed close to the working electrode (noting that macroscopic electrodes may have inhomogeneous potentials). To minimise shielding of the electric field between counter and working electrode and interruption of mass transport processes, a thin Luggin-Haber capillary is often used and a small separation maintained. Understanding of shielding and edge effects is vital when reference electrodes are introduced in situ. This is especially applicable for analysis of energy devices for which constraints on cell design, due to the need to minimise electrolyte resistance and seal the cell, preclude optimal reference electrode positioning 32 , 35 , 36 .

a Illustration (simulated data) of primary (resistive) current and potential distribution in a typical three-electrode cell. The main compartment is cylindrical (4 cm diameter, 1 cm height), filled with electrolyte with conductivity 1.28 S m −1 (0.1 M KCl(aq)). The working electrode (WE) is a 2 mm diameter disc drawing 1 mA (≈ 32 mA cm −2 ) from a faradaic process with infinitely fast kinetics and redox potential 0.2 V vs the reference electrode (RE). The counter electrode (CE) is connected to the main compartment by a porous frit; the RE is connected by a Luggin capillary (green cylinders) whose tip position is offset from the WE by a variable distance. Red lines indicate prevailing current paths; coloured surfaces indicate isopotential contours normal to the current density. b Plot of indicated WE vs RE potential (simulated data). As the Luggin tip is moved away from the WE surface, ohmic losses due to the WE-CE current distribution lead to variation in the indicated WE-RE potential. Appreciable error may arise on an offset length scale comparable to the WE radius.

Quantitative statements about fundamental electrochemical processes based on measured values of current and voltage inevitably rely on models of the system. Such models have assumptions that may be routinely overlooked when following experimental and analysis methods, and that may restrict their application to real-world systems. It is quite possible to make highly precise but meaningless measurements! An often-assumed condition for electrocatalyst analysis is the absence of mass transport limitation. For some reactions, such as the acidic hydrogen oxidation and hydrogen evolution reactions, this state is arguably so challenging to reach at representative conditions that it is impossible to measure true catalyst activity 11 . For example, ex situ thin-film rotating disk electrode measurements routinely fail to predict correct trends in catalyst performance in morphologically complex catalyst layers  as relevant operating conditions (e.g. meaningful current densities) are theoretically inaccessible. This topic has been extensively discussed with some authors directly criticising this technique and exploring alternatives 37 , 38 , and others defending the technique’s applicability for ranking catalysts if scrupulous attention is paid to experimental details 39 ; yet, many reports continue to use this measurement technique blindly with no regard for its applicability. We therefore strongly urge those planning measurements to consider whether their chosen technique is capable of providing sufficient evidence to disprove their hypothesis, even if it has been widely used for similar experiments.

The correct choice of technique should be dependent upon the measurand being probed rather than simply following previous reports. The case of iR correction, where a measurement of the uncompensated resistance is used to correct the applied voltage, is a good example. When the measurand is a material property, such as intrinsic catalyst activity, the uncompensated resistance is a source of error introduced by the experimental method and it should carefully be corrected out (Fig.  1 ). In the case that the uncompensated resistance is intrinsic to the measurand—for instance the operating voltage of an electrolyser cell—iR compensation is inappropriate and only serves to obfuscate. Another example is the choice of ex situ (outside the operating environment), in situ (in the operating environment), and operando (during operation) measurements. While in situ or operando testing allows characterisation under conditions that are more representative of real-world use, it may also yield measurements with increased uncertainty due to the decreased possibility for fine experimental control. Depending on the intended use of the measurement, an informed compromise must be sought between how relevant and how uncertain the resulting measurement will be.

Maximising reproducibility

Most electrochemists assess the repeatability of measurements, performing the same measurement themselves several times. Repeats, where all steps (including sample preparation, where relevant) of a measurement are carried out multiple times, are absolutely crucial for highlighting one-off mistakes (Fig.  2 ). Reproducibility, however, is assessed when comparing results reported by different laboratories. Many readers will be familiar with the variability in key properties reported for various systems e.g. variability in the reported electrochemically active surface area (ECSA) of commercial catalysts, which might reasonably be expected to be constant, suggesting that, in practice, the reproducibility of results cannot be taken for granted. As electrochemistry deals mostly with method-defined measurands, the measurement procedure must be standardised for results to be comparable. Variation in results therefore strongly suggests that measurements are not being performed consistently and that the information typically supplied when publishing experimental methods is insufficient to facilitate reproducibility of electrochemical measurements. Quantitative electrochemical measurements require control over a large range of parameters, many of which are easily overlooked or specified imprecisely when reporting data. An understanding of the crucial parameters and methods for their control is often institutional knowledge, held by expert electrochemists, but infrequently formalised and communicated e.g. through publication of widely adopted standards. This creates challenges to both reproducibility and the corresponding assessment of experimental quality by reviewers. The reporting standards established by various publishers (see Introduction) offer a practical response, but it is still unclear whether these will contain sufficiently granular detail to improve the situation.

The measurements from laboratory 1 show a high degree of repeatability, while the measurements from laboratory 2 do not. Apparently, a mistake has been made in repeat 1, which will need to be excluded from any analysis and any uncertainty analysis, and/or suggests further repeat measurements should be conducted. The error bars are based on an uncertainty with coverage factor ~95% (see below) so the results from the two laboratories are different, i.e. show poor reproducibility. This may indicate differing experimental practice or that some as yet unidentified parameter is influencing the results.

Besides information typically supplied in the description of experimental methods for publication, which, at a minimum, must detail the materials, equipment and measurement methods used to generate the results, we suggest that a much more comprehensive description is often required, especially where measurements have historically poor reproducibility or the presented results differ from earlier reports. Such an expanded ‘supplementary experimental’ section would additionally include any details that could impact the results: for example, material pre-treatment, detailed electrode preparation steps, cleaning procedures, expected electrolyte and gas impurities, electrode preconditioning processes, cell geometry including electrode positions, detail of junctions between electrodes, and any other fine experimental details which might be institutional knowledge but unknown to the (now wide) readership of the electrochemical literature. In all cases any corrections and calculations used should be specified precisely and clearly justified; these may include determinations of properties of the studied system, such as ECSA, or of the environment, such as air pressure. We highlight that knowledge of the ECSA is crucial for conventional reporting of intrinsic electrocatalyst activity, but is often very challenging to measure in a reproducible manner 40 , 41 .

To aid reproducibility we recommend regularly calibrating experimental equipment and doing so in a way that is traceable to primary standards realising the International System of Units (SI) base units. The SI system ensures that measurement units (such as the volt) are uniform globally and invariant over time. Calibration applies to direct experimental indicators, e.g. loads and potentiostats, but equally to supporting tools such as temperature probes, balances, and flow meters. Calibration of reference electrodes is often overlooked even though variations from ideal behaviour can be significant 42 and, as discussed above, are often the limit of accuracy on potential measurement. Sometimes reports will specify internal calibration against a known reaction (such as the onset of the hydrogen evolution reaction), but rarely detail regular comparisons to a local master electrode artefact such as a reference hydrogen electrode or explain how that artefact is traceable, e.g. through control of the filling solution concentration and measurement conditions. If reference is made to a standardised material (e.g. commercial Pt/C) the specified material should be both widely available and the results obtained should be consistent with prior reports.

Beyond calibration and reporting, the best test of reproducibility is to perform intercomparisons between laboratories, either by comparing results to identical experiments reported in the literature or, more robustly, through participation in planned intercomparisons (for example ‘round-robin’ exercises); we highlight a recent study applied to solid electrolyte characterisation as a topical example 43 . Intercomparisons are excellent at establishing the key features of an experimental method and the comparability of results obtained from different methods; moreover they provide a consensus against which other laboratories may compare themselves. However, performing repeat measurements for assessing repeatability and reproducibility cannot estimate uncertainty comprehensively, as it excludes systematic sources of uncertainty.

Assessing measurement uncertainty

Formal uncertainty evaluation is an alien concept to most electrochemists; even the best papers (as well as our own!) typically report only the standard deviation between a few repeats. Metrological best practice dictates that reported values are stated as the combination of a best estimate of the measurand, and an interval, and a coverage factor ( k ) which gives the probability of the true value being within that interval. For example, “the turnover frequency of the electrocatalyst is 1.0 ± 0.2 s −1 ( k  = 2)” 16 means that the scientist (having assumed normally distributed error) is 95% confident that the turnover frequency lies in the range 0.8–1.2 s −1 . Reporting results in such a way makes it immediately clear whether the measurements reliably support the stated conclusions, and enables meaningful comparisons between independent results even if their uncertainties differ (Fig.  3 ). It also encourages honesty and self-reflection about the shortcomings of results, encouraging the development of improved experimental techniques.

a Complete reporting of a measurement includes the best estimate of the measurand and an uncertainty and the probability the true value falls within the uncertainty reported. Here, the percentages indicate that a normal distribution has been assumed. b Horizontal bars indicate 95% confidence intervals from uncertainty analysis. The confidence intervals of measurements 1 and 2 overlap when using k  = 2, so it is not possible to say with 95% confidence that the result of the measurement 2 is higher than measurement 1, but it is possible to say this with 68% confidence, i.e. k  = 1. Measurement 3 has a lower uncertainty, so it is possible to say with 95% confidence that the value is higher than measurement 2.

Constructing such a statement and performing the underlying calculations often appears daunting, not least as there are very few examples for electrochemical systems, with pH measurements being one example to have been treated thoroughly 44 . However, a standard process for uncertainty analysis exists, as briefly outlined graphically in Fig.  4 . We refer the interested reader to both accessible introductory texts 45 and detailed step-by-step guides 16 , 46 . The first steps in the process are to state precisely what is being measured—the measurand—and identify likely sources of uncertainty. Even this qualitative effort is often revealing. Precision in the definition of the measurand (and how it is determined from experimental indicators) clarifies the selection of measurement technique and helps to assess its appropriateness; for example, where the measurand relates only to an instantaneous property of a specific physical object, e.g. the current density of a specific fuel cell at 0.65 V following a standardised protocol, we ignore all variability in construction, device history etc. and no error is introduced by the sample. Whereas, when the measurand is a material property, such as turnover frequency of a catalyst material with a defined chemistry and preparation method, variability related to the material itself and sample preparation will often introduce substantial uncertainty in the final result. In electrochemical measurements, errors may arise from a range of sources including the measurement equipment, fluctuations in operating conditions, or variability in materials and samples. Identifying these sources leads to the design of better-quality experiments. In essence, the subsequent steps in the calculation of uncertainty quantify the uncertainty introduced by each source of error and, by using a measurement model or a sensitivity analysis (i.e. an assessment of how the results are sensitive to variability in input parameters), propagate these to arrive at a final uncertainty on the reported result.

Possible sources of uncertainty are identified, and their standard uncertainty or probability distribution is determined by statistical analysis of repeat measurements (Type A uncertainties) or other evidence (Type B uncertainties). If required, uncertainties are then converted into the same unit as the measurand and adjusted for sensitivity, using a measurement model. Uncertainties are then combined either analytically using a standard approach or numerically to generate an overall estimate of uncertainty for the measurand (as indicated in Fig.  3a ).

Generally, given the historically poor understanding of uncertainty in electrochemistry, we promote increased awareness of uncertainty reporting standards and a focus on reporting measurement uncertainty with a level of detail that is appropriate to the claim made, or the scientific utilisation of the data. For example, where the primary conclusion of a paper relies on demonstrating that a material has the ‘highest ever X’ or ‘X is bigger than Y’ it is reasonable for reviewers to ask authors to quantify how confident they are in their measurement and statement. Additionally, where uncertainties are reported, even with error bars in numerical or graphical data, the method by which the uncertainty was determined should be stated, even if the method is consciously simple (e.g. “error bars indicate the sample standard deviation of n  = 3 measurements carried out on independent electrodes”). Unfortunately, we are aware of only sporadic and incomplete efforts to create formal uncertainty budgets for electrochemical measurements of energy technologies or materials, though work is underway in our group to construct these for some exemplar systems.

Electrochemistry has undoubtedly thrived without significant interaction with formal metrology; we do not urge an abrupt revolution whereby rigorous measurements become devalued if they lack additional arcane formality. Rather, we recommend using the well-honed principles of metrology to illuminate best practice and increase transparency about the strengths and shortcomings of reported experiments. From rethinking experimental design, to participating in laboratory intercomparisons and estimating the uncertainty on key results, the application of metrological principles to electrochemistry will result in more robust science.

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Acknowledgements

This work was supported by the National Measurement System of the UK Department of Business, Energy & Industrial Strategy. Andy Wain, Richard Brown and Gareth Hinds (National Physical Laboratory, Teddington, UK) provided insightful comments on the text.

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Smith, G., Dickinson, E.J.F. Error, reproducibility and uncertainty in experiments for electrochemical energy technologies. Nat Commun 13 , 6832 (2022). https://doi.org/10.1038/s41467-022-34594-x

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Experimental Error

Experimental error is unavoidable during the conduct of any experiment, mainly because of the falsifiability principle of the scientific method.

This article is a part of the guide:

• Significance 2
• Sample Size
• Experimental Probability
• Cronbach’s Alpha
• Systematic Error

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• 1 Inferential Statistics
• 2.1 Bayesian Probability
• 3.1.1 Significance 2
• 3.2 Significant Results
• 3.3 Sample Size
• 3.4 Margin of Error
• 3.5.1 Random Error
• 3.5.2 Systematic Error
• 3.5.3 Data Dredging
• 3.5.4 Ad Hoc Analysis
• 3.5.5 Regression Toward the Mean
• 4.1 P-Value
• 4.2 Effect Size
• 5.1 Philosophy of Statistics
• 6.1.1 Reliability 2
• 6.2 Cronbach’s Alpha

Therefore it is important to take steps to minimize the errors and also to understand them in order to be better able to understand the results of the experiment. This entails a study of the type and degree of errors in experimentation.

Statistical tests contain experimental errors that can be classified as either Type-I or Type-II errors. It is important to study both these effects in order to be able to manage error and report it, so that the conclusion of the experiment can be rightly interpreted.

Type I Error - False Positive Type II Error - False Negative

Type I Error

The Type I error (α-error, false positives) occurs when a the null hypothesis (H 0 ) is rejected in favor of the research hypothesis (H 1 ), when in reality the 'null' is correct. This can be understood in terms of medical tests. For example, suppose there is a test that is used to detect a disease in a person. If a Type I error occurs in the test, it means that the test will say the person is suffering from that disease even though he is healthy.

Type II Error

Type II errors (β-errors, false negatives) on the other hand, imply that we reject the research hypothesis, when in fact it is correct. In the similar example of a medical test for a disease, if a Type-II error occurs, then it means that the test will not detect the disease in the person even though he is actually suffering from it.

Hypothesis Testing

In case of Type-I errors, the research hypothesis is accepted even though the null hypothesis is correct. Type-I errors are a false positive that lead to the rejection of the null hypothesis when in fact it may be true.

When a Type-II error occurs, the research hypothesis is not detected as the correct conclusion and is therefore passed off. In terms of the null hypothesis, this kind of an error might lead to accepting the null hypothesis when in fact it is false.

The significance level refers only to the Type-I error. This is because we ask the question "What is the probability that the correlation we observed is purely by chance?" and when this question yields an answer of below a significance level (typically 5% or 1%), we state that the result wasn't a chance process and that the parameters under study are indeed related.

Reason for Errors

Scientific experiments involve a different type of error analysis than a statistical experiment. In science, experimental errors may be caused due to human inaccuracies like a wrong experimental setup in a science experiment or choosing the wrong set of people for a social experiment .

Systematic error refers to that error which is inherent in the system of experimentation. For example, if you want to calculate the value of acceleration due to gravity by swinging a pendulum , then your result will invariably be affected by air resistance, friction at the point of suspension and finite mass of the thread.

Random errors occur because it is impossible to practically achieve infinite precision. Since the value is higher or lower in a random fashion, averaging several readings will reduce random errors.

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Import error for 'maps' from 'jax.experimental' when trying to run a script on TimesFM [closed]

Working on using the TimesFM time series model to do some forecasting. I'm running a script for the forecasting but it gives this error every time.

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This problem is most likely caused by jax.experimental not having the attribute maps . Double check the spelling, and try to see if maps can be directly accessed by jax.experimental .

jax.experimental.maps was deprecated in JAX v0.4.26 and removed in JAX v0.4.31 .

If this is your own code, you should work on migrating from the deprecated xmap transformation either to shard_map or to vmap with spmd_axis_name .

If this is someone else's code that you don't control, then you should try running it with JAX v0.4.30 or older.

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Applied Optics

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Modeling and simulation of optical system error transmission in the laser tracker

Xiaoxu Qiao, Xiaodong Wang, Jianguo Gong, and Yi Luo

Author Affiliations

Xiaoxu Qiao, Xiaodong Wang, Jianguo Gong, and Yi Luo *

State Key Laboratory of High-Performance Precision Manufacturing, Dalian University of Technology, Dalian 116024, China

* Corresponding author: [email protected]

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• Optical Design and Fabrication
• Nodal aberration theory
• Optical components
• Optical design software
• Optical systems
• Optical testing
• Systems design
• Original Manuscript: May 14, 2024
• Revised Manuscript: July 5, 2024
• Manuscript Accepted: July 22, 2024
• Published: August 7, 2024
• Full Article
• Figures ( 10 )
• Data Availability
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The optical system of the laser tracker utilizes plane mirrors to construct a reflective path, reducing its size and weight. However, maintaining the alignment of the laser with the ideal optical axis during its propagation in the optical system poses significant challenges in the design, fabrication, and assembly of the optical system. This paper explores the principle of error propagation during the assembly process of the optical system and improves the accuracy of the output laser through a numerical simulation and optimization methods. A general error model for the optical system is established to understand the principle of error propagation. A Monte Carlo numerical simulation and sensitivity analysis are used to study the influence of various errors on the accuracy of the output laser. The machining errors are optimized using a simulated annealing method to balance the manufacturing difficulty and system accuracy. The assembly process is also optimized to reduce the degrees of freedom and the number of optical parts required, and verified by experiments. The experimental results indicate that the average position error of the output laser is 15.743 µm, and the average angle error is $1.427^{\prime \prime}$ . This study provides what we believe is a novel approach and methodology for the design and alignment of optical systems.

© 2024 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved.

Zhihao Fan, Xiaokai Mu, Yang Yang, Kaike Yang, Kepeng Sun, Qingchao Sun, Wenjing Ma, and Wei Sun Opt. Express 32 (13) 23916-23931 (2024)

Lili Zhu, Junwen Xue, Jiaojiao Ren, Dandan Zhang, Jian Gu, Jiyang Zhang, and Lijuan Li Appl. Opt. 63 (5) 1377-1384 (2024)

Libin Du, Tong Cui, Xiangqian Meng, Yibo Yuan, Liwei Wang, Zhiwei Shang, Hao Chen, and Hongzhi Huang Appl. Opt. 62 (26) 6939-6951 (2023)

Rui Lu, Jianfu Zhang, Xing Han, Yanpeng Wu, and Lin Li Appl. Opt. 63 (14) 3854-3862 (2024)

Jinkai Wang, Yangyang Bai, Wenhe Zhao, Ningning Xie, and Lizhong Zhang Appl. Opt. 63 (4) 1125-1134 (2024)

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Figures (10)

Equations (18).

Gisele Bennett, Editor-in-Chief

Coordinate System Symbols Table for Optical System

Distribution of the Machining Errors for the Adjustment Seats

Distribution of Assembly Errors for Optical Component i

Values of Elements Representing Fixed Geometric Dimensions in the Matrix

Sensitivity of Machining Errors to Pose Errors of Output Laser

Optimized Machining Errors of the Adjustment Seat According to the Sensitivity

Sensitivity of Optical Component Assembly Errors to the Position and Angle Error of the Output Laser

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Condensed Matter > Statistical Mechanics

Title: time-cost-error trade-off relation in thermodynamics: the third law and beyond.

Abstract: Elucidating the fundamental limitations inherent in physical systems is a central subject in physics. For important thermodynamic operations such as information erasure, cooling, and copying, resources like time and energetic cost must be expended to achieve the desired outcome within a predetermined error margin. In this study, we introduce the concept of {\it separated states}, which consist of fully unoccupied and occupied states. This concept generalizes many critical states involved in relevant thermodynamic operations. We uncover a three-way trade-off relation between {\it time}, {\it cost}, and {\it error} for a general class of thermodynamic operations aimed at creating separated states, simply expressed as $\tau{\cal C}\varepsilon_{\tau}\ge 1-\eta$. This fundamental relation is applicable to diverse thermodynamic operations, including information erasure, cooling, and copying. It provides a profound quantification of the unattainability principle in the third law of thermodynamics in a general form. Building upon this relation, we explore the quantitative limitations governing cooling operations, the preparation of separated states, and a no-go theorem for exact classical copying. Furthermore, we extend these findings to the quantum regime, encompassing both Markovian and non-Markovian dynamics. Specifically, within Lindblad dynamics, we derive a similar three-way trade-off relation that quantifies the cost of achieving a pure state with a given error. The generalization to general quantum dynamics involving a system coupled to a finite bath implies that heat dissipation becomes infinite as the quantum system is exactly cooled down to the ground state or perfectly reset to a pure state, thereby resolving an open question regarding the thermodynamic cost of information erasure.

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