accuracy definition research

Scientific Research and Methodology

5.2 precision and accuracy.

Two issues concerning sampling, raised in Sect. 5.1 , were: which individuals should be in the sample, and how many individuals should be in the sample be. These two issues address two different aspects of sampling: precision and accuracy (Fig. 5.1 ).

Accuracy refers to how close a sample estimate is to the population value (on average ). Precision refers to how close all the possible sample estimates are likely to be (that is, how much variation is likely in the sample estimates).

Using this language:

• The type of sampling (i.e., the way in which the samples in selected) impacts the accuracy of the sample estimate. In other words, the type of sampling impacts the external validity of the study.
• The size of the sample impacts the precision of the sample estimate.

For example, large samples are more likely to be precise estimates because each possible sample value will produced similar estimates, but they may or may not be accurate estimates. Similarly, random samples are likely to produce accurate estimates (and hence the study is more likely to be externally valid), but they may not be precise unless the sample is also large.

FIGURE 5.1: Precision and accuracy: Each coloured dot is like a sample estimate of the population value (shown by the black central dot)

Example 5.2 (Precision and accuracy) To estimate the average age of all Queenslanders , we could ask 9000 Queensland school children (a large sample indeed!).

What Is the Difference Between Accuracy and Precision?

Accuracy is close to a known value; precision measures repeatability

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• B.A., Physics and Mathematics, Hastings College

Accuracy and precision are two important factors to consider when taking data measurements . Both accuracy and precision reflect how close a measurement is to an actual value, but accuracy reflects how close a measurement is to a known or accepted value, while precision reflects how reproducible measurements are, even if they are far from the accepted value.

Key Takeaways: Accuracy Versus Precision

• Accuracy is how close a value is to its true value. An example is how close an arrow gets to the bull's-eye center.
• Precision is how repeatable a measurement is. An example is how close a second arrow is to the first one (regardless of whether either is near the mark).
• Percent error is used to assess whether a measurement is sufficiently accurate and precise.

You can think of accuracy and precision in terms of hitting a bull's-eye. Accurately hitting the target means you are close to the center of the target, even if all the marks are on different sides of the center. Precisely hitting a target means all the hits are closely spaced, even if they are very far from the center of the target. Measurements that are both precise and accurate are repeatable and very near true values.

There are two common definitions of accuracy . In math, science, and engineering, accuracy refers to how close a measurement is to the true value.

The ISO ( International Organization for Standardization ) applies a more rigid definition, where accuracy refers to a measurement with both true and consistent results. The ISO definition means an accurate measurement has no systematic error and no random error. Essentially, the ISO advises that accurate be used when a measurement is both accurate and precise.

Precision is how consistent results are when measurements are repeated. Precise values differ from each other because of random error, which is a form of observational error. 

You can think of accuracy and precision in terms of a basketball player. If the player always makes a basket, even though he strikes different portions of the rim, he has a high degree of accuracy. If he doesn't make many baskets but always strikes the same portion of the rim, he has a high degree of precision. A player whose free throws always make the basket the exact same way has a high degree of both accuracy and precision.

Take experimental measurements for another example of precision and accuracy. You can tell how close a set of measurements are to a true value by averaging them . If you take measurements of the mass of a 50.0-gram standard sample and get values of 47.5, 47.6, 47.5, and 47.7 grams, your scale is precise, but not very accurate. The average of your measurements is 47.6, which is lower than the true value. Yet, your measurements were consistent. If your scale gives you values of 49.8, 50.5, 51.0, and 49.6, it is more accurate than the first balance but not as precise. The average of the measurements is 50.2, but there is a much larger range between them. The more precise scale would be better to use in the lab, providing you made an adjustment for its error. In other words, it's better to calibrate a precise instrument than to use an imprecise, yet accurate one.

Mnemonic to Remember the Difference

An easy way to remember the difference between accuracy and precision is:

• A C curate is C orrect (or C lose to real value)
• P R ecise is R epeating (or R epeatable)

Accuracy, Precision, and Calibration

Do you think it's better to use an instrument that records accurate measurements or one that records precise measurements? If you weigh yourself on a scale three times and each time the number is different, yet it's close to your true weight, the scale is accurate. Yet it might be better to use a scale that is precise, even if it is not accurate. In this case, all the measurements would be very close to each other and "off" from the true value by about the same amount. This is a common issue with scales, which often have a "tare" button to zero them.

While scales and balances might allow you to tare or make an adjustment to make measurements both accurate and precise, many instruments require calibration. A good example is a thermometer . Thermometers often read more reliably within a certain range and give increasingly inaccurate (but not necessarily imprecise) values outside that range. To calibrate an instrument, record how far off its measurements are from known or true values. Keep a record of the calibration to ensure proper readings. Many pieces of equipment require periodic calibration to ensure accurate and precise readings.

Accuracy and precision are only two important concepts used in scientific measurements. Two other important skills to master are significant figures and scientific notation . Scientists use percent error as one method of describing how accurate and precise a value is. It's a simple and useful calculation.

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Definition of accuracy

• accurateness
• preciseness
• rigorousness
• ultraprecision

Examples of accuracy in a Sentence

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'accuracy.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

accur(ate) + -acy

1644, in the meaning defined at sense 1

Dictionary Entries Near accuracy

accumulator

Cite this Entry

“Accuracy.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/accuracy. Accessed 13 May. 2024.

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What are accuracy and precision?

Part of Science Working scientifically Year 5 Year 6

Experiments need to be accurate and precise

It is important that the results from scientific experiments are both accurate and precise . This means that the data collection needs to be accurate and precise as well.

This way, we can be more confident are results are correct.

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Practices of Science: Precision vs. Accuracy

• If the darts are neither close to the bulls-eye, nor close to each other, there is neither accuracy, nor precision (SF Fig. 1.5 A).  
• If all of the darts land very close together, but far from the bulls-eye, there is precision, but not accuracy (SF Fig. 1.5 B).   
• If the darts are all about an equal distance from and spaced equally around the bulls-eye there is mathematical accuracy because the average of the darts is in the bulls-eye. This represents data that is accurate, but not precise (SF Fig. 1.5 C). However, if you were actually playing darts this would not count as a bulls-eye!
• If the darts land close to the bulls-eye and close together, there is both accuracy and precision (SF Fig. 1.5 D). 

SF Fig. 1.5. Dartboards showing different accuracy and precision scenarios.

Image by Byron Inouye

• The oceanographer checks the weather forecast the night before her trip so she knows what to wear on the boat. The TV forecaster says it will be between 26 and 31 degrees (°) Celsius (C) at noon the next day. The actual temperature reading the next day on the boat at noon is 28° C. 
• When the oceanographer’s Global Positioning System (GPS) indicates that she is at the location of the underwater buoy, she anchors the boat and jumps in the water to collect the data logger. However, she can’t see the buoy. The other GPS units belonging to her colleagues on the boat also indicate that they are at the correct location. After an extensive search, the oceanographer finds the buoy 50 meters (m) from the boat. 
• While on the way back to shore, the oceanographer throws in a fishing line to see if she can catch anything for dinner. She is lucky enough to catch a mahi-mahi. When she pulls it out of the water, her colleagues estimate the weight of the fish. Their estimates are 16.1 kilograms (kg), 16.8 kg, and 15.9 kg. When they weigh the fish upon returning to shore, the actual weight is 18.2 kg.  
• Write your own scenario illustrating the difference between accuracy and precision. Swap your scenario with a classmate. Identify your classmate’s scenario measurements as accurate or inaccurate and precise or imprecise.  
• How is this model different from scientists who are measuring a natural phenomenon? 
• Is there a way for scientists to determine how accurate their measurements are? Explain your answer

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[ ak -yer- uh -see ]

• the condition or quality of being true, correct, or exact; freedom from error or defect; precision or exactness; correctness.
• Chemistry, Physics. the extent to which a given measurement agrees with the standard value for that measurement. Compare precision ( def 6 ) .
• Mathematics. the degree of correctness of a quantity, expression, etc. Compare precision ( def 5 ) .

/ ˈækjʊrəsɪ /

• faithful measurement or representation of the truth; correctness; precision
• physics chem the degree of agreement between a measured or computed value of a physical quantity and the standard or accepted value for that quantity

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Other words from.

• hyper·accu·ra·cy noun

Word History and Origins

Origin of accuracy 1

Example Sentences

AIs can do a good job of predicting a few frames into the future, but the accuracy falls off sharply after five or 10 frames, says Athanasios Vlontzos at Imperial College London.

Spokeswoman Nika Edwards said in an email that tenants who lose income can request a decrease in rent, but she acknowledged that HUD does not check the accuracy of rent calculations or monitor enforcement on a local level.

Endangered’s publisher, Marc Specter, notes that the game’s developers consulted with the Center for Biological Diversity to ensure scientific accuracy.

Their method may help bring about new levels of predictive accuracy, which theorists desperately need if they are to move beyond the leading but incomplete model of particle physics.

It’s advisable to check periodically for new categories that would enhance the accuracy of your listing.

CIA goes to great lengths to understand the reliability and accuracy of every source.

The grand jury inquiry affords opportunity to test accuracy of witness accounts.

Did the reporters and editors put much value on on accuracy and objectivity, or were they more a part of the party machine?

The Daily Beast has not verified the accuracy of either of these ads.

As such, they emphatically demonstrate the accuracy of the “no risk to public” trope.

Here convincing proof was given of Mme. Mesdag's accuracy, originality of interpretation, and her skill in the use of color.

The quality of artistic beauty in articulation is very important, beyond the mere accuracy which is ordinarily thought of.

For accuracy, 500 to 1000 leukocytes must be classified; for approximate results, 200 are sufficient.

And he wished also to restore her to her natural setting, with the greatest degree of historic accuracy.

Much has been said of the wonderful accuracy of Stradivari's purfling and that as a purfler he stands unrivalled.

Related Words

• truthfulness

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Meaning of accuracy in English

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• The latest missiles can be fired with incredible accuracy.
• Her paintings are almost photographic in their detail and accuracy.
• All high-tech weaponry demands frequent servicing to ensure accuracy.
• authoritative
• authoritatively
• got it in one! idiom
• in so many words idiom
• strictly speaking idiom
• superaccurate
• synchronization

You can also find related words, phrases, and synonyms in the topics:

accuracy | American Dictionary

Accuracy noun [u] ( correctness ), accuracy noun [u] ( exactness ), accuracy | business english, examples of accuracy, collocations with accuracy.

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National Academies of Sciences, Engineering, and Medicine; Policy and Global Affairs; Committee on Science, Engineering, Medicine, and Public Policy; Committee on Responsible Science. Fostering Integrity in Research. Washington (DC): National Academies Press (US); 2017 Apr 11.

Fostering Integrity in Research.

• Hardcopy Version at National Academies Press

2 Foundations of Integrity in Research: Core Values and Guiding Norms

Problems of scientific freedom and responsibility are not new; one need only consider, as examples, the passionate controversies that were stirred by the work of Galileo and Darwin. In our time, however, such problems have changed in character, and have become far more numerous, more urgent and more complex. Science and its applications have become entwined with the whole fabric of our lives and thoughts. . . . Scientific freedom, like academic freedom, is an acquired right, generally accepted by society as necessary for the advancement of knowledge from which society may benefit. Scientists possess no rights beyond those of other citizens except those necessary to fulfill the responsibility arising from their special knowledge, and from the insight arising from that knowledge. — John Edsall (1975)

Synopsis: The integrity of research is based on adherence to core values—objectivity, honesty, openness, fairness, accountability, and stewardship. These core values help to ensure that the research enterprise advances knowledge. Integrity in science means planning, proposing, performing, reporting, and reviewing research in accordance with these values. Participants in the research enterprise stray from the norms and appropriate practices of science when they commit research misconduct or other misconduct or engage in detrimental research practices.

• TRANSMITTING VALUES AND NORMS IN RESEARCH

The core values and guiding norms of science have been studied and written about extensively, with the work of Robert Merton providing a foundation for subsequent work on the sociology of science ( Merton, 1973 ). Merton posited a set of norms that govern good science: (1) Communalism (common ownership of scientific knowledge), (2) Universalism (all scientists can contribute to the advance of knowledge), (3) Disinterestedness (scientists should work for the good of the scientific enterprise as opposed to personal gain), and (4) Organized Skepticism (results should be examined critically before they are accepted). Research on scientists and scientific organizations has also led to a better understanding of counternorms that appear to conflict with the dominant Mertonian norms but that are recognized as playing an inherent part in the actual practice of science, such as the personal commitment that a scientist may have to a particular hypothesis or theory ( Mitroff, 1974 ).

More recent work on the effectiveness of responsible conduct of research education, covered in more detail in Chapter 9 , explores evidence that at least some scientists may not understand and reflect upon the ethical dimensions of their work ( McCormick et al., 2012 ). Several causes are identified, including a lack of awareness on the part of researchers of the ethical issues that can arise, confidence that they can identify and address these issues without any special training or help, or apprehension that a focus on ethical issues might hinder their progress. An additional challenge arises from the apparent gap “between the normative ideals of science and science's institutional reward system” ( Devereaux, 2014 ). Chapter 6 covers this issue in more detail. Here, it is important to note that identifying and understanding the values and norms of science do not automatically mean that they will be followed in practice. The context in which values and norms are communicated and transmitted in the professional development of scientists is critically important.

Scientists are privileged to have careers in which they explore the frontiers of knowledge. They have greater autonomy than do many other professionals and are usually respected by other members of society. They often are able to choose the questions they want to pursue and the methods used to derive answers. They have rich networks of social relationships that, for the most part, reinforce and further their work. Whether actively involved in research or employed in some other capacity within the research enterprise, scientists are able to engage in an activity about which they are passionate: learning more about the world and how it functions.

In the United States, scientific research in academia emerged during the late 19th century as an “informal, intimate, and paternalistic endeavor” ( NAS-NAE-IOM, 1992 ). Multipurpose universities emphasized teaching, and research was more of an avocation than a profession. Even today, being a scientist and engaging in research does not necessarily entail a career with characteristics traditionally associated with professions such as law, medicine, architecture, some subfields of engineering, and accounting. For example, working as a researcher does not involve state certification of the practitioner's expertise as a requirement to practice, nor does it generally involve direct relationships with fee-paying clients. Many professions also maintain an explicit expectation that practitioners will adhere to a distinctive ethical code ( Wickenden, 1949 ). In contrast, scientists do not have a formal, overarching code of ethics and professional conduct.

However, the nature of professional practice even in the traditional professions continues to evolve ( Evetts, 2013 ). Some scholars assert that the concept of professional work should include all occupations characterized by “expert knowledge, autonomy, a normative orientation grounded in community, and high status, income, and other rewards” ( Gorman and Sandefur, 2011 ). Scientific research certainly shares these characteristics. In this respect, efforts to formalize responsible conduct of research training in the education of researchers often have assumed that this training should be part of the professional development of researchers ( IOM-NRC, 2002 ; NAS-NAE-IOM, 1992 ). However, the training of researchers (and research itself) has retained some “informal, intimate, and paternalistic” features. Attempts to formalize professional development training sometimes have generated resistance in favor of essentially an apprenticeship model with informal, ad hoc approaches to how graduate students and postdoctoral fellows learn how to become professional scientists.

One challenge facing the research enterprise is that informal, ad hoc approaches to scientific professionalism do not ensure that the core values and guiding norms of science are adequately inculcated and sustained. This has become increasingly clear as the changes in the research environment described in Chapter 3 have emerged and taken hold. Indeed, the apparent inadequacy of these older forms of training to the task of socializing and training individuals into responsible research practices is a recurring theme of this report.

Individual scientists work within a much broader system that profoundly influences the integrity of research results. This system, described briefly in Chapter 1 , is characterized by a massive, interconnected web of relationships among researchers, employing institutions, public and private funders, and journals and professional societies. This web comprises unidirectional and bidirectional obligations and responsibilities between the parts of the system. The system is driven by public and private investments and results in various outcomes or products, including research results, various uses of those results, and trained students. However, the system itself has a dynamic that shapes the actions of everyone involved and produces results that reflect the functioning of the system. Because of the large number of relationships between the many players in the web of responsibility, features of one set of relationships may affect other parts of the web. These interdependencies complicate the task of devising interventions and structures that support and encourage the responsible conduct of research.

• THE CORE VALUES OF RESEARCH

The integrity of research is based on the foundational core values of science. The research system could not operate without these shared values that shape the behaviors of all who are involved with the system. Out of these values arise the web of responsibilities that make the system cohere and make scientific knowledge reliable. Many previous guides to responsible conduct in research have identified and described these values ( CCA, 2010 ; ESF-ALLEA, 2011 ; IAC-IAP, 2012 ; ICB, 2010 ; IOM-NRC, 2002 ). This report emphasizes six values that are most influential in shaping the norms that constitute research practices and relationships and the integrity of science:

Objectivity

Accountability, stewardship.

This chapter examines each of these six values in turn to consider how they shape, and are realized in, research practices.

The first of the six values discussed in this report—objectivity—describes the attitude of impartiality with which researchers should strive to approach their work. The next four values—honesty, openness, accountability, and fairness—describe relationships among those involved in the research enterprise. The final value—stewardship—involves the relationship between members of the research enterprise, the enterprise as a whole, and the broader society within which the enterprise is situated. Although we discuss stewardship last, it is an essential value that perpetuates the other values.

The hallmark of scientific thinking that differentiates it from other modes of human inquiry and expression such as literature and art is its dedication to rational and empirical inquiry. In this context, objectivity is central to the scientific worldview. Karl Popper (1999) viewed scientific objectivity as consisting of the freedom and responsibility of the researcher to (1) pose refutable hypotheses, (2) test the hypotheses with the relevant evidence, and (3) state the results clearly and unambiguously to any interested person. The goal is reproducibility, which is essential to advancing knowledge through experimental science. If these steps are followed diligently, Popper suggested, any reasonable second researcher should be able to follow the same steps to replicate the work.

Objectivity means that certain kinds of motivations should not influence a researcher's action, even though others will. For example, if a researcher in an experimental field believes in a particular hypothesis or explanation of a phenomenon, he or she is expected to design experiments that will test the hypothesis. The experiment should be designed in a way that allows the possibility for the hypothesis to be disconfirmed. Scientific objectivity is intended to ensure that scientists' personal beliefs and qualities—motivations, position, material interests, field of specialty, prominence, or other factors—do not introduce biases into their work.

As will be explored in later chapters, in practice it is not that simple. Human judgment and decisions are prone to a variety of cognitive biases and systematic errors in reasoning. Even the best scientific intentions are not always sufficient to ensure scientific objectivity. Scientific objectivity can be compromised accidentally or without recognition by individuals. In addition, broader biases of the reigning scientific paradigm influence the theory and practice of science ( Kuhn, 1962 ). A primary purpose of scientific replication is to minimize the extent to which experimental findings are distorted by biases and errors. Researchers have a responsibility to design experiments in ways that any other person with different motivations, interests, and knowledge could trust the results. Modern problems related to reproducibility are explored later in the report.

In addition, objectivity does not imply or require that researchers can or should be completely neutral or disinterested in pursuing their work. The research enterprise does not function properly without the organized efforts of researchers to convince their scientific audiences. Sometimes researchers are proven correct when they persist in trying to prove theories in the face of evidence that appears to contradict them.

It is important to note, in addition, Popper's suggestion that scientific objectivity consists of not only responsibility but freedom . The scientist must be free from pressures and influences that can bias research results. Objectivity can be compromised when institutional expectations, laboratory culture, the regulatory environment, or funding needs put pressure on the scientist to produce positive results or to produce them under time pressure. Scientists and researchers operate in social contexts, and the incentives and pressures of those contexts can have a profound effect on the exercise of scientific methodology and a researcher's commitment to scientific objectivity.

Scientific objectivity also must coexist with other human motivations that challenge it. As an example of such a challenge, a researcher might become biased in desiring definitive results evaluating the validity of high-profile theories or hypotheses that their experiments were designed to support or refute. Both personal desire to obtain a definitive answer and institutional pressures to produce “significant” conclusions can provide strong motivation to find definitive results in experimental situations. Dedication to scientific objectivity in those settings represents the best guard against scientists finding what they desire instead of what exists. Institutional support of objectivity at every level—from mentors, to research supervisors, to administrators, and to funders—is crucial in counterbalancing the very human tendency to desire definitive outcomes of research.

A researcher's freedom to advance knowledge is tied to his or her responsibility to be honest . Science as an enterprise producing reliable knowledge is based on the assumption of honesty. Science is predicated on agreed-upon systematic procedures for determining the empirical or theoretical basis of a proposition. Dishonest science violates that agreement and therefore violates a defining characteristic of science.

Honesty is the principal value that underlies all of the other relationship values. For example, without an honest foundation, realizing the values of openness, accountability, and fairness would be impossible.

Scientific institutions and stakeholders start with the assumption of honesty. Peer reviewers, granting agencies, journal editors, commercial research and development managers, policy makers, and other players in the scientific enterprise all start with an assumption of the trustworthiness of the reporting scientist and research team. Dishonesty undermines not only the results of the specific research but also the entire scientific enterprise itself, because it threatens the trustworthiness of the scientific endeavor.

Being honest is not always straightforward. It may not be easy to decide what to do with outlier data, for example, or when one suspects fraud in published research. A single outlier data point may be legitimately interpreted as a malfunctioning instrument or a contaminated sample. However, true scientific integrity requires the disclosure of the exclusion of a data point and the effect of that exclusion unless the contamination or malfunction is documented, not merely conjectured. There are accepted statistical methods and standards for dealing with outlier data, although questions are being raised about how often these are followed in certain fields ( Thiese et al., 2015 ).

Dishonesty can take many forms. It may refer to out-and-out fabrication or falsification of data or reporting of results or plagiarism. It includes such things as misrepresentation (e.g., avoiding blame, claiming that protocol requirements have been followed when they have not, or producing significant results by altering experiments that have been previously conducted), nonreporting of phenomena, cherry-picking of data, or overenhancing pictorial representations of data. Honest work includes accurate reporting of what was done, including the methods used to do that work. Thus, dishonesty can encompass lying by omission, as in leaving out data that change the overall conclusions or systematically publishing only trials that yield positive results. The “file drawer” effect was first discussed almost 40 years ago; Robert Rosenthal (1979) presented the extreme view that “journals are filled with the 5 percent of the studies that show Type I errors, while the file drawers are filled with the 95 percent of the studies that show non-significant results.” This hides the possibility of results being published from 1 significant trial in an experiment of 100 trials, as well as experiments that were conducted and then altered in order to produce the desired results. The file drawer effect is a result of publication bias and selective reporting, the probability that a study will be published depending on the significance of its results ( Scargle, 2000 ). As the incentives for researchers to publish in top journals increase, so too do these biases and the file drawer effect.

Another example of dishonesty by omission is failing to report all funding sources where that information is relevant to assessing potential biases that might influence the integrity of the work. Conversely, dishonesty can also include reporting of nonexistent funding sources, giving the impression that the research was conducted with more support and so may have been more thorough than in actuality.

Beyond the individual researcher, those engaged in assessing research, whether those who are funding it or participating in any level of the peer review process, also have fundamental responsibilities of honesty. Most centrally, those assessing the quality of science must be honest in their assessments and aware of and honest in reporting their own conflicts of interest or any cognitive biases that may skew their judgment in self-serving ways. There is also a need to guard against unconscious bias, sometimes by refusing to assess work even when a potential reviewer is convinced that he or she can be objective. Efforts to protect honesty should be reinforced by the organizations and systems within which those assessors function. Universities, research organizations, journals, funding agencies, and professional societies must all work to hold each other to honest interactions without favoritism and with potentially biasing factors disclosed.

Openness is not the same as honesty, but it is predicated on honesty. In the scientific enterprise, openness refers to the value of being transparent and presenting all the information relevant to a decision or conclusion. This is essential so that others in the web of the research enterprise can understand why a decision or conclusion was reached. Openness also means making the data on which a result is based available to others so that they may reproduce and verify results or build on them. In some contexts, openness means listening to conflicting ideas or negative results without allowing preexisting biases or expectations to cloud one's judgment. In this respect, openness reinforces objectivity and the achievement of reliable observations and results.

Openness is an ideal toward which to strive in the research enterprise. It almost always enhances the advance of knowledge and facilitates others in meeting their responsibilities, be it journal editors, reviewers, or those who use the research to build products or as an input to policy making. Researchers have to be especially conscientious about being open, since the incentive structure within science does not always explicitly reward openness and sometimes discourages it. An investigator may desire to keep data private to monopolize the conclusions that can be drawn from those data without fear of competition. Researchers may be tempted to withhold data that do not fit with their hypotheses or conclusions. In the worst cases, investigators may fail to disclose data, code, or other information underlying their published results to prevent the detection of fabrication or falsification.

Openness is an ideal that may not always be possible to achieve within the research enterprise. In research involving classified military applications, sensitive personal information, or trade secrets, researchers may have an obligation not to disseminate data and the results derived from those data. Disclosure of results and underlying data may be delayed to allow time for filing a patent application. These sorts of restrictions are more common in certain research settings—such as commercial enterprises and government laboratories—than they are in academic research institutions performing primarily fundamental work. In the latter, openness in research is a long-held principle shared by the community, and it is a requirement in the United States to avoid privileged access that would undermine the institution's nonprofit status and to maintain the fundamental research exclusion from national security-based restrictions.

As the nature of data changes, so do the demands of achieving openness. For example, modern science is often based on very large datasets and computational implementations that cannot be included in a written manuscript. However, publications describing such results could not exist without the data and code underlying the results. Therefore, as part of the publication process, the authors have an obligation to have the available data and commented code or pseudocode (a high-level description of a program's operating principle) necessary and sufficient to re-create the results listed in the manuscript. Again, in some situations where a code implementation is patentable, a brief delay in releasing the code in order to secure intellectual property protection may be acceptable. When the resources needed to make data and code available are insufficient, authors should openly provide them upon request. Similar considerations apply to such varied forms of data as websites, videos, and still images with associated text or voiceovers.

Central to the functioning of the research enterprise is the fundamental value that members of the community are responsible for and stand behind their work, statements, actions, and roles in the conduct of their work. At its core, accountability implies an obligation to explain and/or justify one's behavior. Accountability requires that individuals be willing and able to demonstrate the validity of their work or the reasons for their actions. Accountability goes hand in hand with the credit researchers receive for their contributions to science and how this credit builds their reputations as members of the research enterprise. Accountability also enables those in the web of relationships to rely on work presented by others as a foundation for additional advances.

Individual accountability builds the trustworthiness of the research enterprise as a whole. Each participant in the research system, including researchers, institutional administrators, sponsors, and scholarly publishers, has obligations to others in the web of science and in return should be able to expect consistent and honest actions by others in the system. Mutual accountability therefore builds trust, which is a consequence of the application of the values described in this report.

The purpose of scientific publishing is to advance the state of knowledge through examination by peers who can assess, test, replicate where appropriate, and build on the work being described. Investigators reporting on their work thus must be accountable for the accuracy of their work. Through this accountability, they form a compact with the users of their work. Readers should be able to trust that the work was performed by the authors as described, with honest and accurate reporting of results. Accountability means that any deviations from the compact would be flagged and explained. Readers then could use these explanations in interpreting and evaluating the work.

Investigators are accountable to colleagues in their discipline or field of research, to the employer and institution at which the work is done, to the funders or other sponsors of the research, to the editors and institutions that disseminate their findings, and to the public, which supports research in the expectation that it will produce widespread benefits. Other participants in the research system have other forms of accountability. Journals are accountable to authors, reviewers, readers, the institutions they represent, and other journals (for the reuse of material, violation of copyright, or other issues of mutual concern). Institutions are accountable to their employees, to students, to the funders of both research and education, and to the communities in which they are located. Organizations that sponsor research are accountable to the researchers whose work they support and to their governing bodies or other sources of support, including the public. These networks of accountability support the web of relationships and responsibilities that define the research enterprise.

The accountability expected of individuals and organizations involved with research may be formally specified in policies or regulations. Accountability under institutional research misconduct policies, for example, could mean that researchers will face reprimand or other corrective actions if they fail to meet their responsibilities.

While responsibilities that are formally defined in policies or regulations are important to accountability in the research enterprise, responsibilities that may not be formally specified should also be included in the concept. For example, senior researchers who supervise others are accountable to their employers and the researchers whom they supervise to conduct themselves as professionals, as this is defined by formal organizational policies. On a less formal level, research supervisors are also accountable for being attentive to the educational and career development needs of students, postdoctoral fellows, and other junior researchers whom they oversee. The same principle holds for individuals working for research institutions, sponsoring organizations, and journals.

The scientific enterprise is filled with professional relationships. Many of them involve judging others' work for purposes of funding, publication, or deciding who is hired or promoted. Being fair in these contexts means making professional judgments based on appropriate and announced criteria, including processes used to determine outcomes. Fairness in adhering to explicit criteria and processes reinforces a system in which the core values can operate and trust among the parties can be maintained.

Fairness takes on another dimension in designing criteria and evaluation mechanisms. Research has demonstrated, for example, that grant proposals in which reviewers were blinded to applicant identity and institution receive systematically different funding decisions compared with the outcomes of unblinded reviews ( Ross et al., 2006 ). Truly blinded reviews may be difficult or impossible in a small field. Nevertheless, to the extent possible, the criteria and mechanisms involved in evaluation must be designed so as to ensure against unfair incentive structures or preexisting cultural biases. Fairness is also important in other review contexts, such as the process of peer reviewing articles and the production of book reviews for publication.

Fairness is a particularly important consideration in the list of authors for a publication and in the citations included in reports of research results. Investigators may be tempted to claim that senior or well-known authors played a larger role than they actually did so that their names may help carry the paper to publication and readership. But such a practice is unfair both to the people who actually did the work and to the honorary author, who may not want to be listed prominently or at all. Similarly, nonattribution of credit for contributions to the reported work or careless or negligent crediting of prior work violates the value of fairness. Best practices in authorship, which are based on the value of fairness, honesty, openness, and accountability, are discussed further in Chapter 9 .

Upholding fairness also requires researchers to acknowledge those whose work contributed to their advances. This is usually done through citing relevant work in reporting results. Also, since research is often a highly competitive activity, sometimes there is a race to make a discovery that results in clear winners and losers. Sometimes two groups of researchers make the same discovery nearly simultaneously. Being fair in these situations involves treating research competitors with generosity and magnanimity.

The importance of fairness is also evident in issues involving the duty of care toward human and animal research subjects. Researchers often depend on the use of human and animal subjects for their research, and they have an obligation to treat those subjects fairly—with respect in the case of human subjects and humanely in the case of laboratory animals. They also have obligations to other living things and to those aspects of the environment that affect humans and other living things. These responsibilities need to be balanced and informed by an appreciation for the potential benefits of research.

The research enterprise cannot continue to function unless the members of that system exhibit good stewardship both toward the other members of the system and toward the system itself. Good stewardship implies being aware of and attending carefully to the dynamics of the relationships within the lab, at the institutional level, and at the broad level of the research enterprise itself. Although we have listed stewardship as the final value in the six we discuss in this report, it supports all the others. Here we take up stewardship within the research enterprise but pause to acknowledge the extension of this value to encompass the larger society.

One area where individual researchers exercise stewardship is by performing service for their institution, discipline, or the broader research enterprise that may not necessarily be recognized or rewarded. These service activities include reviewing, editing, serving on faculty committees, and performing various roles in scientific societies. Senior researchers may also serve as mentors to younger researchers whom they are not directly supervising or formally responsible for. At a broader level, researchers, institutions, sponsors, journals, and societies can contribute to the development and updating of policies and practices affecting research. As will be discussed in Chapter 9 , professional societies perform a valuable service by developing scientific integrity policies for their fields and keeping them updated. Individual journals, journal editors, and member organizations have contributed by developing standards and guidelines in areas such as authorship, data sharing, and the responsibilities of journals when they suspect that submitted work has been fabricated or plagiarized.

Stewardship also involves decisions about support and influences on science. Some aspects of the research system are influenced or determined by outside factors. Public demand, political considerations, concerns about national security, and even the prospects for our species' survival can inform and influence decisions about the amount of public and private resources devoted to the research enterprise. Such forces also play important roles in determining the balance of resources invested in various fields of study (e.g., both among and within federal agencies), as well as the balance of effort devoted to fundamental versus applied work and the use of various funding mechanisms.

In some cases, good stewardship requires attending to situations in which the broader research enterprise may not be operating optimally. Chapter 6 discusses issues where problems have been identified and are being debated, such as workforce imbalances, the poor career prospects of academic researchers in some fields, and the incentive structures of modern research environments.

Stewardship is particularly evident in the commitment of the research enterprise to education, both of the next generation of researchers and of individuals who do not expect to become scientists. In particular, Chapter 10 discusses the need to educate all members of the research enterprise in the responsible conduct of research. Education is one way in which engaging in science provides benefits both to those within the research system and to the general public outside the system.

• A DEFINITION OF RESEARCH INTEGRITY

Making judgments about definitions and terminology as they relate to research integrity and breaches of integrity is a significant component of this committee's statement of task. Practicing integrity in research means planning, proposing, performing, reporting, and reviewing research in accordance with the values described above. These values should be upheld by research institutions, research sponsors, journals, and learned societies as well as by individual researchers and research groups. General norms and specific research practices that conform to these values have developed over time. Sometimes norms and practices need to be updated as technologies and the institutions that compose the research enterprise evolve. There are also disciplinary differences in some specific research practices, but norms and appropriate practices generally apply across science and engineering research fields. As described more fully in Chapter 9 , best practices in research are those actions undertaken by individuals and organizations that are based on the core values of science and enable good research. They should be embraced, practiced, and promoted.

• Cite this Page National Academies of Sciences, Engineering, and Medicine; Policy and Global Affairs; Committee on Science, Engineering, Medicine, and Public Policy; Committee on Responsible Science. Fostering Integrity in Research. Washington (DC): National Academies Press (US); 2017 Apr 11. 2, Foundations of Integrity in Research: Core Values and Guiding Norms.
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• Units Of Measurement
• Accuracy Precision Measurement

Accuracy and Precision - The Art of Measurement

Measurement is essential for us to understand the external world, and through millions of years of life, we have developed a sense of measurement. Measurements require tools that provide scientists with a quantity. The problem here is that the result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is referred to as an error. Accuracy and precision are two important factors to consider while taking measurements. Both these terms reflect how close a measurement is to a known or accepted value. In this article, let us learn in detail about precision and accuracy.

The ability of an instrument to measure the accurate value is known as accuracy. In other words, it is the  the closeness of the measured value to a standard or true value . Accuracy is obtained by taking small readings. The small reading reduces the error of the calculation. The accuracy of the system is classified into three types as follows:

• Point Accuracy

The accuracy of the instrument only at a particular point on its scale is known as point accuracy. It is important to note that this accuracy does not give any information about the general accuracy of the instrument.

• Accuracy as Percentage of Scale Range

The uniform scale range determines the accuracy of a measurement. This can be better understood with the help of the following example: Consider a thermometer having the scale range up to 500 ºC . The thermometer has an accuracy of ±0.5 percent of scale range i.e. 0.005 x 500 = ± 2.5 ºC. Therefore, the reading will have a maximum error of ± 2.5 ºC.

• Accuracy as Percentage of True Value

Such type of accuracy of the instruments is determined by identifying the measured value regarding their true value. The accuracy of the instruments is neglected up to ±0.5 percent from the true value.

The closeness of two or more measurements to each other is known as the precision of a substance. If you weigh a given substance five times and get 3.2 kg each time, then your measurement is very precise but not necessarily accurate. Precision is independent of accuracy. The below examples will tell you about how you can be precise but not accurate and vice versa. Precision is sometimes separated into:

• Repeatability

The variation arising when the conditions are kept identical and repeated measurements are taken during a short time period.

• Reproducibility

The variation arises using the same measurement process among different instruments and operators, and over longer time periods.

Accuracy is the degree of closeness between a measurement and its true value. Precision is the degree to which repeated measurements under the same conditions show the same results.

Since you are here, you might want to check out the following articles:

• Errors in Measurement
• Measurement of length

Accuracy and Precision Examples

The top left image shows the target hit at high precision and accuracy. The top right image shows the target hit at a high accuracy but low precision. The bottom left image shows the target hit at a high precision but low accuracy. The bottom right image shows the target hit at low accuracy and low precision.

More Examples

• If the weather temperature reads 28 °C outside and it is 28 °C outside, then the measurement is said to be accurate. If the thermometer continuously registers the same temperature for several days, the measurement is also precise.
• If you take the measurement of the mass of a body of 20 kg and you get 17.4,17,17.3 and 17.1, your weighing scale is precise but not very accurate. If your scale gives you values of 19.8, 20.5, 21.0, and 19.6, it is more accurate than the first balance but not very precise.

Difference between Accuracy and Precision

In the previous few sections having discussed what each term means, let us now look at their differences.

Practice Questions

Q1) The volume of a liquid is 26 mL. A student measures the volume and finds it to be 26.2 mL, 26.1 mL, 25.9 mL, and 26.3 mL in the first, second, third, and fourth trial, respectively. Which of the following statements is true for his measurements?

a. They are neither precise nor accurate. b. They have poor accuracy. c. They have good precision. d. They have poor precision.

Answer: They have good precision.

Q2) The volume of a liquid is 20.5 mL. Which of the following sets of measurement represents the value with good accuracy? 18.6 mL, 17.8 mL, 19.6 mL, 17.2 mL 19.2 mL, 19.3 mL, 18.8 mL, 18.6 mL 18.9 mL, 19.0 mL, 19.2 mL, 18.8 mL 20.2 mL, 20.5 mL, 20.3 mL, 20.1 mL Answer: The set 20.2 mL, 20.5 mL, 20.3 mL, 20.1 mL represents the value with good accuracy.

Frequently Asked Questions – FAQs

What is meant by accuracy.

Accuracy refers to the closeness of the measured value to a standard or true value.

What is the classification of accuracy of the system?

Accuracy of the system are classified into:

State true or false: Multiple measurements or factors are needed.

What is meant by precision.

Precision is the degree to which repeated measurements under the same conditions show the same results.

What is meant by error

The difference between the actual value and the measured value is known as error.

Write the two factors to be considered while taking measurements?

What is point accuracy, can the results be precise and accurate, define reproducibility., if the player shoots ball into the goal in one shot, he is said to be accurate or precise, watch the video to find out what are base measurements.

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Parameters of the best fitting lunar ellipsoid based on GRAIL’s selenoid model

• Original Study
• Open access
• Published: 27 June 2023
• Volume 58 , pages 139–147, ( 2023 )

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• Kamilla Cziráki   ORCID: orcid.org/0009-0005-4162-4500 1 &
• Gábor Timár   ORCID: orcid.org/0000-0001-9675-6192 1  

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Since the Moon is less flattened than the Earth, most lunar GIS applications use a spherical datum. However, with the renaissance of lunar missions, it seems worthwhile to define an ellipsoid of revolution that better fits the selenoid. The main long-term benefit of this might be to make the lunar adaptation of methods already implemented in terrestrial GNSS and gravimetry easier and somewhat more accurate. In our work, we used the GRGM 1200A Lunar Geoid (Goossens et al. in A global degree and order 1200 model of the lunar gravity field using GRAIL mission data. In: Lunar and planetary science conference, Houston, TX, Abstract #1484, 2016; Lemoine et al. in Geophys Res Lett 41:3382–3389. http://dx.doi.org/10.1002/2014GL060027 , 2014), a 660th degree and order potential surface, developed in the frame of the GRAIL project. Samples were taken from the potential surface along a mesh that represents equal area pieces of the surface, using a Fibonacci sphere. We tried Fibonacci spheres with several numbers of points and also separately examined the effect of rotating the network for a given number of points on the estimated parameters. We estimated the best-fitting rotation ellipsoid’s semi-major axis and flatness data by minimizing the selenoid undulation values at the network points, which were obtained for a = 1,737,576.6 m and f = 0.000305. This parameter pair is already obtained for a 10,000 point grid, while the case of reducing the points of the mesh to 3000 does not cause a deviation in the axis data of more than 10 cm. As expected, the absolute value of the selenoid undulations have decreased compared to the values taken with respect to the spherical basal surface, but significant extreme values still remained as well.

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1 Introduction

The theoretical shape of a celestial body is defined by Gauss ( 1828 ) as the level surface of its gravitational field for a specific potential value. On the Earth, this potential surface is aligned with the mean sea level, but on other celestial bodies, such as the Moon, in the absence of sea, a potential value is chosen, compared to which about half of the Moon's surface is higher and half is lower. The resulting W = 2,821,713.3 m 2 * s −2 (Martinec & Pěč 1988 ) is the potential value for the selenoid surface. In addition to the theoretical definition of the shape of a celestial body, it is also essential to have a reference surface (a sphere or rotation ellipsoid in general) that can be described easily, with a few parameters, whose centre lies at the centre of mass of the body and which deviates as little as possible from the potential surface.

In our work, we have created a lunar rotation ellipsoid fitting this criteria using the GRGM 1200A Lunar Geoid from the GRAIL mission. To do this, we sampled the selenoid with a mesh of points representing the same areas and then used a program to find the semi-major axis-semi-minor axis pair that gave the best fit.

Such a rotating ellipsoid has significant practical benefits. It can be used well as a geodetic datum, and thus has a significant role as a base surface for various Geographic Informaton System (GIS) applications. What makes the problem of selenodetical datums relevant now is that lunar exploration is experiencing a renaissance, with not only the US but also the EU (in cooperation with the US) and China planning to send a series of space probes and landers to the Moon. NASA's Artemis program began this year, and its goal is to land on the Moon again, and perform research there. The Chinese Chang'e program has been going on for more than a decade, and its ultimate goal is to land a man on the far side of the Moon. During these missions, more time than ever before is planned to be spent on the lunar surface, during which positioning, and hence the lunar equivalent of Global Navigation Satellite System (GNSS), will play a major role. Since the basic data for GNSS applications is always the ellipsoid of revolution that best approximates the geometric shape on Earth, the simplest lunar application of these requires the parameters of the ellipsoidal approximation of the solenoid.

For LunaNet (NASA 2022 ), which NASA plans to create for GIS applications, the reference surface is currently planned to be the Lunar Reference Frame Standard, a sphere with a radius of 1737.4 km (NASA 2008 ). However, in the future, mainly in the process of the migration of gravimetric applications developed on Earth and the equations of the terrestrial GNSS, it is conceivable that a rotational ellipsoid will replace it.

For the estimation of the ellipsoidal parameters, we used the latest and highest resolution selenoid-model, the GRGM 1200A Lunar Geoid, based on GRAIL (Fig. 1 ). This includes the coefficients of the spherical harmonical functions up to 660th degree and order (Goossens et al. 2016 ; Lemoine et al. 2014 ) and also provides direct data from the far side of the Moon.

GRGM 1200A above the r  = 1737.4 km reference sphere displayed in QGIS Source : Goossens et al. ( 2016 ), Lemoine et al. ( 2014 ), QGIS

Besides the selenoid model, sampling points were needed to perform the estimation. The aim was to define points that represent equal areas on a spherical surface.

The points were determined using the Fibonacci sphere (Fig. 2 ). This is a set of points that maps the Fibonacci spiral onto a sphere. To create this, the z-axis was first divided into as many parts as the number of sample points defined. This gives the latitudes, and the longitudes were determined by rotations calculated with the Fibonacci number (Appendix A).

Display of the 3000-point Fibonacci sphere in QGIS

This was then used to sample the selenoid model. The resulting database contains the coordinates of the points and the altitude of the selenoid above a reference surface (a sphere of radius 1737.4 km). From this, the ellipsoidal approximation was made in a Python program. The idea is to find a semi-major axis-semi-minor axis pair for which the sum of the squares of the undulations of the selenoid at the sample points is minimal.

Equation  1 : Method to calculate the best-fitting ellipsoid for an N-point Fibonacci sphere.

Since the semi-major axis-major axis pair cannot be any value, both must be in the vicinity of the lunar reference sphere, all calculations were only performed between 1734 and 39 kms in the program. In addition, several iterative steps were used, first determining the minimum at a resolution of only 100 m, then at a higher resolution in the vicinity of the resulting minimum ± 100 m, and so on down to a resolution of 10 cm, saving a lot of time.

This program was not only used once on a single grid, but we also tested how the number of points on the Fibonacci sphere affected the results, using 100, 300, 1000, 3000, 5000, 10,000 and 100,000 point spheres, performing the steps of the calculation described above.

The semi-major axes, semi-minor axes and flatnesses of the ellipsoids thus obtained are shown in the table below. The seven results are based on samples of 100, 300, 1000, 3000, 5000, 10,000 and 100,000 points, with an accuracy of 10 cm (Table 1 ).

The parameters estimated from 100,000 sampled points have the highest accuracy (this result is the same as the parameters obtained for 10,000 points), so the half major axis of the rotation ellipsoid optimally fitting the Moon has a major axis of 1,737,576.6 m, a minor axis of 1,737,046.8 m and a flatness of 3.05 * 10 –4 (Cziráki & Timár 2023 ).

5 Discussion

5.1 comparison of the most accurate result with other reference surfaces.

As shown in Table 2 , the parameters of the rotation ellipsoid we have estimated differ by 176.6 and 353.2 m from the best fitting sphere. The flattening is 3/10,000, which is quite small, the same value on Earth is roughly 33/10,000 (World Geodetic System, WGS84), a significantly larger deviation from the sphere. This is due to the Earth's much faster axial rotation, which not only affects its gravitational field but also distorts its physical shape.

Our results were also compared to the triaxial ellipsoid created in 2010. This is based on the Chang'e 1 and Lunar Prospector (Konopliv et al. 2001 ) data CE-1-LAM-LEVEL (Wang et al. 2010 ). As can be seen from Table 2 , the differences are 125.9, − 12.8 and − 112.8 ms respectively. These may be due to the fact that the two models describe different geometric bodies. Another important difference is that the selenoid models used to calculate the parameters of the two ellipsoids are not the same. In the case of CE-1-LAM-LEVEL, the primary source of the selenoid model is the 180 degree and order model of the Lunar Prospector mission (Wang et al. 2010 ), which is lower in resolution than the GRGM 1200A model we used, and the Lunar Prospector model does not have direct data from the far side of the Moon, while GRAIL provided data from the far side with the same resolution as the near side. The differences between the two ellipsoids could therefore be due to these factors.

5.2 Variation of results as a function of Fibonacci spheres with different numbers of points

A central element of our research was that we carried out the estimation with meshes containing different numbers of points. These also provide information on the minimum number of points at which the result does not differ or differs only marginally from the most accurate estimate.

This was not expected for the samples with few points, as they represented too large areas (e.g., 379,323 km 2 for the 100-point sphere). Nevertheless, no deviation of more than 1 m was observed in our original sample. In order to determine the uncertainties, the 100-point sphere was rotated above the selenoid model every 30 degrees, the results of which are presented in Sect.  5.3 .

The resolution of the GRGM 1200A selenoid model is roughly 16.54 km, so the aim was to perform the calculation with a Fibonacci sphere of similar resolution. The 100,000-point Fibonacci sphere is the closest to achieving this, with 1 point representing 379 km 2 . Of course, the two resolutions are not exactly the same, as the selenoid model does not provide data in equal areas, but they are still similar in magnitude.

An important result, however, is that the result obtained with a 100,000 point sphere is already obtained with a sample with a tenth of a point, and from the 3000 point sample onwards, there is no difference of more than 10 cm.

5.3 Effect of rotating the Fibonacci sphere on the results

The results obtained are influenced to some extent by the position of the sampling points relative to the selenoid model. The significance of this decreases as the number of points increases and becomes negligible. In order to quantify this effect, the 100- and 5000-point Fibonacci spheres were rotated every 30 degrees above the selenoid and the approximation was also performed with these samples.

As expected, there are significant variations (Figs. 3 ,  4 ), especially among the 100-point samples, with a deviation of 5.56 m for the semi-major axis and 7.54 m for the semi-minor axis, and differences of more than 10 m. At 5000 points, the differences are almost negligible, with a standard deviation of 0.09 m for the semi-major axis and 0.14 m for the semi-minor axis, and all differences are within 0.5 m.

Length of the semi-major axis of the fitted ellipsoid for spheres with different positions

Length of the semi-minor axis of the fitted ellipsoid for spheres with different positions

5.4 Histogram of selenoid undulations on the resulting ellipsoid

The histogram of the selenoid undulations is of interest for the resulting ellipsoid, and for the selenoid model as well. This is shown with 5 m intervals in Fig.  5 .

Histogram of the selenoid undulations at the points of a Fibonacci sphere of 100,000 points, and the density function of the normal distribution ( m  = −0.0147 m, σ  = 110 m) fitted to it

Fortunately, the expected value of the fitted normal distribution is 0, but there are significant outliers. These are mainly indicative of inequalities in the potential surface of the selenoid. For example, mass concentrations, mascons (Muller & Sjorgen 1968 ), which cause a variation of several hundred mgal, have such an effect on the shape of the potential surfaces, and it is therefore not surprising that such excursions are observed.

It is also interesting to note that the extremes are much larger than, for example, on the Earth, where the geoidundulation values do not exceed ± 110 m. And for the Moon, these values reach as high as ± 450 m. This is mainly due to the absence of active planeto-dynamics, which does not allow the formation of mass accretions of this magnitude on Earth. Without these, anomalies can persist on the Moon, which can affect the shape of the potential surfaces to such a large extent.

5.5 Map representation of the selenoid undulations

As can be seen in Fig.  6 , the outliers are localised, mainly in the vicinity of certain craters. The positive anomalies mainly occur near the two craters on the Moon's near side, Mare Serenitatis and Mare Imbrium. The largest negative anomalies are found at the South Pole Aitken and at the Mare Orientale craters.

The undulations of the selenoid visualised in QGIS

On the map representation of the undulations, the outliers characterise the selenoid model, not the ellipsoid, since the localised large anomalies are not significantly reduced by just one extra parameter. More information about the effect of the ellipsoid can be obtained by comparing Fig.  6 with Fig.  1 , which is basically a map of the selenoidal undulations with respect to the reference sphere. This shows that the ellipsoid has reduced the undulations somewhat in general, but most strikingly, it has almost completely eliminated the anomalies at the poles. This was expected, as the rotation ellipsoid should show the greatest improvement in this area compared to the sphere.

5.6 Definition of the ideal ellipsoid on Earth

Using the methods described in Chapter 3, and with some minor modifications, the ideally fitting ellipsoid of the geoid was also calculated. This is, of course, a very widely used existing ellipsoid, WGS84 (DMA 1984 ). The primary aim of the calculation is therefore not to create a "new" ellipsoid, but rather to see how well our method can reproduce the parameters of WGS84, essentially to check the calculation method used for the Moon.

The input data used was the geoid model grid of EGM96 (Lemoine et al. 1996 ), downloaded from https://www.agisoft.com/downloads/geoids/ . This takes into account the coefficients of the spherical harmonic functions up to 360 degrees and order. The geoid was sampled using the 100,000-point Fibonacci sphere.

The ideal ellipsoid for the geoid was also calculated to an accuracy of 10 cm. This, as expected, differs only minimally from the parameters of WGS 84. Its semi-major axis is 6,378,137.0 m and its semi-minor axis is 6,356,752.3142 m. For the ellipsoid we obtained, the former is 6,378,136.4 m and the latter 6,356,751.7 m.

6 Conclusion

In our research, we performed an ellipsoidal approximation of the lunar gravitational field, we searched for the rotational ellipsoid that deviates least from the Moon's specific potential surface, which defines the selenoid. In doing so, we sampled the GRGM1200A selenoid model using a Fibonacci sphere with points representing equal areas, and then with this database, we estimated the parameters of the ellipsoid using a program based on a least-squares approximation of the selenoid undulations.

The resulting ellipsoid has its centre at the centre of mass of the Moon, a semi-major axis of 1,737,576.6 m, a semi-minor axis of 1,737,046.8 m and a flattening of 0.000305. The ellipsoidal parameters were determined with an accuracy of 10 cm, using multiple samples.

In our research, we also performed the calculation for the Earth geoid, with the aim of showing that this method can give a good estimate of the ideally fitting ellipsoid. Since the parameters of this ellipsoid on Earth are known, we compared our results to it, and they were very close, with a deviation of only 60 cm.

In the future, we would like to extend our research to the Earth, and investigate the differences in best fitting ellipsoids using varying geoid models.

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Supported by the ÚNKP-22-6 New National Excellence Program of the Ministry for Culture and Innovation from the source of the National Research, Development and Innovation Fund.

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Cziráki, K., Timár, G. Parameters of the best fitting lunar ellipsoid based on GRAIL’s selenoid model. Acta Geod Geophys 58 , 139–147 (2023). https://doi.org/10.1007/s40328-023-00415-w

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Received : 20 March 2023

Accepted : 29 May 2023

Published : 27 June 2023

Issue Date : June 2023

DOI : https://doi.org/10.1007/s40328-023-00415-w

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