Cognitivism (psychology)

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https://en.wikipedia.org/wiki/Cognitivism_(psychology)

In psychology, cognitivism is a theoretical framework for understanding the mind that gained credence in the 1950s. The movement was a response to behaviorism, which cognitivists said neglected to explain cognition. Cognitive psychology derived its name from the Latin cognoscere, referring to knowing and information, thus cognitive psychology is an information-processing psychology derived in part from earlier traditions of the investigation of thought and problem solving.

Behaviorists acknowledged the existence of thinking but identified it as a behavior. Cognitivists argued that the way people think impacts their behavior and therefore cannot be a behavior in and of itself. Cognitivists later claimed that thinking is so essential to psychology that the study of thinking should become its own field. However, cognitivists typically presuppose a specific form of mental activity, of the kind advanced by computationalism.

Cognitivism has more recently been challenged by postcognitivism.

Cognitive development

The process of assimilating and expanding our intellectual horizon is termed as cognitive development. We have a complex physiological structure that absorbs a variety of stimuli from the environment, stimuli being the interactions that are able to produce knowledge and skills. Parents process knowledge informally in the home while teachers process knowledge formally in school. Knowledge should be pursued with zest and zeal; if not, then learning becomes a burden.

Attention

Attention is the first part of cognitive development. It pertains to a person's ability to focus and sustain concentration. Attention can also be how focus minded an individual is and having their full concentration on one thing. It is differentiated from other temperamental characteristics like persistence and distractibility in the sense that the latter modulates an individual's daily interaction with the environment. Attention, on the other hand, involves his behavior when performing specific tasks. Learning, for instance, takes place when the student gives attention towards the teacher. Interest and effort closely relate to attention. Attention is an active process which involves numerous outside stimuli. The attention of an organism at any point in time involves three concentric circles; beyond awareness, margin, and focus. Individuals have a mental capacity; there are only so many things someone can focus on at one time.

A theory of cognitive development called information processing holds that memory and attention are the foundation of cognition. It is suggested that children's attention is initially selective and is based on situations that are important to their goals. This capacity increases as the child grows older since they are more able to absorb stimuli from tasks. Another conceptualization classified attention into mental attention and perceptual attention. The former is described as the executive-driven attentional "brain energy" that activates task-relevant processes in the brain while the latter are immediate or spontaneous attention driven by novel perceptual experiences.

Process of learning

Cognitive theory mainly stresses the acquisition of knowledge and growth of the mental structure. Cognitive theory tends to focus on conceptualizing the student's learning process: how information is received; how information is processed and organized into existing schema; how information is retrieved upon recall. In other words, cognitive theory seeks to explain the process of knowledge acquisition and the subsequent effects on the mental structures within the mind. Learning is not about the mechanics of what a learner does, but rather a process depending on what the learner already knows (existing information) and their method of acquiring new knowledge (how they integrate new information into their existing schemas). Knowledge acquisition is an activity consisting of internal codification of mental structures within the student's mind. Inherent to the theory, the student must be an active participant in their own learning process. Cognitive approaches mainly focus on the mental activities of the learner like mental planning, goal setting, and organizational strategies. In cognitive theories not only the environmental factors and instructional components play an important role in learning. There are additional key elements like learning to code, transform, rehearse, and store and retrieve the information. The learning process includes learner's thoughts, beliefs, and attitude values.

Role of memory

Memory plays a vital role in the learning process. Information is stored within memory in an organised, meaningful manner. Here, teacher and designers play different roles in the learning process. Teachers supposedly facilitate learning and the organization of information in an optimal way. Whereas designers supposedly use advanced techniques (such as analogies, mnemonic devices, and hierarchical relationships) to help learners acquire new information to add to their prior knowledge. Forgetting is described as an inability to retrieve information from memory. Memory loss may be a mechanism used to discard situationally irrelevant information by assessing the relevance of newly acquired information.

Process of transfer

According to cognitive theory, if a learner knows how to implement knowledge in different contexts and conditions, then we can say that transfer has occurred. Understanding is composed of knowledge - in the form of rules, concepts and discrimination. Knowledge stored in memory is important, but the use of such knowledge is also important. Prior knowledge will be used for identifying similarities and differences between itself and novel information.

Types of learning explained in detail by this position

Cognitive theory mostly explains complex forms of learning in terms of reasoning, problem solving and information processing. Emphasis must be placed on the fact that the goal of all aforementioned viewpoints is considered to be the same - the transfer of knowledge to the student in the most efficient and effective manner possible. Simplification and standardization are two techniques used to enhance the effectiveness and efficiency of knowledge transfer. Knowledge can be analysed, decomposed and simplified into basic building blocks. There is a correlation with the behaviorist model of the knowledge transfer environment. Cognitivists stress the importance of efficient processing strategies.

Basic principles of the cognitive theory and relevance to instructional design

A behaviorist uses feedback (reinforcement) to change the behavior in the desired direction, while the cognitivist uses the feedback for guiding and supporting the accurate mental connections. For different reasons learners' task analyzers are critical to both cognitivists and behaviorists. Cognitivists look at the learner's predisposition to learning (How does the learner activate, maintain, and direct their learning?). Additionally, cognitivists examine the learners' 'how to design' instruction that it can be assimilated. (i.e., what about the learner's existing mental structures?) In contrast, the behaviorists look to determine where the lesson should begin (i.e., at what level the learners are performing successfully?) and what are the most effective reinforcements (i.e., What are the consequences that are most desired by the learner?).

There are some specific assumptions or principles that direct the instructional design: active involvement of the learner in the learning process, learner control, metacognitive training (e.g., self-planning, monitoring, and revising techniques), the use of hierarchical analyses to identify and illustrate prerequisite relationships (cognitive task analysis procedure), facilitating optimal processing of structuring, organizing and sequencing information (use of cognitive strategies such as outlining, summaries, synthesizers, advance organizers etc.), encouraging the students to make connections with previously learned material, and creating learning environments (recall of prerequisite skills; use of relevant examples, analogies).

Structuring instruction

Cognitive theories emphasize mainly on making knowledge meaningful and helping learners to organize and relate new information to existing knowledge in memory. Instruction should be based on students' existing schema or mental structures, to be effective. The organisation of information is connected in such a manner that it should relate to the existing knowledge in some meaningful way. Examples of cognitive strategies include the use of analogies and metaphors, framing, outlining the mnemonics, concept mapping, advance organizers, and so forth. The cognitive theory mainly emphasizes the major tasks of the teacher / designer and includes analyzing various learning experiences to the learning situation, which can impact learning outcomes of different individuals. Organizing and structuring the new information to connect the learners' previously acquired knowledge abilities and experiences. The new information is effectively and efficiently assimilated/accommodated within the learners cognitive structure.

Theoretical approach

Cognitivism has two major components, one methodological, the other theoretical. Methodologically, cognitivism has a positivist approach and says that psychology can be (in principle) fully explained by the use of the scientific method, there is speculation on whether or not this is true. This is also largely a reductionist goal, with the belief that individual components of mental function (the 'cognitive architecture') can be identified and meaningfully understood. The second says that cognition contains discrete and internal mental states (representations or symbols) that can be changed using rules or algorithms.

Cognitivism became the dominant force in psychology in the late-20th century, replacing behaviorism as the most popular paradigm for understanding mental function. Cognitive psychology is not a wholesale refutation of behaviorism, but rather an expansion that accepts that mental states exist. This was due to the increasing criticism towards the end of the 1950s of simplistic learning models. One of the most notable criticisms was Noam Chomsky's argument that language could not be acquired purely through conditioning, and must be at least partly explained by the existence of internal mental states.

The main issues that interest cognitive psychologists are the inner mechanisms of human thought and the processes of knowing. Cognitive psychologists have attempted to shed some light on the alleged mental structures that stand in a causal relationship to our physical actions.

Criticisms of psychological cognitivism

In the 1990s, various new theories emerged that challenged cognitivism and the idea that thought was best described as computation. Some of these new approaches, often influenced by phenomenological and postmodern philosophy, include situated cognition, distributed cognition, dynamicism and embodied cognition. Some thinkers working in the field of artificial life (for example Rodney Brooks) have also produced non-cognitivist models of cognition. On the other hand, much of early cognitive psychology, and the work of many currently active cognitive psychologists, does not treat cognitive processes as computational. The idea that mental functions can be described as information processing models has been criticised by philosopher John Searle and mathematician Roger Penrose who both argue that computation has some inherent shortcomings which cannot capture the fundamentals of mental processes.

• Penrose uses Gödel's incompleteness theorem (which states that there are mathematical truths which can never be proven in a sufficiently strong mathematical system; any sufficiently strong system of axioms will also be incomplete) and Turing's halting problem (which states that there are some things which are inherently non-computable) as evidence for his position.
• Searle has developed two arguments, the first (well known through his Chinese room thought experiment) is the 'syntax is not semantics' argument—that a program is just syntax, while understanding requires semantics; therefore programs (hence cognitivism) cannot explain understanding. Such an argument presupposes the controversial notion of a private language. The second, which Searle now prefers but is less well known, is his 'syntax is not physics' argument—nothing in the world is intrinsically a computer program except as applied, described, or interpreted by an observer, so either everything can be described as a computer and trivially a brain can but then this does not explain any specific mental processes, or there is nothing intrinsic in a brain that makes it a computer (program). Many oppose these views and have criticized his arguments, which have created significant disagreement. Both points, Searle claims, refute cognitivism.

Another argument against cognitivism is the problems of Ryle's Regress or the homunculus fallacy. Cognitivists have offered a number of arguments attempting to refute these attacks.

Tokenism

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Tokenism

In sociology, tokenism is the social practice of making a perfunctory and symbolic effort towards the equitable inclusion of members of a minority group, especially by recruiting people from under-represented social-minority groups in order for the organization to give the public appearance of racial and gender equality, usually within a workplace or a school. The sociological purpose of tokenism is to give the appearance of inclusivity to a workplace or a school that is not as culturally diverse (racial, religious, sexual, etc.) as the rest of society.

History

The social concept and the employment practice of tokenism became understood in the popular culture of the United States in the late 1950s. In the face of racial segregation, tokenism emerged as a solution that though earnest in effort, only acknowledged an issue without actually solving it. In the book Why We Can't Wait (1964), civil rights activist Martin Luther King Jr. discussed the subject of tokenism, and how it constitutes a minimal acceptance of black people to the mainstream of U.S. society.

When asked about the gains of the Civil Rights Movement in 1963, human rights activist Malcolm X answered: "Tokenism is hypocrisy. One little student in the University of Mississippi, that's hypocrisy. A handful of students in Little Rock, Arkansas, is hypocrisy. A couple of students going to school in Georgia is hypocrisy. Integration in America is hypocrisy in the rawest form. And the whole world can see it. All this little tokenism that is dangled in front of the Negro and then he's told, 'See what we're doing for you, Tom.' Why the whole world can see that this is nothing but hypocrisy? All you do is make your image worse; you don't make it better." Malcolm X highlights that tokenism is used as a tool by America to improve its image but fails in its attempts. For instance, in 1954, the United States ruled segregation in public schools unconstitutional through the Brown v. Board of Education case. Malcolm X references Little Rock, Arkansas, where nine students sought to fight for their rights to attend school. On September 4, 1957, Arkansas National Guard troops were sent around Central High School to prevent the entry of nine African American students into an all-white school, defying federal law. President Eisenhower federalized the Arkansas National Guard and enforced federal troops to uphold the law. While this marked the day that ignited change within Arkansas' school system for African-American children, desegregation did not constitute equality. All nine of the students were brutally bullied by white students and this behavior was encouraged by the school's administration. Malcolm X's example of Little Rock exemplifies how tokenism can be intended to create the impression of social inclusiveness and diversity without bringing about any significant changes to the inclusion of underrepresented groups.

In psychology

In the field of psychology, the broader definition of tokenism is a situation in which a member of a distinctive category is treated differently from other people. The characteristics that make the person of interest a token can be perceived as either a handicap or an advantage, as supported by Václav Linkov. In a positive light, these distinct people can be seen as experts in their racial/cultural category, valued skills, or a different perspective on a project. In contrast, tokenism is most often seen as a handicap due to the ostracism of a selected sample of a minority group. Linkov also attributes drawbacks in psychology to Cultural and Numerical Tokenism, instances that have shifted where value of expertise is placed and its effect on proliferating information that is not representative of all the possible facts.

In the workplace

A Harvard Business School professor, Rosabeth Moss Kanter, asserted back in 1977 that a token employee is usually part of a "socially-skewed group" of employees who belong to a minority group that constitutes less than 15% of the total employee population of the workplace.

By definition, token employees in a workplace are known to be few; hence, their alleged high visibility among the staff subjects them to greater pressure to perform their work at higher production standards of quality and volume and to behave in the expected, stereotypical manner. Given the smallness of the group of token employees in a workplace, the individual identity of each token person is usually disrespected by the dominant group, who apply a stereotype role to them as a means of social control in the workplace. In order to avoid tokenism within the workplace, diversity and inclusion must be integrated to foster an environment where people feel connected and included. Employees must be hired on the basis of their capabilities rather than their gender, ethnicity, race, and sexuality.

Tokenism can also have an impact on mental health in the workplace. According to one study, racial minorities also experience heightened performance pressures related to their race and gender; however, many reported that racial problems were more common than gender problems. Being a token makes one appear more visible within the workplace, placing more scrutiny and pressure for them to represent an entire group. Anxiety, stress, exhaustion, guilt, shame and burnout can arise from overworking in efforts to become a good representative of their identity group.

In professor Kanter's work on tokenism and gender, she found that the problems experienced by women in typically male-dominated occupations were due solely to the skewed proportions of men and women in these occupations. For example, women are often underrepresented within the STEM field, where women also sometimes face more hostile working environments where discrimination and sexual harassment are more frequent. Women in STEM may experience greater performance pressure to work harder in a male-dominated field while also experiencing social isolation from the males within their workplace. The pressure to perform better can be influenced by the stereotype of women being less competent in mathematics and science. These non-inclusive measures contribute to the lack of women in STEM.

Professor Kanter found that being a token evoked three behaviour consequences of visibility, polarization, and assimilation. Firstly, tokens often felt that they were being watched all the time, leading to the feeling of more pressure to perform well. In attempts to perform well, tokens will feel the need to work harder and strive for perfection. Secondly, polarization implies that the dominant group are uncomfortable around tokens or feel threatened by them due to their differences. As a result, tokens may experience social isolation from the exclusion of the majority group. Finally, tokens will feel the need to assimilate to the stereotyped caricature of their roles. For instance, women will feel forced to perform the “suitable behaviour" of a woman in reinforcing the behaviour of stereotypes attached to which they are associated with.

There has been much debate surrounding the concept of tokenism behind women directors on corporate boards. Since men disproportionately occupy the majority of board seats globally, governments and corporations have attempted to address this inequitable distribution of seats through reform measures. Reform measures include legislation mandating gender representation on corporate boards of directors, which has been the focus of societal and political debates. All-male boards typically recruit women to improve specialized skills and to bring different values to decision making. In particular, women introduce useful female leadership qualities and skills like risk averseness, less radical decision-making, and more sustainable investment strategies. However, the mandate of gender diversity may also harm women. Some critics of the mandate believe that it makes women seem like "space fillers," which undermines the qualifications that women can bring to their jobs.

In politics

In politics, allegations of tokenism may occur when a political party puts forward candidates from under-represented groups, such as women or racial minorities, in races that the party has little or no chance of winning, while making limited or no effort to ensure that such candidates have similar opportunity to win the nomination in races where the party is safe or favoured. The "token" candidates are frequently submitted as paper candidates, while nominations in competitive or safe seats continue to favor members of the majority group.

The end result of such an approach is that the party's slate of candidates maintains the appearance of diversity, but members of the majority group remain overrepresented in the party's caucus after the election—and thus little to no substantive progress toward greater inclusion of underrepresented groups has actually occurred.

Legal scholar David Schraub writes about the use of "dissident minorities" by political movements to give themselves a veneer of legitimacy while promoting policies that the majority of the minority group opposes. He uses the examples of Anti-Zionist Jews and African-American conservatives, both of which dissent from their demographic group's consensus position on matters critical to their group's collective liberation or interests. These "dissidents" from minority groups are accused of either allowing the majority to tokenize them, or willingly tokenizing themselves as a shield against complaints and accusations made by the rest of that minority, and an excuse for the majority to avoid addressing or considering the concerns of the minority in question. Sometimes they may actively work to exclude non-dissident members of their group, to preserve their social and political power within the movement they support. Schraub contends that the majority of the movement dissident minorities support values them not for their contributions but for their identity, since more weight is given to people of minority background when talking about issues concerning that minority. If they break ranks and criticize their political movement, they often find themselves shunned, since they are no longer a reliable token.

In fiction

See also: Magical Negro and Stereotype § Role in art and culture

In fiction, token characters represent groups which vary from the norm (usually defined as a white, heterosexual male) and are otherwise excluded from the story. The token character can be based on ethnicity (e.g. black, Hispanic, Asian), religion (e.g. Jewish, Muslim), sexual orientation (e.g., gay), gender (typically a female character in a predominantly male cast) or disability. Token characters are usually background characters, and, as such, are usually disposable and are eliminated from the narrative early in the story, in order to enhance the drama, while conserving the main characters.

In television

Tokenism, in a television setting, can be any act of putting a minority into the mix to create some sort of publicly viewed diversity. A racial divide in TV has been present since the first television show that hired minorities, Amos 'n' Andy (1928–1960), in 1943. Regardless of whether a token character may be stereotypical or not, tokenism can initiate a whole biased perception that may conflict with how people see a specific race, culture, gender or ethnicity. From The Huffington Post, America Ferrera states: “Tokenism is about inserting diverse characters because you feel you have to; true diversity means writing characters that aren't just defined by the color of their skin, and casting the right actor for the role".

Ethnic and racial representation in television has been proven as an educational basis to inform mass audiences. However, tokenism leads to a narrow representation of minority groups, and this trend often leads to minority characters being exposed in negative or stereotypical fashions. Research done as early as the 1970s suggests an early recognition and disapproval of tokenism and its effects on perceptions of minority groups—specifically, perceptions of African Americans. Tokenism seemed to be used as a quick fix for the complete void of major/recurring minority roles in television, but its skewed representation lacked room for thoroughly independent and positive roles. Throughout that decade, major broadcast networks including NBC and ABC held a collective 10:1 ratio of white characters to black characters, a much smaller margin of which had recurring African American characters. At that, the representation of African American women was much slimmer. The use of these token characters often portrayed African American people to stand in sidekick positions to their white counterparts. Research completed on token ethnic characters into the new millennium has found that the representation of males has grown in numbers, but has not improved in negative portrayal. Statistics on token ethnic characters still suggest toxic masculinity in African American males; threateningly powerful stereotypes of African American women; hyper-sexuality of African American and Asian women; and effeminate characteristics in Asian men and men of other racial minorities.

In the media

Just like television, tokenism in the media has changed over time to coincide with real-life events. During the years of 1946–87, The New Yorker was analyzed to determine how often and in what situations black people were being portrayed in the magazine's cartoon section. Over the 42 years of research, there was only one U.S. black main character in a cartoon where race was not the main theme, race was actually completely irrelevant. All cartoons from the earliest times depicted black people in the U.S. in stereotypical roles. In the late 1960s and early 1970s, cartoons were mostly racially themed, and depicted black people in "token" roles where they are only there to create a sense of inclusion.

Tokenism appears in advertising as well as other subdivisions of major media. Tokenism is interpreted as reinforcing subtle representations of minorities in commercials. Studies have shown that, among other racial minorities, Asian Americans are targeted by advertising companies to fulfill casting diversity, but are the most likely ethnic minority to be placed in the backgrounds of advertisements.

Black characters being the first characters to die was first identified in Hollywood horror movies of the 1930s, notes writer Renee Cozier. The Oscars ceremonies have received criticism over a lack of representation of people of color, as critics have pointed towards a lack of minorities nominated for awards, particularly in 2015 and 2016, when not a single actor of color was nominated. Around this time, minorities accounted for 12.9% of lead roles in 163 films surveyed in 2014, according to the 2016 Hollywood Diversity Report.

Film examples

Since the release of the original three Star Wars films and the later three prequels, there has been much discussion, on Twitter and Reddit especially, of this use of tokenism. The character of Lando Calrissian (portrayed by Billy Dee Williams) and Mace Windu (portrayed by Samuel L. Jackson) have been cited as two human characters of a racial minority that appear on screen. Lando was one of the first developed black characters in a science-fiction film at the time. Loyola Marymount University Professor of African American Studies, Adilifu Nama, has stated that this character is "a form of tokenism that placed one of the most optimistic faces on racial inclusion in a genre that had historically excluded Black representation."

When the first film of the newest installment of the franchise, The Force Awakens, was released in 2015, the conversation shifted. Where in the past two trilogies the main three characters were two white men and a white woman, in the new trilogy the main trio consists of a black man (John Boyega), a Hispanic man (Oscar Isaac), and a white woman (Daisy Ridley).

Directed by Ryan Coogler, the film Black Panther portrays the heroes of the fictional African kingdom of Wakanda as godlike. They possess otherworldly sophistication by virtue of their blackness, in contrast to longstanding tendencies in mainstream film toward tokenism, stereotyping, and victimhood in depictions of people of African descent. The superhero the Black Panther, a.k.a. King T’Challa, learns to stand in solidarity with the oppressed, even those in whose oppression he has been unwittingly complicit, such as the children of the African diaspora. As a result, the film can function as catalyst for reflection on the part of viewers in terms of how they might perceive more clearly the complexity, variety, and ambiguity represented by blackness, whether others’ or their own, and how they, too, might identify with the Other.

In G.B.F., directed by Darren Stein, the film tells the journey of two closeted gay teens, Tanner and Brent, on their quest to popularity in high school. The film explores the theme of tokenism through demonstrating the desire of a homosexual male best friend by typically heterosexual women. The three most popular girls in school: Fawcett Brooks, Caprice Winters, and 'Shley Osgood believe that the key to winning the prom queen title is through acquiring a gay best friend. In media, gay best friends are displayed as sassy, effeminate, fashionable, and flamboyant, making them act as a stock character accessory to the main character. While Tanner and Brent plan to become popular through exposing their sexuality, the girls are disappointed to find out that Tanner contradicts the stereotypical gay men they have seen in television. The film shows how harmful it can be to associate gay stereotypes with gay characters.

Film critic Armond White cited the Sight and Sound Greatest Films of All Time poll in 2022 as an example of tokenism. He wrote that the poll had become "a referendum on political correctness" which "prefers feminist, black, queer politics—not cinephilia.

Christianity and domestic violence

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Christianity_and_domestic_violence

Christianity and domestic violence deals with the debate in Christian communities about the recognition and response to domestic violence, which is complicated by a culture of silence and acceptance among abuse victims. There are some Bible verses that abusers use to justify discipline of their wives.

Abuse within marriage

Main article: Gender roles in Christianity

Christian groups and authorities generally condemn domestic violence as inconsistent with the general Christian duty to love others and to the scriptural relationship between husband and wife.

Relationship between husband and wife

According to the U.S. Conference of Catholic Bishops, "Men who abuse often use Ephesians 5:22, taken out of context, to justify their behavior, but the passage (v. 21-33) refers to the mutual submission of husband and wife out of love for Christ. Husbands should love their wives as they love their own body, as Christ loves the Church."

Some Christian theologians, such as the Rev. Marie Fortune and Mary Pellauer, have raised the question of a close connection between patriarchal Christianity and domestic violence and abuse. Steven Tracy, author of "Patriarchy and Domestic Violence" writes: "While patriarchy may not be the overarching cause of all abuse, it is an enormously significant factor, because in traditional patriarchy males have a disproportionate share of power... So while patriarchy is not the sole explanation for violence against women, we would expect that male headship would be distorted by insecure, unhealthy men to justify their domination and abuse of women."

Few empirical studies have examined the relationship between religion and domestic violence. According to Dutton, no single-factor explanation for wife assault was sufficient to explain the available data. A study by Dutton and Browning in the same year found that misogyny is correlated with only a minority of abusive male partners. Campbell's study in 1992 found no evidence of greater violence towards women in more patriarchal cultures. Pearson's study in 1997 observed, "Studies of male batterers have failed to confirm that these men are more conservative or sexist about marriage than nonviolent men".

Responding to Domestic Abuse, a report issued by the Church of England in 2006, suggests that patriarchy should be replaced rather than reinterpreted: "Following the pattern of Christ means that patterns of domination and submission are being transformed in the mutuality of love, faithful care and sharing of burdens. 'Be subject to one another out of reverence for Christ'(Ephesians 5.21). Although strong patriarchal tendencies have persisted in Christianity, the example of Christ carries the seeds of their displacement by a more symmetrical and respectful model of male–female relations."

Bible

Main article: The Bible and violence

Bible verses are often used to justify domestic abuse, such as those that refer to male superiority and female submission. Others counter that the use of violence is a misinterpreted view of the male role. For instance, Eve (Genesis 2-3), is seen by some Christians to be disobedient to patriarchal God and man, and to many a generalized symbol of womanhood that must be submissive and subject to discipline, while others disagree with this interpretation.

Christian domestic discipline

A subculture known as Christian domestic discipline (CDD) promotes spanking of wives by their husbands as a form of punishment. While its advocates rely on Biblical interpretations to support the practice, advocates for victims of domestic violence describe CDD as a form of abuse and controlling behavior. Others describe the practice as a simple sexual fetish and an outlet for sadomasochistic desires. Christian conservative radio host Bryan Fischer said to the Huffington Post that it was a "horrifying trend – bizarre, twisted, unbiblical and un-Christian".

Responses to abuse

There are a variety of responses by Christian leaders to how victims should handle abuse:

• Marjorie Proctor-Smith in Violence against women and children: a Christian Theological Sourcebook states that domestic physical, psychological or sexual violence is a sin. It victimizes family members dependent on a man and violates trust needed for healthy, equitable and cooperative relationships. She finds that domestic violence is a symptom of sexism, a social sin.
• The U.S. Conference of Catholic Bishops said in 2002, "As pastors of the Catholic Church in the United States, we state as clearly and strongly as we can that violence against women, inside or outside the home, is never justified."
• The Church of England's report, Responding to Domestic Abuse advises that Christian pastors and counselors should not advise victims to make forgiving the perpetrator the top priority "when the welfare and safety of the person being abused are at stake."
• One mid-1980s survey of 5,700 pastors found that 26 percent of pastors ordinarily would tell a woman being abused that she should continue to submit and to "trust that God would honor her action by either stopping the abuse or giving her the strength to endure it" and that 71 percent of pastors would never advise a battered wife to leave her husband or separate because of abuse.

A contributing factor to the disparity of responses to abuse is lack of training; many Christian seminaries had not educated future church leaders about how to manage violence against women. Once pastors began receiving training, and announced their participation in domestic violence educational programs, they immediately began receiving visits from women church members who had been subject to violence. The first Theological Education and Domestic Violence Conference, sponsored by the Center for the Prevention of Sexual and Domestic Violence, was held in 1985 to identify topics that should be covered in seminaries. First, church leaders will encounter sexual and domestic violence and they need to know what community resources are available. Secondly, they need to focus on ending the violence, rather than on keeping families together.

The American religious news-magazine Christianity Today has published articles lamenting U.S. churches for possibly making domestic abuse worse "not in incidence, but in response" due to inadequate understandings. In December 2017, academic W. Bradford Wilcox wrote for the publication, "Domestic violence is still present in church-going homes... some local churches, clergy, and counselors fail to address abuse head-on for fear of breaking up a marriage." He also argued, "Others steer clear of addressing the topic from the pulpit or in adult education for fear of broaching an uncomfortable subject. This silence around domestic violence has to end."

Research into incidents of domestic violence

In the 1970s, when multiple programs were initiated to train church leaders about domestic violence, the response "But no one ever comes to me with this problem" often came up to frustrate efforts. Church leaders frequently believed that if no one reached out for assistance within their congregations that there was no problem for them to deal with; however, women often withheld discussing their problems over concern that it would not be handled appropriately. When women increasingly became pastors over the 20th century, many of them found that much of their time became devoted to handling domestic abuse and other forms of violence against women; "crisis intervention" became a vital topic for them.

In terms of regions of the United States, research has speculated that local societies with a general culture of violence and low socio-economic status, which may also be at least nominally religious, are more likely to produce abusive men. The role of working-class Scots-Irish American culture in areas such as Appalachia is cited.

Differing viewpoints between husband and wife may be an aggravating factor in terms of abuse, particularly when women hold beliefs in contrast to more ideologically hardline men.

Significant figures

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Significant_figures

Significant figures, also referred to as significant digits or sig figs, are specific digits within a number written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcome of a measurement (such as length, pressure, volume, or mass), if the number of digits exceeds what the measurement instrument can resolve, only the number of digits within the resolution's capability are dependable and therefore considered significant.

For instance, if a length measurement yields 114.8 mm, using a ruler with the smallest interval between marks at 1 mm, the first three digits (1, 1, and 4, representing 114 mm) are certain and constitute significant figures. Further, digits that are uncertain yet meaningful are also included in the significant figures. In this example, the last digit (8, contributing 0.8 mm) is likewise considered significant despite its uncertainty. Therefore, this measurement contains four significant figures.

Another example involves a volume measurement of 2.98 L with an uncertainty of ± 0.05 L. The actual volume falls between 2.93 L and 3.03 L. Even if certain digits are not completely known, they are still significant if they are meaningful, as they indicate the actual volume within an acceptable range of uncertainty. In this case, the actual volume might be 2.94 L or possibly 3.02 L, so all three digits are considered significant. Thus, there are three significant figures in this example.

The following types of digits are not considered significant:

• Leading zeros. For instance, 013 kg has two significant figures—1 and 3—while the leading zero is insignificant since it does not impact the mass indication; 013 kg is equivalent to 13 kg, rendering the zero unnecessary. Similarly, in the case of 0.056 m, there are two insignificant leading zeros since 0.056 m is the same as 56 mm, thus the leading zeros do not contribute to the length indication.
• Trailing zeros when they serve as placeholders. In the measurement 1500 m, when the measurement resolution is 100 m, the trailing zeros are insignificant as they simply stand for the tens and ones places. In this instance, 1500 m indicates the length is approximately 1500 m rather than an exact value of 1500 m.
• Spurious digits that arise from calculations resulting in a higher precision than the original data or a measurement reported with greater precision than the instrument's resolution.

A zero after a decimal (e.g., 1.0) is significant, and care should be used when appending such a decimal of zero. Thus, in the case of 1.0, there are two significant figures, whereas 1 (without a decimal) has one significant figure.

Among a number's significant digits, the most significant digit is the one with the greatest exponent value (the leftmost significant digit/figure), while the least significant digit is the one with the lowest exponent value (the rightmost significant digit/figure). For example, in the number "123" the "1" is the most significant digit, representing hundreds (10²), while the "3" is the least significant digit, representing ones (10⁰).

To avoid conveying a misleading level of precision, numbers are often rounded. For instance, it would create false precision to present a measurement as 12.34525 kg when the measuring instrument only provides accuracy to the nearest gram (0.001 kg). In this case, the significant figures are the first five digits (1, 2, 3, 4, and 5) from the leftmost digit, and the number should be rounded to these significant figures, resulting in 12.345 kg as the accurate value. The rounding error (in this example, 0.00025 kg = 0.25 g) approximates the numerical resolution or precision. Numbers can also be rounded for simplicity, not necessarily to indicate measurement precision, such as for the sake of expediency in news broadcasts.

Significance arithmetic encompasses a set of approximate rules for preserving significance through calculations. More advanced scientific rules are known as the propagation of uncertainty.

Radix 10 (base-10, decimal numbers) is assumed in the following. (See unit in the last place for extending these concepts to other bases.)

Identifying significant figures

Rules to identify significant figures in a number

Digits in light blue are significant figures; those in black are not.

Identifying the significant figures in a number requires knowing which digits are meaningful, which requires knowing the resolution with which the number is measured, obtained, or processed. For example, if the measurable smallest mass is 0.001 g, then in a measurement given as 0.00234 g the "4" is not useful and should be discarded, while the "3" is useful and should often be retained.

• Non-zero digits within the given measurement or reporting resolution are significant.
  • 91 has two significant figures (9 and 1) if they are measurement-allowed digits.
  • 123.45 has five significant digits (1, 2, 3, 4 and 5) if they are within the measurement resolution. If the resolution is, say, 0.1, then the 5 shows that the true value to 4 sig figs is equally likely to be 123.4 or 123.5.
• Zeros between two significant non-zero digits are significant (significant trapped zeros).
  • 101.12003 consists of eight significant figures if the resolution is to 0.00001.
  • 125.340006 has seven significant figures if the resolution is to 0.0001: 1, 2, 5, 3, 4, 0, and 0.
• Zeros to the left of the first non-zero digit (leading zeros) are not significant.
  • If a length measurement gives 0.052 km, then 0.052 km = 52 m so 5 and 2 are only significant; the leading zeros appear or disappear, depending on which unit is used, so they are not necessary to indicate the measurement scale.
  • 0.00034 has 2 significant figures (3 and 4) if the resolution is 0.00001.
• Zeros to the right of the last non-zero digit (trailing zeros) in a number with the decimal point are significant if they are within the measurement or reporting resolution.
  • 1.200 has four significant figures (1, 2, 0, and 0) if they are allowed by the measurement resolution.
  • 0.0980 has three significant digits (9, 8, and the last zero) if they are within the measurement resolution.
  • 120.000 consists of six significant figures (1, 2, and the four subsequent zeroes) if, as before, they are within the measurement resolution.
• Trailing zeros in an integer may or may not be significant, depending on the measurement or reporting resolution.
  • 45,600 has 3, 4 or 5 significant figures depending on how the last zeros are used. For example, if the length of a road is reported as 45600 m without information about the reporting or measurement resolution, then it is not clear if the road length is precisely measured as 45600 m or if it is a rough estimate. If it is the rough estimation, then only the first three non-zero digits are significant since the trailing zeros are neither reliable nor necessary; 45600 m can be expressed as 45.6 km or as 4.56 × 10⁴ m in scientific notation, and neither expression requires the trailing zeros.
• An exact number has an infinite number of significant figures.
  • If the number of apples in a bag is 4 (exact number), then this number is 4.0000... (with infinite trailing zeros to the right of the decimal point). As a result, 4 does not impact the number of significant figures or digits in the result of calculations with it.
• A mathematical or physical constant has significant figures to its known digits.
  • π is a specific real number with several equivalent definitions. All of the digits in its exact decimal expansion 3.14159265358979323... are significant. Although many properties of these digits are known — for example, they do not repeat, because π is irrational — not all of the digits are known. As of March 2024, more than 102 trillion digits have been calculated. A 102 trillion-digit approximation has 102 trillion significant digits. In practical applications, far fewer digits are used. The everyday approximation 3.14 has three significant figures and 7 correct binary digits. The approximation 22/7 has the same three correct decimal digits but has 10 correct binary digits. Most calculators and computer programs can handle the 16-digit expansion 3.141592653589793, which is sufficient for interplanetary navigation calculations.
  • The Planck constant is ${\displaystyle h=6.62607015\times {10}^{-34}\mathrm{J}\cdot \mathrm{s}}$ and is defined as an exact value so that it is more properly defined as ${\displaystyle h=6.62607015(0)\times {10}^{-34}\mathrm{J}\cdot \mathrm{s}}$.

Ways to denote significant figures in an integer with trailing zeros

The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if the number 1300 is precise to the nearest unit (just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundreds due to rounding or uncertainty. Many conventions exist to address this issue. However, these are not universally used and would only be effective if the reader is familiar with the convention:

• An overline, sometimes also called an overbar, or less accurately, a vinculum, may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten).

• Less often, using a closely related convention, the last significant figure of a number may be underlined; for example, "1300" has two significant figures.

• A decimal point may be placed after the number; for example "1300." indicates specifically that trailing zeros are meant to be significant.

As the conventions above are not in general use, the following more widely recognized options are available for indicating the significance of number with trailing zeros:

• Eliminate ambiguous or non-significant zeros by changing the unit prefix in a number with a unit of measurement. For example, the precision of measurement specified as 1300 g is ambiguous, while if stated as 1.30 kg it is not. Likewise 0.0123 L can be rewritten as 12.3 mL.

• Eliminate ambiguous or non-significant zeros by using Scientific Notation: For example, 1300 with three significant figures becomes 1.30×10³. Likewise 0.0123 can be rewritten as 1.23×10⁻². The part of the representation that contains the significant figures (1.30 or 1.23) is known as the significand or mantissa. The digits in the base and exponent (10³ or 10⁻²) are considered exact numbers so for these digits, significant figures are irrelevant.

• Explicitly state the number of significant figures (the abbreviation s.f. is sometimes used): For example "20 000 to 2 s.f." or "20 000 (2 sf)".

• State the expected variability (precision) explicitly with a plus–minus sign, as in 20 000 ± 1%. This also allows specifying a range of precision in-between powers of ten.

Rounding to significant figures

Rounding to significant figures is a more general-purpose technique than rounding to n digits, since it handles numbers of different scales in a uniform way. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant figures (5 and 2). This reflects the fact that the significance of the error is the same in both cases, relative to the size of the quantity being measured.

To round a number to n significant figures:

1. If the n + 1 digit is greater than 5 or is 5 followed by other non-zero digits, add 1 to the n digit. For example, if we want to round 1.2459 to 3 significant figures, then this step results in 1.25.
2. If the n + 1 digit is 5 not followed by other digits or followed by only zeros, then rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant figures:
   • Round half away from zero rounds up to 1.3. This is the default rounding method implied in many disciplines^{[citation needed]} if the required rounding method is not specified.
   • Round half to even, which rounds to the nearest even number. With this method, 1.25 is rounded down to 1.2. If this method applies to 1.35, then it is rounded up to 1.4. This is the method preferred by many scientific disciplines, because, for example, it avoids skewing the average value of a long list of values upwards.
3. For an integer in rounding, replace the digits after the n digit with zeros. For example, if 1254 is rounded to 2 significant figures, then 5 and 4 are replaced to 0 so that it will be 1300. For a number with the decimal point in rounding, remove the digits after the n digit. For example, if 14.895 is rounded to 3 significant figures, then the digits after 8 are removed so that it will be 14.9.

In financial calculations, a number is often rounded to a given number of places. For example, to two places after the decimal separator for many world currencies. This is done because greater precision is immaterial, and usually it is not possible to settle a debt of less than the smallest currency unit.

In UK personal tax returns, income is rounded down to the nearest pound, whilst tax paid is calculated to the nearest penny.

As an illustration, the decimal quantity 12.345 can be expressed with various numbers of significant figures or decimal places. If insufficient precision is available then the number is rounded in some manner to fit the available precision. The following table shows the results for various total precision at two rounding ways (N/A stands for Not Applicable).

Precision	Rounded to significant figures	Rounded to decimal places
6	12.3450	12.345000
5	12.345	12.34500
4	12.34 or 12.35	12.3450
3	12.3	12.345
2	12	12.34 or 12.35
1	10	12.3
0	—	12

Another example for 0.012345. (Remember that the leading zeros are not significant.)

Precision	Rounded to significant figures	Rounded to decimal places
7	0.01234500	0.0123450
6	0.0123450	0.012345
5	0.012345	0.01234 or 0.01235
4	0.01234 or 0.01235	0.0123
3	0.0123	0.012
2	0.012	0.01
1	0.01	0.0
0	—	0

The representation of a non-zero number x to a precision of p significant digits has a numerical value that is given by the formula:

${\displaystyle {10}^n\cdot \mathrm{round}\left(\frac{x}{{10}^n}\right)}$

where

${\displaystyle n=\lfloor {\log }_{10}(|x|)\rfloor +1-p}$

which may need to be written with a specific marking as detailed above to specify the number of significant trailing zeros.

Writing uncertainty and implied uncertainty

Significant figures in writing uncertainty

It is recommended for a measurement result to include the measurement uncertainty such as ${\displaystyle x_{best}\pm {\sigma }_x}$, where x_best and σ_x are the best estimate and uncertainty in the measurement respectively. x_best can be the average of measured values and σ_x can be the standard deviation or a multiple of the measurement deviation. The rules to write ${\displaystyle x_{best}\pm {\sigma }_x}$are:

• σ_x should usually be quoted to only one or two significant figures, as more precision is unlikely to be reliable or meaningful:
  • 1.79 ± 0.06 (correct), 1.79 ± 0.96 (correct), 1.79 ± 1.96 (incorrect).
• The digit positions of the last significant figures in x_best and σ_x are the same, otherwise the consistency is lost. For example, "1.79 ± 0.067" is incorrect, as it does not make sense to have more accurate uncertainty than the best estimate.
  • 1.79 ± 0.06 (correct), 1.79 ± 0.96 (correct), 1.79 ± 0.067 (incorrect).

Implied uncertainty

Persnickety sign implies 14.6 is speed violation

Uncertainty may be implied by the last significant figure if it is not explicitly expressed. The implied uncertainty is ± the half of the minimum scale at the last significant figure position. For example, if the mass of an object is reported as 3.78 kg without mentioning uncertainty, then ± 0.005 kg measurement uncertainty may be implied. If the mass of an object is estimated as 3.78 ± 0.07 kg, so the actual mass is probably somewhere in the range 3.71 to 3.85 kg, and it is desired to report it with a single number, then 3.8 kg is the best number to report since its implied uncertainty ± 0.05 kg gives a mass range of 3.75 to 3.85 kg, which is close to the measurement range. If the uncertainty is a bit larger, i.e. 3.78 ± 0.09 kg, then 3.8 kg is still the best single number to quote, since if "4 kg" was reported then a lot of information would be lost.

If there is a need to write the implied uncertainty of a number, then it can be written as ${\displaystyle x\pm {\sigma }_x}$with stating it as the implied uncertainty (to prevent readers from recognizing it as the measurement uncertainty), where x and σ_x are the number with an extra zero digit (to follow the rules to write uncertainty above) and the implied uncertainty of it respectively. For example, 6 kg with the implied uncertainty ± 0.5 kg can be stated as 6.0 ± 0.5 kg.

Arithmetic

As there are rules to determine the significant figures in directly measured quantities, there are also guidelines (not rules) to determine the significant figures in quantities calculated from these measured quantities.

Significant figures in measured quantities are most important in the determination of significant figures in calculated quantities with them. A mathematical or physical constant (e.g., π in the formula for the area of a circle with radius r as πr²) has no effect on the determination of the significant figures in the result of a calculation with it if its known digits are equal to or more than the significant figures in the measured quantities used in the calculation. An exact number such as ½ in the formula for the kinetic energy of a mass m with velocity v as ½mv² has no bearing on the significant figures in the calculated kinetic energy since its number of significant figures is infinite (0.500000...).

The guidelines described below are intended to avoid a calculation result more precise than the measured quantities, but it does not ensure the resulted implied uncertainty close enough to the measured uncertainties. This problem can be seen in unit conversion. If the guidelines give the implied uncertainty too far from the measured ones, then it may be needed to decide significant digits that give comparable uncertainty.

Multiplication and division

For quantities created from measured quantities via multiplication and division, the calculated result should have as many significant figures as the least number of significant figures among the measured quantities used in the calculation. For example,

• 1.234 × 2 = 2.468 ≈ 2
• 1.234 × 2.0 = 2.468 ≈ 2.5
• 0.01234 × 2 = 0.02468 ≈ 0.02
• 0.012345678 / 0.00234 = 5.2759 ≈ 5.28

with one, two, and one significant figures respectively. (2 here is assumed not an exact number.) For the first example, the first multiplication factor has four significant figures and the second has one significant figure. The factor with the fewest or least significant figures is the second one with only one, so the final calculated result should also have one significant figure.

Exception

For unit conversion, the implied uncertainty of the result can be unsatisfactorily higher than that in the previous unit if this rounding guideline is followed; For example, 8 inch has the implied uncertainty of ± 0.5 inch = ± 1.27 cm. If it is converted to the centimeter scale and the rounding guideline for multiplication and division is followed, then 20.32 cm ≈ 20 cm with the implied uncertainty of ± 5 cm. If this implied uncertainty is considered as too overestimated, then more proper significant digits in the unit conversion result may be 20.32 cm ≈ 20. cm with the implied uncertainty of ± 0.5 cm.

Another exception of applying the above rounding guideline is to multiply a number by an integer, such as 1.234 × 9. If the above guideline is followed, then the result is rounded as 1.234 × 9.000.... = 11.106 ≈ 11.11. However, this multiplication is essentially adding 1.234 to itself 9 times such as 1.234 + 1.234 + … + 1.234 so the rounding guideline for addition and subtraction described below is more proper rounding approach. As a result, the final answer is 1.234 + 1.234 + … + 1.234 = 11.106 = 11.106 (one significant digit increase).

Addition and subtraction of significant figures

For quantities created from measured quantities via addition and subtraction, the last significant figure position (e.g., hundreds, tens, ones, tenths, hundredths, and so forth) in the calculated result should be the same as the leftmost or largest digit position among the last significant figures of the measured quantities in the calculation. For example,

• 1.234 + 2 = 3.234 ≈ 3
• 1.234 + 2.0 = 3.234 ≈ 3.2
• 0.01234 + 2 = 2.01234 ≈ 2
• 12000 + 77 = 12077 ≈ 12000

with the last significant figures in the ones place, tenths place, ones place, and thousands place respectively. (2 here is assumed not an exact number.) For the first example, the first term has its last significant figure in the thousandths place and the second term has its last significant figure in the ones place. The leftmost or largest digit position among the last significant figures of these terms is the ones place, so the calculated result should also have its last significant figure in the ones place.

The rule to calculate significant figures for multiplication and division are not the same as the rule for addition and subtraction. For multiplication and division, only the total number of significant figures in each of the factors in the calculation matters; the digit position of the last significant figure in each factor is irrelevant. For addition and subtraction, only the digit position of the last significant figure in each of the terms in the calculation matters; the total number of significant figures in each term is irrelevant. However, greater accuracy will often be obtained if some non-significant digits are maintained in intermediate results which are used in subsequent calculations.

Logarithm and antilogarithm

The base-10 logarithm of a normalized number (i.e., a × 10^b with 1 ≤ a < 10 and b as an integer), is rounded such that its decimal part (called mantissa) has as many significant figures as the significant figures in the normalized number.

• log₁₀(3.000 × 10⁴) = log₁₀(10⁴) + log₁₀(3.000) = 4.000000... (exact number so infinite significant digits) + 0.4771212547... = 4.4771212547 ≈ 4.4771.

When taking the antilogarithm of a normalized number, the result is rounded to have as many significant figures as the significant figures in the decimal part of the number to be antiloged.

• 10^4.4771 = 29998.5318119... = 30000 = 3.000 × 10⁴.

Transcendental functions

If a transcendental function ${\displaystyle f(x)}$ (e.g., the exponential function, the logarithm, and the trigonometric functions) is differentiable at its domain element 'x', then its number of significant figures (denoted as "significant figures of ${\displaystyle f(x)}$") is approximately related with the number of significant figures in x (denoted as "significant figures of x") by the formula

${\displaystyle (\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{f}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{f}(\mathrm{x}))\approx (\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{f}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{x})-{\log }_{10}\left(\left|\frac{df(x)}{dx}\frac{x}{f(x)}\right|\right)}$,

where ${\displaystyle \left|\frac{df(x)}{dx}\frac{x}{f(x)}\right|}$ is the condition number.

Round only on the final calculation result

When performing multiple stage calculations, do not round intermediate stage calculation results; keep as many digits as is practical (at least one more digit than the rounding rule allows per stage) until the end of all the calculations to avoid cumulative rounding errors while tracking or recording the significant figures in each intermediate result. Then, round the final result, for example, to the fewest number of significant figures (for multiplication or division) or leftmost last significant digit position (for addition or subtraction) among the inputs in the final calculation.

• (2.3494 + 1.345) × 1.2 = 3.6944 × 1.2 = 4.43328 ≈ 4.4.
• (2.3494 × 1.345) + 1.2 = 3.159943 + 1.2 = 4.359943 ≈ 4.4.

Estimating an extra digit

When using a ruler, initially use the smallest mark as the first estimated digit. For example, if a ruler's smallest mark is 0.1 cm, and 4.5 cm is read, then it is 4.5 (±0.1 cm) or 4.4 cm to 4.6 cm as to the smallest mark interval. However, in practice a measurement can usually be estimated by eye to closer than the interval between the ruler's smallest mark, e.g. in the above case it might be estimated as between 4.51 cm and 4.53 cm.

It is also possible that the overall length of a ruler may not be accurate to the degree of the smallest mark, and the marks may be imperfectly spaced within each unit. However assuming a normal good quality ruler, it should be possible to estimate tenths between the nearest two marks to achieve an extra decimal place of accuracy. Failing to do this adds the error in reading the ruler to any error in the calibration of the ruler.

Estimation in statistic

Main article: Estimation

When estimating the proportion of individuals carrying some particular characteristic in a population, from a random sample of that population, the number of significant figures should not exceed the maximum precision allowed by that sample size.

Relationship to accuracy and precision in measurement

Main article: Accuracy and precision

Traditionally, in various technical fields, "accuracy" refers to the closeness of a given measurement to its true value; "precision" refers to the stability of that measurement when repeated many times. Thus, it is possible to be "precisely wrong". Hoping to reflect the way in which the term "accuracy" is actually used in the scientific community, there is a recent standard, ISO 5725, which keeps the same definition of precision but defines the term "trueness" as the closeness of a given measurement to its true value and uses the term "accuracy" as the combination of trueness and precision. (See the accuracy and precision article for a full discussion.) In either case, the number of significant figures roughly corresponds to precision, not to accuracy or the newer concept of trueness.

In computing

Main article: Floating-point arithmetic

Computer representations of floating-point numbers use a form of rounding to significant figures (while usually not keeping track of how many), in general with binary numbers. The number of correct significant figures is closely related to the notion of relative error (which has the advantage of being a more accurate measure of precision, and is independent of the radix, also known as the base, of the number system used).

Electronic calculators supporting a dedicated significant figures display mode are relatively rare.

Among the calculators to support related features are the Commodore M55 Mathematician (1976) and the S61 Statistician (1976), which support two display modes, where DISP+n will give n significant digits in total, while DISP+.+n will give n decimal places.

The Texas Instruments TI-83 Plus (1999) and TI-84 Plus (2004) families of graphical calculators support a Sig-Fig Calculator mode in which the calculator will evaluate the count of significant digits of entered numbers and display it in square brackets behind the corresponding number. The results of calculations will be adjusted to only show the significant digits as well.

For the HP 20b/30b-based community-developed WP 34S (2011) and WP 31S (2014) calculators significant figures display modes SIG+n and SIG0+n (with zero padding) are available as a compile-time option. The SwissMicros DM42-based community-developed calculators WP 43C (2019) / C43 (2022) / C47 (2023) support a significant figures display mode as well.