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5,100 | In the 19th century, the internal development of geometry (pure mathematics) lead to define and study non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einste... | https://en.wikipedia.org/wiki?curid=18831 |
5,101 | A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon formula_5 In both cases, the equations of the theories had unexplained solutions, which led to conjecture the existence of a unknown ... | https://en.wikipedia.org/wiki?curid=18831 |
5,102 | The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that math... | https://en.wikipedia.org/wiki?curid=18831 |
5,103 | Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views. | https://en.wikipedia.org/wiki?curid=18831 |
5,104 | Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics. | https://en.wikipedia.org/wiki?curid=18831 |
5,105 | There is no general consensus about a definition of mathematics or its epistemological statusthat is, its place among other human activities. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art... | https://en.wikipedia.org/wiki?curid=18831 |
5,106 | Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from re... | https://en.wikipedia.org/wiki?curid=18831 |
5,107 | Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theoremassertions whose validity relies on a proof, that is, a purely-logical deduction. Others take the perspective that mathematics is an investigation of axiomatic set theory... | https://en.wikipedia.org/wiki?curid=18831 |
5,108 | Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of inference rules, without any use of empirical evidence and intuition. Rigorous reasoning is not specific to mathematics, but, in mathematics, the stan... | https://en.wikipedia.org/wiki?curid=18831 |
5,109 | The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhan... | https://en.wikipedia.org/wiki?curid=18831 |
5,110 | At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russel's paradox). This was solved by the inclusion of axioms with the apodictic inference rules o... | https://en.wikipedia.org/wiki?curid=18831 |
5,111 | Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof. | https://en.wikipedia.org/wiki?curid=18831 |
5,112 | Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of ... | https://en.wikipedia.org/wiki?curid=18831 |
5,113 | Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia. Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE. The oldest known mathematics tex... | https://en.wikipedia.org/wiki?curid=18831 |
5,114 | Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium. Formal instruction in pedagogy began with Jesuit schools in the 16th and 17th century. Most mathematical curriculum remained at a basic and practical level until the nineteenth century, when it began to... | https://en.wikipedia.org/wiki?curid=18831 |
5,115 | During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics. Some students studying math may develop an apprehension ... | https://en.wikipedia.org/wiki?curid=18831 |
5,116 | The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems ... | https://en.wikipedia.org/wiki?curid=18831 |
5,117 | Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzles. This aspect of mathematical activity is emphasized in recreational mathematics. | https://en.wikipedia.org/wiki?curid=18831 |
5,118 | Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to "elegance", which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in "A Mathematician's Apology" expressed the belief that the aesthetic considerations are, in ... | https://en.wikipedia.org/wiki?curid=18831 |
5,119 | Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are "created" (as in art) or "discovered" (as in science). The popularity of re... | https://en.wikipedia.org/wiki?curid=18831 |
5,120 | In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as intuitionism, which primarily identifies mathematics with certain creative processes in the mind. Intuitionism is in turn one flavor of a stance known as constructivism, which only considers a mathematical object... | https://en.wikipedia.org/wiki?curid=18831 |
5,121 | In the end, neither constructivism nor intuitionism displaced classical mathematics or achieved mainstream acceptance. However, these programs have motivated specific developments, such as intuitionistic logic and other foundational insights, which are appreciated in their own right. | https://en.wikipedia.org/wiki?curid=18831 |
5,122 | The most prestigious award in mathematics is the Fields Medal, established in 1936 and awarded every four years (except around World War II) to up to four individuals. It is considered the mathematical equivalent of the Nobel Prize. | https://en.wikipedia.org/wiki?curid=18831 |
5,123 | A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list has achieved great celebrity among mathematicians, and, , at least thirteen of the problems (depending how some are interpreted) have been solved. | https://en.wikipedia.org/wiki?curid=18831 |
5,124 | A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. To date, only one of these problems, the Poincaré conjecture, has ... | https://en.wikipedia.org/wiki?curid=18831 |
5,125 | Body mass index (BMI) is a value derived from the mass (weight) and height of a person. The BMI is defined as the body mass divided by the square of the body height, and is expressed in units of kg/m, resulting from mass in kilograms and height in metres. | https://en.wikipedia.org/wiki?curid=4788 |
5,126 | The BMI may be determined using a table or chart which displays BMI as a function of mass and height using contour lines or colours for different BMI categories, and which may use other units of measurement (converted to metric units for the calculation). | https://en.wikipedia.org/wiki?curid=4788 |
5,127 | The BMI is a convenient rule of thumb used to broadly categorize a person as "underweight", "normal weight", "overweight", or "obese" based on tissue mass (muscle, fat, and bone) and height. Major adult BMI classifications are underweight (under 18.5 kg/m), normal weight (18.5 to 24.9), overweight (25 to 29.9), and obe... | https://en.wikipedia.org/wiki?curid=4788 |
5,128 | BMIs under 20 and over 25 have been associated with higher all-cause mortality, with the risk increasing with distance from the 20–25 range. | https://en.wikipedia.org/wiki?curid=4788 |
5,129 | Adolphe Quetelet, a Belgian astronomer, mathematician, statistician, and sociologist, devised the basis of the BMI between 1830 and 1850 as he developed what he called "social physics". The modern term "body mass index" (BMI) for the ratio of human body weight to squared height was coined in a paper published in the Ju... | https://en.wikipedia.org/wiki?curid=4788 |
5,130 | The interest in an index that measures body fat came with observed increasing obesity in prosperous Western societies. Keys explicitly judged BMI as appropriate for "population" studies and inappropriate for individual evaluation. Nevertheless, due to its simplicity, it has come to be widely used for preliminary diagno... | https://en.wikipedia.org/wiki?curid=4788 |
5,131 | The BMI is expressed in kg/m, resulting from mass in kilograms and height in metres. If pounds and inches are used, a conversion factor of 703 (kg/m)/(lb/in) is applied. When the term BMI is used informally, the units are usually omitted. | https://en.wikipedia.org/wiki?curid=4788 |
5,132 | BMI provides a simple numeric measure of a person's "thickness" or "thinness", allowing health professionals to discuss weight problems more objectively with their patients. BMI was designed to be used as a simple means of classifying average sedentary (physically inactive) populations, with an average body composition... | https://en.wikipedia.org/wiki?curid=4788 |
5,133 | A common use of the BMI is to assess how far an individual's body weight departs from what is normal for a person's height. The weight excess or deficiency may, in part, be accounted for by body fat (adipose tissue) although other factors such as muscularity also affect BMI significantly (see discussion below and overw... | https://en.wikipedia.org/wiki?curid=4788 |
5,134 | The WHO regards an adult BMI of less than 18.5 as underweight and possibly indicative of malnutrition, an eating disorder, or other health problems, while a BMI of 25 or more is considered overweight and 30 or more is considered obese. In addition to the principle, international WHO BMI cut-off points (16, 17, 18.5, 25... | https://en.wikipedia.org/wiki?curid=4788 |
5,135 | BMI is used differently for children and youth. It is calculated in the same way as for adults but then compared to typical values for other children or youth of the same age. Instead of comparison against fixed thresholds for underweight and overweight, the BMI is compared against the percentiles for children of the s... | https://en.wikipedia.org/wiki?curid=4788 |
5,136 | A BMI that is less than the 5th percentile is considered underweight and above the 95th percentile is considered obese. Children with a BMI between the 85th and 95th percentile are considered to be overweight. | https://en.wikipedia.org/wiki?curid=4788 |
5,137 | Studies in Britain from 2013 have indicated that females between the ages 12 and 16 had a higher BMI than males of the same age by 1.0 kg/m on average. | https://en.wikipedia.org/wiki?curid=4788 |
5,138 | These recommended distinctions along the linear scale may vary from time to time and country to country, making global, longitudinal surveys problematic. People from different populations and descent have different associations between BMI, percentage of body fat, and health risks, with a higher risk of type 2 diabetes... | https://en.wikipedia.org/wiki?curid=4788 |
5,139 | A 2000 study from the Japan Society for the Study of Obesity (JASSO) presents the following table of BMI categories: | https://en.wikipedia.org/wiki?curid=4788 |
5,140 | In Singapore, the BMI cut-off figures were revised in 2005 by the Health Promotion Board (HPB), motivated by studies showing that many Asian populations, including Singaporeans, have a higher proportion of body fat and increased risk for cardiovascular diseases and diabetes mellitus, compared with general BMI recommend... | https://en.wikipedia.org/wiki?curid=4788 |
5,141 | In the UK, NICE guidance recommends prevention of type 2 diabetes should start at a BMI of 30 in White and 27.5 in Black African, African-Caribbean, South Asian, and Chinese populations. | https://en.wikipedia.org/wiki?curid=4788 |
5,142 | New research based on a large sample of almost 1.5 million people in England found that some ethnic groups would benefit from prevention at or above a BMI of (rounded): | https://en.wikipedia.org/wiki?curid=4788 |
5,143 | In 1998, the U.S. National Institutes of Health brought U.S. definitions in line with World Health Organization guidelines, lowering the normal/overweight cut-off from a BMI of 27.8 (men) and 27.3 (women) to a BMI of 25. This had the effect of redefining approximately 25 million Americans, previously "healthy", to "ove... | https://en.wikipedia.org/wiki?curid=4788 |
5,144 | This can partially explain the increase in the "overweight" diagnosis in the past 20 years, and the increase in sales of weight loss products during the same time. WHO also recommends lowering the normal/overweight threshold for southeast Asian body types to around BMI 23, and expects further revisions to emerge from c... | https://en.wikipedia.org/wiki?curid=4788 |
5,145 | A survey in 2007 showed 63% of Americans were then overweight or obese, with 26% in the obese category (a BMI of 30 or more). By 2014, 37.7% of adults in the United States were obese, 35.0% of men and 40.4% of women; class 3 obesity (BMI over 40) values were 7.7% for men and 9.9% for women. The U.S. National Health and... | https://en.wikipedia.org/wiki?curid=4788 |
5,146 | The BMI ranges are based on the relationship between body weight and disease and death. Overweight and obese individuals are at an increased risk for the following diseases: | https://en.wikipedia.org/wiki?curid=4788 |
5,147 | Among people who have never smoked, overweight/obesity is associated with 51% increase in mortality compared with people who have always been a normal weight. | https://en.wikipedia.org/wiki?curid=4788 |
5,148 | The BMI is generally used as a means of correlation between groups related by general mass and can serve as a vague means of estimating adiposity. The duality of the BMI is that, while it is easy to use as a general calculation, it is limited as to how accurate and pertinent the data obtained from it can be. Generally,... | https://en.wikipedia.org/wiki?curid=4788 |
5,149 | This general correlation is particularly useful for consensus data regarding obesity or various other conditions because it can be used to build a semi-accurate representation from which a solution can be stipulated, or the RDA for a group can be calculated. Similarly, this is becoming more and more pertinent to the gr... | https://en.wikipedia.org/wiki?curid=4788 |
5,150 | Cross-sectional studies indicated that sedentary people can decrease BMI by becoming more physically active. Smaller effects are seen in prospective cohort studies which lend to support active mobility as a means to prevent a further increase in BMI. | https://en.wikipedia.org/wiki?curid=4788 |
5,151 | In France, Italy, and Spain, legislation has been introduced banning the usage of fashion show models having a BMI below 18. In Israel, a BMI below 18.5 is banned. This is done to fight anorexia among models and people interested in fashion. | https://en.wikipedia.org/wiki?curid=4788 |
5,152 | A study published by "Journal of the American Medical Association" ("JAMA") in 2005 showed that "overweight" people had a death rate similar to "normal" weight people as defined by BMI, while "underweight" and "obese" people had a higher death rate. | https://en.wikipedia.org/wiki?curid=4788 |
5,153 | A study published by "The Lancet" in 2009 involving 900,000 adults showed that "overweight" and "underweight" people both had a mortality rate higher than "normal" weight people as defined by BMI. The optimal BMI was found to be in the range of 22.5–25. The average BMI of athletes is 22.4 for women and 23.6 for men. | https://en.wikipedia.org/wiki?curid=4788 |
5,154 | High BMI is associated with type 2 diabetes only in people with high serum gamma-glutamyl transpeptidase. | https://en.wikipedia.org/wiki?curid=4788 |
5,155 | In an analysis of 40 studies involving 250,000 people, patients with coronary artery disease with "normal" BMIs were at higher risk of death from cardiovascular disease than people whose BMIs put them in the "overweight" range (BMI 25–29.9). | https://en.wikipedia.org/wiki?curid=4788 |
5,156 | One study found that BMI had a good general correlation with body fat percentage, and noted that obesity has overtaken smoking as the world's number one cause of death. But it also notes that in the study 50% of men and 62% of women were obese according to body fat defined obesity, while only 21% of men and 31% of wome... | https://en.wikipedia.org/wiki?curid=4788 |
5,157 | A 2010 study that followed 11,000 subjects for up to eight years concluded that BMI is not a good measure for the risk of heart attack, stroke or death. A better measure was found to be the waist-to-height ratio. A 2011 study that followed 60,000 participants for up to 13 years found that waist–hip ratio was a better p... | https://en.wikipedia.org/wiki?curid=4788 |
5,158 | The exponent in the denominator of the formula for BMI is arbitrary. The BMI depends upon weight and the "square" of height. Since mass increases to the "third power" of linear dimensions, taller individuals with exactly the same body shape and relative composition have a larger BMI. BMI is proportional to the mass and... | https://en.wikipedia.org/wiki?curid=4788 |
5,159 | However, many taller people are not just "scaled up" short people but tend to have narrower frames in proportion to their height. Carl Lavie has written that "The B.M.I. tables are excellent for identifying obesity and body fat in large populations, but they are far less reliable for determining fatness in individuals.... | https://en.wikipedia.org/wiki?curid=4788 |
5,160 | For US adults, exponent estimates range from 1.92 to 1.96 for males and from 1.45 to 1.95 for females. | https://en.wikipedia.org/wiki?curid=4788 |
5,161 | The BMI overestimates roughly 10% for a large (or tall) frame and underestimates roughly 10% for a smaller frame (short stature). In other words, people with small frames would be carrying more fat than optimal, but their BMI indicates that they are "normal". Conversely, large framed (or tall) individuals may be quite ... | https://en.wikipedia.org/wiki?curid=4788 |
5,162 | For example, a height/weight chart may say the ideal weight (BMI 21.5) for a man is . But if that man has a slender build (small frame), he may be overweight at and should reduce by 10% to roughly (BMI 19.4). In the reverse, the man with a larger frame and more solid build should increase by 10%, to roughly (BMI 23.7).... | https://en.wikipedia.org/wiki?curid=4788 |
5,163 | Accurate frame size calculators use several measurements (wrist circumference, elbow width, neck circumference, and others) to determine what category an individual falls into for a given height. The BMI also fails to take into account loss of height through ageing. In this situation, BMI will increase without any corr... | https://en.wikipedia.org/wiki?curid=4788 |
5,164 | Assumptions about the distribution between muscle mass and fat mass are inexact. BMI generally overestimates adiposity on those with more lean body mass (e.g., athletes) and underestimates excess adiposity on those with less lean body mass. | https://en.wikipedia.org/wiki?curid=4788 |
5,165 | A study in June 2008 by Romero-Corral et al. examined 13,601 subjects from the United States' third National Health and Nutrition Examination Survey (NHANES III) and found that BMI-defined obesity (BMI ≥ 30) was present in 21% of men and 31% of women. Body fat-defined obesity was found in 50% of men and 62% of women. W... | https://en.wikipedia.org/wiki?curid=4788 |
5,166 | Body composition for athletes is often better calculated using measures of body fat, as determined by such techniques as skinfold measurements or underwater weighing and the limitations of manual measurement have also led to new, alternative methods to measure obesity, such as the body volume indicator. | https://en.wikipedia.org/wiki?curid=4788 |
5,167 | It is not clear where on the BMI scale the threshold for "overweight" and "obese" should be set. Because of this, the standards have varied over the past few decades. Between 1980 and 2000 the U.S. Dietary Guidelines have defined overweight at a variety of levels ranging from a BMI of 24.9 to 27.1. In 1985 the National... | https://en.wikipedia.org/wiki?curid=4788 |
5,168 | In 1998, an NIH report concluded that a BMI over 25 is overweight and a BMI over 30 is obese. In the 1990s the World Health Organization (WHO) decided that a BMI of 25 to 30 should be considered overweight and a BMI over 30 is obese, the standards the NIH set. This became the definitive guide for determining if someone... | https://en.wikipedia.org/wiki?curid=4788 |
5,169 | The current WHO and NIH ranges of "normal" weights are proved to be associated with decreased risks of some diseases such as diabetes type II; however using the same range of BMI for men and women is considered arbitrary and makes the definition of underweight quite unsuitable for men. | https://en.wikipedia.org/wiki?curid=4788 |
5,170 | One study found that the vast majority of people labelled 'overweight' and 'obese' according to current definitions do not in fact face any meaningful increased risk for early death. In a quantitative analysis of several studies, involving more than 600,000 men and women, the lowest mortality rates were found for peopl... | https://en.wikipedia.org/wiki?curid=4788 |
5,171 | The corpulence index uses an exponent of 3 rather than 2. The corpulence index yields valid results even for very short and very tall people, which is a problem with BMI — for example, an ideal body weight for a person 152.4 cm tall (106 lb) will render BMI of 20.74 and CI of 13.6, while for a person 200 cm tall (277 l... | https://en.wikipedia.org/wiki?curid=4788 |
5,172 | A new formula for computing Body Mass Index that accounts for the distortions of the traditional BMI formula for shorter and taller individuals has been proposed by Nick Trefethen, Professor of numerical analysis at the University of Oxford: | https://en.wikipedia.org/wiki?curid=4788 |
5,173 | The scaling factor of 1.3 was determined to make the proposed new BMI formula align with the traditional BMI formula for adults of average height, while the exponent of 2.5 is a compromise between the exponent of 2 in the traditional formula for BMI and the exponent of 3 that would be expected for the scaling of weight... | https://en.wikipedia.org/wiki?curid=4788 |
5,174 | BMI Prime, a modification of the BMI system, is the ratio of actual BMI to upper limit optimal BMI (currently defined at 25 kg/m), i.e., the actual BMI expressed as a proportion of upper limit optimal. The ratio of actual body weight to body weight for upper limit optimal BMI (25 kg/m) is equal to BMI Prime. BMI Prime ... | https://en.wikipedia.org/wiki?curid=4788 |
5,175 | For instance, a person with BMI 34 kg/m has a BMI Prime of 34/25 = 1.36, and is 36% over their upper mass limit. In South East Asian and South Chinese populations (see § international variations), BMI Prime should be calculated using an upper limit BMI of 23 in the denominator instead of 25. BMI Prime allows easy compa... | https://en.wikipedia.org/wiki?curid=4788 |
5,176 | Waist circumference is a good indicator of visceral fat, which poses more health risks than fat elsewhere. According to the U.S. National Institutes of Health (NIH), waist circumference in excess of for men and for (non-pregnant) women is considered to imply a high risk for type 2 diabetes, dyslipidemia, hypertension, ... | https://en.wikipedia.org/wiki?curid=4788 |
5,177 | Waist-to-hip circumference ratio has also been used, but has been found to be no better than waist circumference alone, and more complicated to measure. | https://en.wikipedia.org/wiki?curid=4788 |
5,178 | A related indicator is waist circumference divided by height. The values indicating increased risk are: greater than 0.5 for people under 40 years of age, 0.5 to 0.6 for people aged 40–50, and greater than 0.6 for people over 50 years of age. | https://en.wikipedia.org/wiki?curid=4788 |
5,179 | The Surface-based Body Shape Index (SBSI) is far more rigorous and is based upon four key measurements: the body surface area (BSA), vertical trunk circumference (VTC), waist circumference (WC) and height (H). Data on 11,808 subjects from the National Health and Human Nutrition Examination Surveys (NHANES) 1999–2004, s... | https://en.wikipedia.org/wiki?curid=4788 |
5,180 | Within some medical contexts, such as familial amyloid polyneuropathy, serum albumin is factored in to produce a modified body mass index (mBMI). The mBMI can be obtained by multiplying the BMI by serum albumin, in grams per litre. | https://en.wikipedia.org/wiki?curid=4788 |
5,181 | In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wid... | https://en.wikipedia.org/wiki?curid=27590 |
5,182 | Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation. | https://en.wikipedia.org/wiki?curid=27590 |
5,183 | The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the varianc... | https://en.wikipedia.org/wiki?curid=27590 |
5,184 | The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and... | https://en.wikipedia.org/wiki?curid=27590 |
5,185 | In science, it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in the findings). By convention, only effects more than two standard errors away from a null expectation are considered "statistically significant",... | https://en.wikipedia.org/wiki?curid=27590 |
5,186 | When only a sample of data from a population is available, the term "standard deviation of the sample" or "sample standard deviation" can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the "population standard deviation" (the standard dev... | https://en.wikipedia.org/wiki?curid=27590 |
5,187 | Suppose that the entire population of interest is eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the square root of the average of the squared deviations of the values subtracted from their average value. The marks of a class of eight students (th... | https://en.wikipedia.org/wiki?curid=27590 |
5,188 | This formula is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some large parent population (for example, they were 8 students randomly and independently chosen from a class of 2 million), then one divides by instead of in the denom... | https://en.wikipedia.org/wiki?curid=27590 |
5,189 | If the population of interest is approximately normally distributed, the standard deviation provides information on the proportion of observations above or below certain values. For example, the average height for adult men in the United States is about 70 inches, with a standard deviation of around 3 inches. This mean... | https://en.wikipedia.org/wiki?curid=27590 |
5,190 | The standard deviation of a probability distribution is the same as that of a random variable having that distribution. | https://en.wikipedia.org/wiki?curid=27590 |
5,191 | Not all random variables have a standard deviation. If the distribution has fat tails going out to infinity, the standard deviation might not exist, because the integral might not converge. The normal distribution has tails going out to infinity, but its mean and standard deviation do exist, because the tails diminish ... | https://en.wikipedia.org/wiki?curid=27590 |
5,192 | In the case where "X" takes random values from a finite data set "x", "x", …, "x", with each value having the same probability, the standard deviation is | https://en.wikipedia.org/wiki?curid=27590 |
5,193 | If, instead of having equal probabilities, the values have different probabilities, let "x" have probability "p", "x" have probability "p", …, "x" have probability "p". In this case, the standard deviation will be | https://en.wikipedia.org/wiki?curid=27590 |
5,194 | The standard deviation of a continuous real-valued random variable "X" with probability density function "p"("x") is | https://en.wikipedia.org/wiki?curid=27590 |
5,195 | and where the integrals are definite integrals taken for "x" ranging over the set of possible values of the random variable "X". | https://en.wikipedia.org/wiki?curid=27590 |
5,196 | In the case of a parametric family of distributions, the standard deviation can be expressed in terms of the parameters. For example, in the case of the log-normal distribution with parameters "μ" and "σ", the standard deviation is | https://en.wikipedia.org/wiki?curid=27590 |
5,197 | One can find the standard deviation of an entire population in cases (such as standardized testing) where every member of a population is sampled. In cases where that cannot be done, the standard deviation "σ" is estimated by examining a random sample taken from the population and computing a statistic of the sample, w... | https://en.wikipedia.org/wiki?curid=27590 |
5,198 | Unlike in the case of estimating the population mean, for which the sample mean is a simple estimator with many desirable properties (unbiased, efficient, maximum likelihood), there is no single estimator for the standard deviation with all these properties, and unbiased estimation of standard deviation is a very techn... | https://en.wikipedia.org/wiki?curid=27590 |
5,199 | The formula for the "population" standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by "s", is known as the "uncorrected sample... | https://en.wikipedia.org/wiki?curid=27590 |
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