Ronald Fisher
Sir Ronald Aylmer Fisher was a British statistician and geneticist. For his work in statistics, he has been described as "a genius who single-handedly created the foundations for modern statistical science" and "the single most important figure in 20th century statistics". In genetics, his work used mathematics to combine natural selection. For his contributions to biology, Fisher has been called "the greatest of Darwin’s successors". From 1919 onward, he worked at the Rothamsted Experimental Station for 14 years, he established his reputation there in the following years as a biostatistician. He is known as one of the three principal founders of population genetics, he outlined Fisher's principle, the Fisherian runaway and sexy son hypothesis theories of sexual selection. His contributions to statistics include the maximum likelihood, fiducial inference, the derivation of various sampling distributions, founding principles of the design of experiments, much more. Fisher held strong views on race.
Throughout his life, he was a prominent supporter of eugenics, an interest which led to his work on statistics and genetics. Notably, he was a dissenting voice in UNESCO's statement The Race Question, insisting on racial differences. Fisher was born in East Finchley in London, into a middle-class household, he was one of twins, with the other twin being still-born and grew up the youngest, with three sisters and one brother. From 1896 until 1904 they lived at Inverforth House in London, where English Heritage installed a blue plaque in 2002, before moving to Streatham, his mother, died from acute peritonitis when he was 14, his father lost his business 18 months later. Lifelong poor eyesight caused his rejection by the British Army for World War I, but developed his ability to visualize problems in geometrical terms, not in writing mathematical solutions, or proofs, he entered Harrow School won the school's Neeld Medal in mathematics. In 1909, he won a scholarship to study Mathematics at Cambridge.
In 1912, he gained a First in Astronomy. In 1915 he published a paper The evolution of sexual preference on sexual mate choice. During 1913–1919, Fisher worked for six years as a statistician in the City of London and taught physics and maths at a sequence of public schools, at the Thames Nautical Training College, at Bradfield College. There he settled with Eileen Guinness, with whom he had two sons and six daughters. In 1918 he published "The Correlation Between Relatives on the Supposition of Mendelian Inheritance", in which he introduced the term variance and proposed its formal analysis, he put forward a genetics conceptual model showing that continuous variation amongst phenotypic traits measured by biostatisticians could be produced by the combined action of many discrete genes and thus be the result of Mendelian inheritance. This was the first step towards establishing population genetics and quantitative genetics, which demonstrated that natural selection could change allele frequencies in a population, resulting in reconciling its discontinuous nature with gradual evolution.
Joan Box, Fisher's biographer and daughter says that Fisher had resolved this problem in 1911. In 1919, he began working at the Rothamsted Experimental Station for 14 years, where he analysed its immense data from crop experiments since the 1840s, developed the analysis of variance. In 1919, he was offered a position at the Galton Laboratory in University College London led by Karl Pearson, but instead accepted a temporary job at Rothamsted in Harpenden to investigate the possibility of analysing the vast amount of crop data accumulated since 1842 from the "Classical Field Experiments", he analysed the data recorded over many years and in 1921, published Studies in Crop Variation, his first application of the analysis of variance ANOVA. In 1928, Joseph Oscar Irwin began a three-year stint at Rothamsted and became one of the first people to master Fisher's innovations. Between 1912 and 1922 Fisher recommended and vastly popularized Maximum likelihood. Fisher's 1924 article On a distribution yielding the error functions of several well known statistics presented Pearson's chi-squared test and William Gosset's Student's t-distribution in the same framework as the Gaussian distribution and is where he developed Fisher's z-distribution a new statistical method used decades as the F distribution.
He pioneered the principles of the design of experiments and the statistics of small samples and the analysis of real data. In 1925 he published Statistical Methods for Research Workers, one of the 20th century's most influential books on statistical methods. Fisher's method is a technique for data fusion or "meta-analysis"; this book popularized the p-value, plays a central role in his approach. Fisher proposes the level p=0.05, or a 1 in 20 chance of being exceeded by chance, as a limit for statistical significance, applies this to a normal distribution, thus yielding the rule of two standard deviations for statistical significance. The 1.96, the approximate value of the 97.5 percentile point of the normal distribution used in probability and statistics originated in this book. "The value for which P=.05, or 1 in 20, is 1.96 or nearly 2
Median
The median is the value separating the higher half from the lower half of a data sample. For a data set, it may be thought of as the "middle" value. For example, in the data set, the median is 6, the fourth largest, the fifth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is likely to fall above or below it; the median is a used measure of the properties of a data set in statistics and probability theory. The basic advantage of the median in describing data compared to the mean is that it is not skewed so much by large or small values, so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets which vary a mean may be skewed by a small number of high or low values. Median income, for example, may be a better way to suggest; because of this, the median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large or small result.
The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest. If there is an odd number of numbers, the middle one is picked. For example, consider the list of numbers 1, 3, 3, 6, 7, 8, 9This list contains seven numbers; the median is the fourth of them, 6. If there is an number of observations there is no single middle value. For example, in the data set 1, 2, 3, 4, 5, 6, 8, 9the median is the mean of the middle two numbers: this is / 2, 4.5.. The formula used to find the index of the middle number of a data set of n numerically ordered numbers is / 2; this either gives the halfway point between the two middle values. For example, with 14 values, the formula will give an index of 7.5, the median will be taken by averaging the seventh and eighth values. So the median can be represented by the following formula: m e d i a n = a ⌈ # x ÷ 2 ⌉ + a ⌈ # x ÷ 2 + 1 ⌉ 2 One can find the median using the Stem-and-Leaf Plot. There is no accepted standard notation for the median, but some authors represent the median of a variable x either as x͂ or as μ1/2 sometimes M.
In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced. The median is used for skewed distributions, which it summarizes differently from the arithmetic mean. Consider the multiset; the median is 2 in this case, it might be seen as a better indication of central tendency than the arithmetic mean of 4. The median is a popular summary statistic used in descriptive statistics, since it is simple to understand and easy to calculate, while giving a measure, more robust in the presence of outlier values than is the mean; the cited empirical relationship between the relative locations of the mean and the median for skewed distributions is, not true. There are, various relationships for the absolute difference between them. With an number of observations no value need be at the value of the median. Nonetheless, the value of the median is uniquely determined with the usual definition. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid.
In a population, at most half have values less than the median and at most half have values greater than it. If each group contains less than half the population some of the population is equal to the median. For example, if a < b < c the median of the list is b, and, if a < b < c < d the median of the list is the mean of b and c. Indeed, as it is based on the middle data in a group, it is not necessary to know the value of extreme results in order to calculate a median. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated; the median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g. because they may be measurement errors. A median is only defined on ordered one-dimensional data, is independent of any distance metric. A geometric median, on the other hand, is defined in any number of dimensions.
The median is one of a number of ways
Extreme weather
Extreme weather includes unexpected, unpredictable, severe or unseasonal weather. Extreme events are based on a location’s recorded weather history and defined as lying in the most unusual ten percent. In recent years some extreme weather events have been attributed to human-induced global warming, with studies indicating an increasing threat from extreme weather in the future. According to IPCC estimates of annual losses have ranged since 1980 from a few billion to above US$200 billion, with the highest value for 2005; the global weather-related disaster losses, such as loss of human lives, cultural heritage, ecosystem services, are difficult to value and monetize, thus they are poorly reflected in estimates of losses. Heat waves are periods of heat index. Definitions of a heatwave vary because of the variation of temperatures in different geographic locations. Excessive heat is accompanied by high levels of humidity, but can be catastrophically dry; because heat waves are not visible as other forms of severe weather are, like hurricanes and thunderstorms, they are one of the less known forms of extreme weather.
Severe heat weather can damage populations and crops due to potential dehydration or hyperthermia, heat cramps, heat expansion and heat stroke. Dried soils are more susceptible to erosion. Outbreaks of wildfires can increase in frequency as dry vegetation has increased likeliness of igniting; the evaporation of bodies of water can be devastating to marine populations, decreasing the size of the habitats available as well as the amount of nutrition present within the waters. Livestock and other animal populations may decline as well. During excessive heat plants shut their leaf pores, a protective mechanism to conserve water but curtails plants' absorption capabilities; this leaves more pollution and ozone in the air, which leads to a higher mortality in the population. It has been estimated that extra pollution during the hot summer 2006 in cost 460 lives; the European heat waves from summer 2003 are estimated to have caused 30,000 excess deaths, due to heat stress and air pollution. Over 200 U.
S cities have registered new record high temperatures. The worst heatwave in the USA killed more than 5000 people directly; the worst heat wave in Australia occurred in 1938-39 and killed 438. The second worst was in 1896. Power outages can occur within areas experiencing heat waves due to the increased demand for electricity; the urban heat island effect can increase temperatures overnight. A cold wave is a weather phenomenon, distinguished by a cooling of the air; as used by the U. S. National Weather Service, a cold wave is a rapid fall in temperature within a 24-hour period requiring increased protection to agriculture, industry and social activities; the precise criterion for a cold wave is determined by the rate at which the temperature falls, the minimum to which it falls. This minimum temperature is dependent on the geographical time of year. Cold waves are capable of occurring any geological location and are formed by large cool air masses that accumulate over certain regions, caused by movements of air streams.
A cold wave can cause injury to livestock and wildlife. Exposure to cold mandates greater caloric intake for all animals, including humans, if a cold wave is accompanied by heavy and persistent snow, grazing animals may be unable to reach necessary food and water, die of hypothermia or starvation. Cold waves necessitate the purchase of fodder for livestock at considerable cost to farmers. Human populations can be inflicted with frostbites when exposed for extended periods of time to cold and may result in the loss of limbs or damage to internal organs. Extreme winter cold causes poorly insulated water pipes to freeze; some poorly protected indoor plumbing may rupture as frozen water expands within them, causing property damage. Fires, become more hazardous during extreme cold. Water mains may break and water supplies may become unreliable, making firefighting more difficult. Cold waves that bring unexpected freezes and frosts during the growing season in mid-latitude zones can kill plants during the early and most vulnerable stages of growth.
This results in crop failure as plants are killed. Such cold waves have caused famines. Cold waves can cause soil particles to harden and freeze, making it harder for plants and vegetation to grow within these areas. One extreme was the so-called Year Without a Summer of 1816, one of several years during the 1810s in which numerous crops failed during freakish summer cold snaps after volcanic eruptions reduced incoming sunlight. In general climate models show that with climate change, the planet will experience more extreme weather. In particular temperature record highs outpace record lows and some types of extreme weather such as extreme heat, intense precipitation, drought have become more frequent and severe in recent decades; some studies assert a connection between warming arctic temperatures and thus a vanishing cryosphere to extreme weather in mid-latitudes. In the PNAS, Steven C. Sherwood and Matthew Huber state that humans and other mammals cannot tolerate a wet-bulb temperature of over 35 °C for extended periods, that this "would begin to occur with global-mean warming of about 7 °C...
With 11–12 °C warming, such regions would spread to encompass the majority of the human population
Evolution
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes that are passed on from parent to offspring during reproduction. Different characteristics tend to exist within any given population as a result of mutation, genetic recombination and other sources of genetic variation. Evolution occurs when evolutionary processes such as natural selection and genetic drift act on this variation, resulting in certain characteristics becoming more common or rare within a population, it is this process of evolution that has given rise to biodiversity at every level of biological organisation, including the levels of species, individual organisms and molecules. The scientific theory of evolution by natural selection was proposed by Charles Darwin and Alfred Russel Wallace in the mid-19th century and was set out in detail in Darwin's book On the Origin of Species. Evolution by natural selection was first demonstrated by the observation that more offspring are produced than can survive.
This is followed by three observable facts about living organisms: 1) traits vary among individuals with respect to their morphology and behaviour, 2) different traits confer different rates of survival and reproduction and 3) traits can be passed from generation to generation. Thus, in successive generations members of a population are more to be replaced by the progenies of parents with favourable characteristics that have enabled them to survive and reproduce in their respective environments. In the early 20th century, other competing ideas of evolution such as mutationism and orthogenesis were refuted as the modern synthesis reconciled Darwinian evolution with classical genetics, which established adaptive evolution as being caused by natural selection acting on Mendelian genetic variation. All life on Earth shares a last universal common ancestor that lived 3.5–3.8 billion years ago. The fossil record includes a progression from early biogenic graphite, to microbial mat fossils, to fossilised multicellular organisms.
Existing patterns of biodiversity have been shaped by repeated formations of new species, changes within species and loss of species throughout the evolutionary history of life on Earth. Morphological and biochemical traits are more similar among species that share a more recent common ancestor, can be used to reconstruct phylogenetic trees. Evolutionary biologists have continued to study various aspects of evolution by forming and testing hypotheses as well as constructing theories based on evidence from the field or laboratory and on data generated by the methods of mathematical and theoretical biology, their discoveries have influenced not just the development of biology but numerous other scientific and industrial fields, including agriculture and computer science. The proposal that one type of organism could descend from another type goes back to some of the first pre-Socratic Greek philosophers, such as Anaximander and Empedocles; such proposals survived into Roman times. The poet and philosopher Lucretius followed Empedocles in his masterwork De rerum natura.
In contrast to these materialistic views, Aristotelianism considered all natural things as actualisations of fixed natural possibilities, known as forms. This was part of a medieval teleological understanding of nature in which all things have an intended role to play in a divine cosmic order. Variations of this idea became the standard understanding of the Middle Ages and were integrated into Christian learning, but Aristotle did not demand that real types of organisms always correspond one-for-one with exact metaphysical forms and gave examples of how new types of living things could come to be. In the 17th century, the new method of modern science rejected the Aristotelian approach, it sought explanations of natural phenomena in terms of physical laws that were the same for all visible things and that did not require the existence of any fixed natural categories or divine cosmic order. However, this new approach was slow to take root in the biological sciences, the last bastion of the concept of fixed natural types.
John Ray applied one of the more general terms for fixed natural types, "species," to plant and animal types, but he identified each type of living thing as a species and proposed that each species could be defined by the features that perpetuated themselves generation after generation. The biological classification introduced by Carl Linnaeus in 1735 explicitly recognised the hierarchical nature of species relationships, but still viewed species as fixed according to a divine plan. Other naturalists of this time speculated on the evolutionary change of species over time according to natural laws. In 1751, Pierre Louis Maupertuis wrote of natural modifications occurring during reproduction and accumulating over many generations to produce new species. Georges-Louis Leclerc, Comte de Buffon suggested that species could degenerate into different organisms, Erasmus Darwin proposed that all warm-blooded animals could have descended from a single microorganism; the first full-fledged evolutionary scheme was Jean-Baptiste Lamarck's "transmutation" theory of 1809, which envisaged spontaneous generation continually producing simple forms of life that developed greater complexity in parallel lineages with an inherent progressive tendency, postulated that on a local level, these lineages adapted to the environment by inheriting changes caused by their use or disuse in parents.
These ideas were cond
Flood
A flood is an overflow of water that submerges land, dry. In the sense of "flowing water", the word may be applied to the inflow of the tide. Floods are an area of study of the discipline hydrology and are of significant concern in agriculture, civil engineering and public health. Flooding may occur as an overflow of water from water bodies, such as a river, lake, or ocean, in which the water overtops or breaks levees, resulting in some of that water escaping its usual boundaries, or it may occur due to an accumulation of rainwater on saturated ground in an areal flood. While the size of a lake or other body of water will vary with seasonal changes in precipitation and snow melt, these changes in size are unlikely to be considered significant unless they flood property or drown domestic animals. Floods can occur in rivers when the flow rate exceeds the capacity of the river channel at bends or meanders in the waterway. Floods cause damage to homes and businesses if they are in the natural flood plains of rivers.
While riverine flood damage can be eliminated by moving away from rivers and other bodies of water, people have traditionally lived and worked by rivers because the land is flat and fertile and because rivers provide easy travel and access to commerce and industry. Some floods develop while others such as flash floods can develop in just a few minutes and without visible signs of rain. Additionally, floods can be local, impacting a neighborhood or community, or large, affecting entire river basins; the word "flood" comes from a word common to Germanic languages. Floods can happen on flat or low-lying areas when water is supplied by rainfall or snowmelt more than it can either infiltrate or run off; the excess accumulates in place, sometimes to hazardous depths. Surface soil can become saturated, which stops infiltration, where the water table is shallow, such as a floodplain, or from intense rain from one or a series of storms. Infiltration is slow to negligible through frozen ground, concrete, paving, or roofs.
Areal flooding begins in flat areas like floodplains and in local depressions not connected to a stream channel, because the velocity of overland flow depends on the surface slope. Endorheic basins may experience areal flooding during periods when precipitation exceeds evaporation. Floods occur in all types of river and stream channels, from the smallest ephemeral streams in humid zones to normally-dry channels in arid climates to the world's largest rivers; when overland flow occurs on tilled fields, it can result in a muddy flood where sediments are picked up by run off and carried as suspended matter or bed load. Localized flooding may be caused or exacerbated by drainage obstructions such as landslides, debris, or beaver dams. Slow-rising floods most occur in large rivers with large catchment areas; the increase in flow may be the result of sustained rainfall, rapid snow melt, monsoons, or tropical cyclones. However, large rivers may have rapid flooding events in areas with dry climate, since they may have large basins but small river channels and rainfall can be intense in smaller areas of those basins.
Rapid flooding events, including flash floods, more occur on smaller rivers, rivers with steep valleys, rivers that flow for much of their length over impermeable terrain, or normally-dry channels. The cause may be localized convective precipitation or sudden release from an upstream impoundment created behind a dam, landslide, or glacier. In one instance, a flash flood killed eight people enjoying the water on a Sunday afternoon at a popular waterfall in a narrow canyon. Without any observed rainfall, the flow rate increased from about 50 to 1,500 cubic feet per second in just one minute. Two larger floods occurred at the same site within a week, but no one was at the waterfall on those days; the deadly flood resulted from a thunderstorm over part of the drainage basin, where steep, bare rock slopes are common and the thin soil was saturated. Flash floods are the most common flood type in normally-dry channels in arid zones, known as arroyos in the southwest United States and many other names elsewhere.
In that setting, the first flood water to arrive is depleted. The leading edge of the flood thus advances more than and higher flows; as a result, the rising limb of the hydrograph becomes quicker as the flood moves downstream, until the flow rate is so great that the depletion by wetting soil becomes insignificant. Flooding in estuaries is caused by a combination of sea tidal surges caused by winds and low barometric pressure, they may be exacerbated by high upstream river flow. Coastal areas may be flooded by storm events at sea, resulting in waves over-topping defenses or in severe cases by tsunami or tropical cyclones. A storm surge, from either a tropical cyclone or an extratropical cyclone, falls within this category. Research from the NHC explains: "Storm surge is an abnormal rise of water generated by a storm and above the predicted astronomical tides. Storm surge should not be confused with storm tide, defined as the water level rise due to the combination of storm surge and the astronomical tide.
This rise in water level can cause extreme flooding in coastal areas when storm surge coincides with normal high tide, resulting in storm tides reaching up to 20 feet or more in some cases." Urban flooding is the inundation of land or property in a built environment in more densely populated areas, caused by rainfall overwhelmi
100-year flood
A one-hundred-year flood is a flood event that has a 1% probability of occurring in any given year. The 100-year flood is referred to as the 1% flood, since its annual exceedance probability is 1%. For river systems, the 100-year flood is expressed as a flowrate. Based on the expected 100-year flood flow rate, the flood water level can be mapped as an area of inundation; the resulting floodplain map is referred to as the 100-year floodplain. Estimates of the 100-year flood flowrate and other streamflow statistics for any stream in the United States are available. In the UK The Environment Agency publishes a comprehensive map of all areas at risk of a 1 in 100 year flood. Areas near the coast of an ocean or large lake can be flooded by combinations of tide, storm surge, waves. Maps of the riverine or coastal 100-year floodplain may figure in building permits, environmental regulations, flood insurance. A common misunderstanding is that a 100-year flood is to occur only once in a 100-year period.
In fact, there is a 63.4% chance of one or more 100-year floods occurring in any 100-year period. On the Danube River at Passau, the actual intervals between 100-year floods during 1501 to 2013 ranged from 37 to 192 years; the probability Pe that one or more floods occurring during any period will exceed a given flood threshold can be expressed, using the binomial distribution, as P e = 1 − n where T is the threshold return period, n is the number of years in the period. The probability of exceedance Pe is described as the natural, inherent, or hydrologic risk of failure. However, the expected value of the number of 100-year floods occurring in any 100-year period is 1. Ten-year floods have a 10% chance of occurring in any given year; the percent chance of an X-year flood occurring in a single year is 100/X. A similar analysis is applied to coastal flooding or rainfall data; the recurrence interval of a storm is identical to that of an associated riverine flood, because of rainfall timing and location variations among different drainage basins.
The field of extreme value theory was created to model rare events such as 100-year floods for the purposes of civil engineering. This theory is most applied to the maximum or minimum observed stream flows of a given river. In desert areas where there are only ephemeral washes, this method is applied to the maximum observed rainfall over a given period of time; the extreme value analysis only considers the most extreme event observed in a given year. So, between the large spring runoff and a heavy summer rain storm, whichever resulted in more runoff would be considered the extreme event, while the smaller event would be ignored in the analysis. There are a number of assumptions that are made to complete the analysis that determines the 100-year flood. First, the extreme events observed in each year must be independent from year to year. In other words, the maximum river flow rate from 1984 cannot be found to be correlated with the observed flow rate in 1985, which cannot be correlated with 1986, so forth.
The second assumption is that the observed extreme events must come from the same probability distribution function. The third assumption is that the probability distribution relates to the largest storm that occurs in any one year; the fourth assumption is that the probability distribution function is stationary, meaning that the mean, standard deviation and maximum and minimum values are not increasing or decreasing over time. This concept is referred to as stationarity; the first assumption is but not always valid and should be tested on a case by case basis. The second assumption is valid if the extreme events are observed under similar climate conditions. For example, if the extreme events on record all come from late summer thunderstorms, or from snow pack melting this assumption should be valid. If, there are some extreme events taken from thunder storms, others from snow pack melting, others from hurricanes this assumption is most not valid; the third assumption is only a problem when trying to forecast maximum flow event.
Since this is not a goal in extreme analysis, or in civil engineering design the situation presents itself. The final assumption about stationarity is difficult to test from data for a single site because of the large uncertainties in the longest flood records. More broadly, substantial evidence of climate change suggests that the probability distribution is changing and that managing flood risks in the future will become more difficult; the simplest implication of this is that not all of the historical data are, or can be, considered valid as input into the extreme event analysis. When these assumptions are violated there is an unknown amount of uncertainty introduced into the reported value of what the 100-year flood means in terms of rainfall intensity, or flood depth; when all of the inputs are known the uncertainty can be measured in the form of a confidence interval. For example, one mi
Outlier
In statistics, an outlier is an observation point, distant from other observations. An outlier may be due to variability in the measurement or it may indicate experimental error. An outlier can cause serious problems in statistical analyses. Outliers can occur by chance in any distribution, but they indicate either measurement error or that the population has a heavy-tailed distribution. In the former case one wishes to discard them or use statistics that are robust to outliers, while in the latter case they indicate that the distribution has high skewness and that one should be cautious in using tools or intuitions that assume a normal distribution. A frequent cause of outliers is a mixture of two distributions, which may be two distinct sub-populations, or may indicate'correct trial' versus'measurement error'. In most larger samplings of data, some data points will be further away from the sample mean than what is deemed reasonable; this can be due to incidental systematic error or flaws in the theory that generated an assumed family of probability distributions, or it may be that some observations are far from the center of the data.
Outlier points can therefore indicate faulty data, erroneous procedures, or areas where a certain theory might not be valid. However, in large samples, a small number of outliers is to be expected. Outliers, being the most extreme observations, may include the sample maximum or sample minimum, or both, depending on whether they are high or low. However, the sample maximum and minimum are not always outliers because they may not be unusually far from other observations. Naive interpretation of statistics derived from data sets. For example, if one is calculating the average temperature of 10 objects in a room, nine of them are between 20 and 25 degrees Celsius, but an oven is at 175 °C, the median of the data will be between 20 and 25 °C but the mean temperature will be between 35.5 and 40 °C. In this case, the median better reflects the temperature of a randomly sampled object than the mean; as illustrated in this case, outliers may indicate data points that belong to a different population than the rest of the sample set.
Estimators capable of coping with outliers are said to be robust: the median is a robust statistic of central tendency, while the mean is not. However, the mean is a more precise estimator. In the case of distributed data, the three sigma rule means that 1 in 22 observations will differ by twice the standard deviation or more from the mean, 1 in 370 will deviate by three times the standard deviation. In a sample of 1000 observations, the presence of up to five observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice the expected number and hence within 1 standard deviation of the expected number – see Poisson distribution – and not indicate an anomaly. If the sample size is only 100, just three such outliers are reason for concern, being more than 11 times the expected number. In general, if the nature of the population distribution is known a priori, it is possible to test if the number of outliers deviate from what can be expected: for a given cutoff of a given distribution, the number of outliers will follow a binomial distribution with parameter p, which can be well-approximated by the Poisson distribution with λ = pn.
Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, p is 0.3%, thus for 1000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a Poisson distribution with λ = 3. Outliers can have many anomalous causes. A physical apparatus for taking measurements may have suffered a transient malfunction. There may have been an error in data transcription. Outliers arise due to changes in system behaviour, fraudulent behaviour, human error, instrument error or through natural deviations in populations. A sample may have been contaminated with elements from outside the population being examined. Alternatively, an outlier could be the result of a flaw in the assumed theory, calling for further investigation by the researcher. Additionally, the pathological appearance of outliers of a certain form appears in a variety of datasets, indicating that the causative mechanism for the data might differ at the extreme end. There is no rigid mathematical definition of.
There are various methods of outlier detection. Some are graphical such as normal probability plots. Others are model-based. Box plots are a hybrid. Model-based methods which are used for identification assume that the data are from a normal distribution, identify observations which are deemed "unlikely" based on mean and standard deviation: Chauvenet's criterion Grubbs's test for outliers Dixon's Q test ASTM E178 Standard Practice for Dealing With Outlying Observations Mahalanobis distance and leverage are used to detect outliers in the development of linear regression models. Subspace and correlation based techniques for high-dimensional numerical data It is proposed to determine in a series of m observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are
![G(z)\propto \exp \left[-(1+\zeta z)^{-1/\zeta }\right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c04e064d0109cfceba396d85551c5b6ecd1e10b)
