In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be undefined. For a unimodal distribution, negative skew indicates that the tail is on the left side of the distribution, positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value means. Consider the two distributions in the figure just below. Within each graph, the values on the right side of the distribution taper differently from the values on the left side; these tapering sides are called tails, they provide a visual means to determine which of the two kinds of skewness a distribution has: negative skew: The left tail is longer. The distribution is said to be left-skewed, left-tailed, or skewed to the left, despite the fact that the curve itself appears to be skewed or leaning to the right. A left-skewed distribution appears as a right-leaning curve.
Positive skew: The right tail is longer. The distribution is said to be right-skewed, right-tailed, or skewed to the right, despite the fact that the curve itself appears to be skewed or leaning to the left. A right-skewed distribution appears as a left-leaning curve. Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the numeric sequence, whose values are evenly distributed around a central value of 50. We can transform this sequence into a negatively skewed distribution by adding a value far below the mean, e.g.. We can make the sequence positively skewed by adding a value far above the mean, e.g.. The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, for positive skew. In the older notion of nonparametric skew, defined as / σ, where μ is the mean, ν is the median, σ is the standard deviation, the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than the median, while negative/left nonparametric skew means the mean is less than the median.
However, the modern definition of skewness and the traditional nonparametric definition do not in general have the same sign: while they agree for some families of distributions, they differ in general, conflating them is misleading. If the distribution is symmetric the mean is equal to the median, the distribution has zero skewness. If the distribution is both symmetric and unimodal the mean = median = mode; this is the case of a coin toss or the series 1,2,3,4... Note, that the converse is not true in general, i.e. zero skewness does not imply that the mean is equal to the median. A 2005 journal article points out: Many textbooks, teach a rule of thumb stating that the mean is right of the median under right skew, left of the median under left skew; this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy. Most though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal.
Such distributions not only contradict the textbook relationship between mean and skew, they contradict the textbook interpretation of the median. The skewness of a random variable X is the third standardized moment γ1, defined as: γ 1 = E = μ 3 σ 3 = E 3 / 2 = κ 3 κ 2 3 / 2 where μ is the mean, σ is the standard deviation, E is the expectation operator, μ3 is the third central moment, κt are the tth
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Per capita income
Per capita income or average income measures the average income earned per person in a given area in a specified year. It is calculated by dividing the area's total income by its total population. Per capita income is national income divided by population size. Per capita income is used to measure an area's average income and compare the wealth of different populations. Per capita income is used to measure a country's standard of living, it is expressed in terms of a used international currency such as the euro or United States dollar, is useful because it is known, is calculable from available gross domestic product and population estimates, produces a useful statistic for comparison of wealth between sovereign territories. This helps to ascertain a country's development status, it is one of the three measures for calculating the Human Development Index of a country. In the United States, it is defined by the U. S. Census Bureau as the following: "Per capita income is the mean money income received in the past 12 months computed for every man and child in a geographic area."
Critics claim that per capita income has several weaknesses in measuring prosperity: Comparisons of per capita income over time need to consider inflation. Without adjusting for inflation, figures tend to overstate the effects of economic growth. International comparisons can be distorted by cost of living differences not reflected in exchange rates. Where the objective is to compare living standards between countries, adjusting for differences in purchasing power parity will more reflect what people are able to buy with their money, it does not reflect income distribution. If a country's income distribution is skewed, a small wealthy class can increase per capita income while the majority of the population has no change in income. In this respect, median income is more useful when measuring of prosperity than per capita income, as it is less influenced by outliers. Non-monetary activity, such as barter or services provided within the family, is not counted; the importance of these services varies among economies.
Per capita income does not consider whether income is invested in factors to improve the area's development, such as health, education, or infrastructure. List of countries by average wage List of countries by GDP per capita—GDP at market or government official exchange rates per inhabitant List of countries by GDP per capita—GDP calculated at purchasing power parity exchange per inhabitant List of countries by GNI per capita List of countries by GNI per capita List of countries by income equality Total personal income
In probability theory, the normal distribution is a common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate; the normal distribution is useful because of the central limit theorem. In its most general form, under some conditions, it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are distributed; the normal distribution is sometimes informally called the bell curve.
However, many other distributions are bell-shaped. The probability density of the normal distribution is f = 1 2 π σ 2 e − 2 2 σ 2 where μ is the mean or expectation of the distribution, σ is the standard deviation, σ 2 is the variance; the simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, it is described by this probability density function: φ = 1 2 π e − 1 2 x 2 The factor 1 / 2 π in this expression ensures that the total area under the curve φ is equal to one; the factor 1 / 2 in the exponent ensures that the distribution has unit variance, therefore unit standard deviation. This function is symmetric around x = 0, where it attains its maximum value 1 / 2 π and has inflection points at x = + 1 and x = − 1. Authors may differ on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance σ 2 = 1 / 2, φ = e − x 2 π Stigler goes further, defining the standard normal with variance σ 2 = 1 /: φ = e − π x 2 Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ and translated by μ: f = 1 σ φ.
The probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a standard normal deviate X = σ Z + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a normal deviate with parameters μ and σ 2 Z = / σ
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
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
In statistics, quality assurance, survey methodology, sampling is the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt for the samples to represent the population in question. Two advantages of sampling are lower cost and faster data collection than measuring the entire population; each observation measures one or more properties of observable bodies distinguished as independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design in stratified sampling. Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is used for gathering information about a population. Acceptance sampling is used to determine if a production lot of material meets the governing specifications. Successful statistical practice is based on focused problem definition. In sampling, this includes defining the "population".
A population can be defined as including all people or items with the characteristic one wishes to understand. Because there is rarely enough time or money to gather information from everyone or everything in a population, the goal becomes finding a representative sample of that population. Sometimes what defines. For example, a manufacturer needs to decide whether a batch of material from production is of high enough quality to be released to the customer, or should be sentenced for scrap or rework due to poor quality. In this case, the batch is the population. Although the population of interest consists of physical objects, sometimes it is necessary to sample over time, space, or some combination of these dimensions. For instance, an investigation of supermarket staffing could examine checkout line length at various times, or a study on endangered penguins might aim to understand their usage of various hunting grounds over time. For the time dimension, the focus may be on discrete occasions.
In other cases, the examined'population' may be less tangible. For example, Joseph Jagger studied the behaviour of roulette wheels at a casino in Monte Carlo, used this to identify a biased wheel. In this case, the'population' Jagger wanted to investigate was the overall behaviour of the wheel, while his'sample' was formed from observed results from that wheel. Similar considerations arise when taking repeated measurements of some physical characteristic such as the electrical conductivity of copper; this situation arises when seeking knowledge about the cause system of which the observed population is an outcome. In such cases, sampling theory may treat the observed population as a sample from a larger'superpopulation'. For example, a researcher might study the success rate of a new'quit smoking' program on a test group of 100 patients, in order to predict the effects of the program if it were made available nationwide. Here the superpopulation is "everybody in the country, given access to this treatment" – a group which does not yet exist, since the program isn't yet available to all.
Note that the population from which the sample is drawn may not be the same as the population about which information is desired. There is large but not complete overlap between these two groups due to frame issues etc.. Sometimes they may be separate – for instance, one might study rats in order to get a better understanding of human health, or one might study records from people born in 2008 in order to make predictions about people born in 2009. Time spent in making the sampled population and population of concern precise is well spent, because it raises many issues and questions that would otherwise have been overlooked at this stage. In the most straightforward case, such as the sampling of a batch of material from production, it would be most desirable to identify and measure every single item in the population and to include any one of them in our sample. However, in the more general case this is not possible or practical. There is no way to identify all rats in the set of all rats. Where voting is not compulsory, there is no way to identify which people will vote at a forthcoming election.
These imprecise populations are not amenable to sampling in any of the ways below and to which we could apply statistical theory. As a remedy, we seek a sampling frame which has the property that we can identify every single element and include any in our sample; the most straightforward type of frame is a list of elements of the population with appropriate contact information. For example, in an opinion poll, possible sampling frames include an electoral register and a telephone directory. A probability sample is a sample in which every unit in the population has a chance of being selected in the sample, this probability can be determined; the combination of these traits makes it possible to produce unbiased estimates of population totals, by weighting sampled units according to their probability of selection. Example: We want to estimate the total income of adults living in a given street. We visit each household in that street, identify all adults living there, randomly select one adult from each household..
We interview the selected person and find their income