1.
Londen
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London /ˈlʌndən/ is the capital and most populous city of England and the United Kingdom. Standing on the River Thames in the south east of the island of Great Britain and it was founded by the Romans, who named it Londinium. Londons ancient core, the City of London, largely retains its 1. 12-square-mile medieval boundaries. London is a global city in the arts, commerce, education, entertainment, fashion, finance, healthcare, media, professional services, research and development, tourism. It is crowned as the worlds largest financial centre and has the fifth- or sixth-largest metropolitan area GDP in the world, London is a world cultural capital. It is the worlds most-visited city as measured by international arrivals and has the worlds largest city airport system measured by passenger traffic, London is the worlds leading investment destination, hosting more international retailers and ultra high-net-worth individuals than any other city. Londons universities form the largest concentration of education institutes in Europe. In 2012, London became the first city to have hosted the modern Summer Olympic Games three times, London has a diverse range of people and cultures, and more than 300 languages are spoken in the region. Its estimated mid-2015 municipal population was 8,673,713, the largest of any city in the European Union, Londons urban area is the second most populous in the EU, after Paris, with 9,787,426 inhabitants at the 2011 census. The citys metropolitan area is the most populous in the EU with 13,879,757 inhabitants, the city-region therefore has a similar land area and population to that of the New York metropolitan area. London was the worlds most populous city from around 1831 to 1925, Other famous landmarks include Buckingham Palace, the London Eye, Piccadilly Circus, St Pauls Cathedral, Tower Bridge, Trafalgar Square, and The Shard. The London Underground is the oldest underground railway network in the world, the etymology of London is uncertain. It is an ancient name, found in sources from the 2nd century and it is recorded c.121 as Londinium, which points to Romano-British origin, and hand-written Roman tablets recovered in the city originating from AD 65/70-80 include the word Londinio. The earliest attempted explanation, now disregarded, is attributed to Geoffrey of Monmouth in Historia Regum Britanniae and this had it that the name originated from a supposed King Lud, who had allegedly taken over the city and named it Kaerlud. From 1898, it was accepted that the name was of Celtic origin and meant place belonging to a man called *Londinos. The ultimate difficulty lies in reconciling the Latin form Londinium with the modern Welsh Llundain, which should demand a form *lōndinion, from earlier *loundiniom. The possibility cannot be ruled out that the Welsh name was borrowed back in from English at a later date, and thus cannot be used as a basis from which to reconstruct the original name. Until 1889, the name London officially applied only to the City of London, two recent discoveries indicate probable very early settlements near the Thames in the London area

2.
Wiskundige
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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems

3.
Statistiek
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes

4.
Kansrekening
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events, two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, a great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory and this culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of space, introduced by Richard von Mises. This became the mostly undisputed axiomatic basis for modern probability theory, most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment. The power set of the space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results, one collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the set of the sample space of die rolls. In this case, is the event that the die falls on some odd number, If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one, the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6 and this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, discrete probability theory deals with events that occur in countable sample spaces. Modern definition, The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by Ω

5.
Chi-kwadraattoets
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A chi-squared test, also written as χ2 test, is any statistical hypothesis test wherein the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. Without other qualification, chi-squared test often is used as short for Pearsons chi-squared test, chi-squared tests are often constructed from a sum of squared errors, or through the sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, a chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent. The chi-squared test is used to determine there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. One test statistic follows a chi-squared distribution exactly is the test that the variance of a normally distributed population has a given value based on a sample variance. Such tests are uncommon in practice because the variance of the population is usually unknown. When the chi-squared test is mentioned without any modifiers or other precluding contexts, for an exact test used in place of the chi-squared test, see Fishers exact test. This assumption is not quite correct, and introduces some error and this reduces the chi-squared value obtained and thus increases its p-value. For example, a process might have been in stable condition for a long period. Suppose that a variant of the process is being tested, giving rise to a sample of n product items whose variation is to be tested. The test statistic T in this instance could be set to be the sum of squares about the sample mean, then T has a chi-squared distribution with n −1 degrees of freedom. For example, if the size is 21, the acceptance region for T with a significance level of 5% is between 9.59 and 34.17. Suppose there is a city of 1 million residents with four neighborhoods, A, B, C, a random sample of 650 residents of the city is taken and their occupation is recorded as white collar, blue collar, or no collar. The null hypothesis is that each neighborhood of residence is independent of the persons occupational classification. The data are tabulated as, Let us take the living in neighborhood A,150. Similarly we take 349/650 to estimate what proportion of the 1 million people are white-collar workers, by the assumption of independence under the hypothesis we should expect the number of white-collar workers in neighborhood A to be 150 ×349650 ≈80.54. Then in that cell of the table, we have 2 expected =280.54, the sum of these quantities over all of the cells is the test statistic. Under the null hypothesis, it has approximately a chi-squared distribution whose number of degrees of freedom are = =6, if the test statistic is improbably large according to that chi-squared distribution, then one rejects the null hypothesis of independence

6.
Standaardafwijking
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In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. The standard deviation of a variable, statistical population, data set. It is algebraically simpler, though in practice less robust, than the absolute deviation. A useful property of the deviation is that, unlike the variance. There are also other measures of deviation from the norm, including mean absolute deviation, in addition to expressing the variability of a population, the standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the standard deviation in the results if the same poll were to be conducted multiple times. This derivation of a deviation is often called the standard error of the estimate or standard error of the mean when referring to a mean. It is computed as the deviation of all the means that would be computed from that population if an infinite number of samples were drawn. It is very important to note that the deviation of a population. The reported margin of error of a poll is computed from the error of the mean and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval. The standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment. For a finite set of numbers, the deviation is found by taking the square root of the average of the squared deviations of the values from their average value. For example, the marks of a class of eight students are the eight values,2,4,4,4,5,5,7,9. These eight data points have the mean of 5,2 +4 +4 +4 +5 +5 +7 +98 =5 and this formula is valid only if the eight values with which we began form the complete population. If the values instead were a sample drawn from some large parent population. In that case the result would be called the standard deviation. Dividing by n −1 rather than by n gives an estimate of the variance of the larger parent population. This is known as Bessels correction, as a slightly more complicated real-life example, the average height for adult men in the United States is about 70 inches, with a standard deviation of around 3 inches

7.
Normale verdeling
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In probability theory, the normal distribution is a very 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. The normal distribution is useful because of the limit theorem. Physical quantities that are expected to be the sum of independent processes often have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are normally 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 =12 π σ2 e −22 σ2 Where, μ is mean or expectation of the distribution. σ is standard deviation σ2 is variance A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate. The simplest case of a distribution is known as the standard normal distribution. The factor 1 /2 in the exponent ensures that the distribution has unit variance and 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 also on which normal distribution should be called the standard one, the probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a normal deviate, then X = Zσ + μ will have a normal distribution with expected value μ. Conversely, if X is a normal deviate, then Z = /σ will have a standard normal distribution. Every normal distribution is the exponential of a function, f = e a x 2 + b x + c where a is negative. In this form, the mean value μ is −b/, for the standard normal distribution, a is −1/2, b is zero, and c is − ln /2. The standard Gaussian distribution is denoted with the Greek letter ϕ. The alternative form of the Greek phi letter, φ, is used quite often. The normal distribution is often denoted by N. Thus when a random variable X is distributed normally with mean μ and variance σ2, some authors advocate using the precision τ as the parameter defining the width of the distribution, instead of the deviation σ or the variance σ2

8.
Kurtosis
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In probability theory and statistics, kurtosis is a measure of the tailedness of the probability distribution of a real-valued random variable. Depending on the measure of kurtosis that is used, there are various interpretations of kurtosis. The standard measure of kurtosis, originating with Karl Pearson, is based on a version of the fourth moment of the data or population. This number is related to the tails of the distribution, not its peak, hence, for this measure, higher kurtosis is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations. The kurtosis of any normal distribution is 3. It is common to compare the kurtosis of a distribution to this value, distributions with kurtosis less than 3 are said to be platykurtic, although this does not imply the distribution is flat-topped as sometimes reported. Rather, it means the distribution produces fewer and less extreme outliers than does the normal distribution, an example of a platykurtic distribution is the uniform distribution, which does not produce outliers. Distributions with kurtosis greater than 3 are said to be leptokurtic and it is also common practice to use an adjusted version of Pearsons kurtosis, the excess kurtosis, which is the kurtosis minus 3, to provide the comparison to the normal distribution. Some authors use kurtosis by itself to refer to the excess kurtosis, for the reason of clarity and generality, however, this article follows the non-excess convention and explicitly indicates where excess kurtosis is meant. Alternative measures of kurtosis are, the L-kurtosis, which is a version of the fourth L-moment. These are analogous to the measures of skewness that are not based on ordinary moments. The kurtosis is the fourth standardized moment, defined as Kurt = μ4 σ4 = E 2, several letters are used in the literature to denote the kurtosis. A very common choice is κ, which is fine as long as it is clear that it does not refer to a cumulant, other choices include γ2, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis. The kurtosis is bounded below by the squared skewness plus 1, μ4 σ4 ≥2 +1, the lower bound is realized by the Bernoulli distribution. There is no limit to the excess kurtosis of a general probability distribution. A reason why some authors favor the excess kurtosis is that cumulants are extensive, formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. Xn be independent random variables for which the fourth moment exists, the excess kurtosis of Y is Kurt −3 =12 ∑ i =1 n σ i 4 ⋅, where σ i is the standard deviation of X i. In particular if all of the Xi have the same variance, the reason not to subtract off 3 is that the bare fourth moment better generalizes to multivariate distributions, especially when independence is not assumed

9.
Histogram
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A histogram is a graphical representation of the distribution of numerical data. It is an estimate of the probability distribution of a variable and was first introduced by Karl Pearson. It is a kind of bar graph, to construct a histogram, the first step is to bin the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable, the bins must be adjacent, and are often of equal size. If the bins are of size, a rectangle is erected over the bin with height proportional to the frequency — the number of cases in each bin. A histogram may also be normalized to display relative frequencies and it then shows the proportion of cases that fall into each of several categories, with the sum of the heights equaling 1. However, bins need not be of equal width, in that case, the vertical axis is then not the frequency but frequency density — the number of cases per unit of the variable on the horizontal axis. Examples of variable bin width are displayed on Census bureau data below, as the adjacent bins leave no gaps, the rectangles of a histogram touch each other to indicate that the original variable is continuous. Histograms give a sense of the density of the underlying distribution of the data. The total area of a used for probability density is always normalized to 1. If the length of the intervals on the x-axis are all 1, a histogram can be thought of as a simplistic kernel density estimation, which uses a kernel to smooth frequencies over the bins. This yields a smoother probability density function, which will in general more accurately reflect distribution of the underlying variable, the density estimate could be plotted as an alternative to the histogram, and is usually drawn as a curve rather than a set of boxes. Another alternative is the average shifted histogram, which is fast to compute, the histogram is one of the seven basic tools of quality control. Histograms are sometimes confused with bar charts, a histogram is used for continuous data, where the bins represent ranges of data, while a bar chart is a plot of categorical variables. Some authors recommend that bar charts have gaps between the rectangles to clarify the distinction, the etymology of the word histogram is uncertain. Sometimes it is said to be derived from the Ancient Greek ἱστός – anything set upright and it is also said that Karl Pearson, who introduced the term in 1891, derived the name from historical diagram. This is a toy example, The words used to describe the patterns in a histogram are, symmetric, skewed left or right and it is a good idea to plot the data using several different bin widths to learn more about it. Here is an example on tips given in a restaurant, here are a couple more examples, The U. S. Census Bureau found that there were 124 million people who work outside of their homes

10.
Correlatiecoëfficiënt
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In statistics, dependence or association is any statistical relationship, whether causal or not, between two random variables or bivariate data. Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring, and the correlation between the demand for a product and its price, correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a day based on the correlation between electricity demand and weather. In this example there is a relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship, formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence, however, when used in a technical sense, correlation refers to any of several specific types of relationship between mean values. There are several correlation coefficients, often denoted ρ or r, the most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables. Other correlation coefficients have been developed to be more robust than the Pearson correlation – that is, mutual information can also be applied to measure dependence between two variables. It is obtained by dividing the covariance of the two variables by the product of their standard deviations, karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton. The Pearson correlation is defined only if both of the deviations are finite and nonzero. It is a corollary of the Cauchy–Schwarz inequality that the correlation cannot exceed 1 in absolute value, the correlation coefficient is symmetric, corr = corr. As it approaches zero there is less of a relationship, the closer the coefficient is to either −1 or 1, the stronger the correlation between the variables. If the variables are independent, Pearsons correlation coefficient is 0, for example, suppose the random variable X is symmetrically distributed about zero, and Y = X2. Then Y is completely determined by X, so that X and Y are perfectly dependent, however, in the special case when X and Y are jointly normal, uncorrelatedness is equivalent to independence. If we have a series of n measurements of X and Y written as xi, N, then the sample correlation coefficient can be used to estimate the population Pearson correlation r between X and Y. If x and y are results of measurements that contain measurement error, for the case of a linear model with a single independent variable, the coefficient of determination is the square of r, Pearsons product-moment coefficient. If, as the one variable increases, the other decreases, to illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers. As we go from each pair to the pair x increases