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| {{about|correlation and dependence in statistical data||correlation (disambiguation)}}
| | Shedding weight is really a objective for lots of people and they also turn this into objective for many distinct factors. In addition to the apparent benefits, you may want to match elegant clothing, look really good for special occasions, or maintain off some of the effects of growing older on your bones. Regardless of what your inspiration, our ideas will help lighting the path of fat loss.<br><br>View your calorie consumption every single day. If you can remove unhealthy fats, practice it. Swapping is also a good plan. Substitute the unhealthy fats with delicious, healthier alternatives.<br><br>To actually remain focused on your weight loss plan, get anyone inside your family enthusiastic about maintaining a healthy diet food items. Make your same snack foods and foods for each individual surviving in your family like that, you won't attempt to enjoy one of several unhealthy food that they are involving in.<br><br>Among the finest ways to start your weight loss program is to discover methods to decrease the stress in your life. Anxiety might cause cortisol ranges to go up while keeping glucose levels volatile, thus contributing to needless excess weight. Stress also can get you to more likely to overeat if you are annoyed about one thing.<br><br>Obtain physical exercise boots that feel good in your feet. You should take into consideration the fact that unwell-fitted shoes or boots could cause injuries. They want not be expensive or recommended by way of a sports superstar, just clever shoes or boots that are great for well and feel [http://www.latestsms.in/latest-sardar-jokes-2013.htm Good Morning SMS].<br><br>One of the better ways to assist you to slim down is always to harness the effectiveness of visualization. By imagining what we want out physiques to appear and feel like in the foreseeable future, we'll be significantly better prepared to adhere to your workout goals. Visualization really is key when slimming down.<br><br>Should you be trying to lose weight, use the Internet! Sure, you can study about diet programs online but it's much better should you be a part of an online excess weight-damage local community or discussion board. Diet plan discussion boards can help you keep motivated and provide you the ability to help other folks by revealing your own personal activities. There's an internet group for nearly each diet program.<br><br>Clear your pearly whites late at night! It may well seem strange, however, if you clean your the teeth just before heading to bed, that sneaky night goody that can pile on the body weight might not be as tempting. By cleansing your the teeth, subconsciously you might be informing the body which you have concluded ingesting during the day.<br><br>Once you dine out with buddies, possess a skinny cappuccino as an alternative to delicacy. This way you will not think that you are just sitting down there viewing when they enjoy. In addition, you will know you happen to be saving a whole lot of unhealthy calories sipping on the wonderful, comfortable, very low-caloric drink as opposed to picking that substantial-excess fat, calories-filled component of cheesecake.<br><br>Monitor your ultimate goal! Take a look at the following tips when you feel a need to seize a bit of determination and success will be in your potential. Make up your mind to accept the first actions and start in your journey. You'll seem better, feel good and you will definitely achieve a sense of fulfillment that will provide you with an enormous improve for your self esteem. |
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| In [[statistics]], '''dependence''' is any statistical relationship between two [[random variable]]s or two sets of [[data]]. '''Correlation''' refers to any of a broad class of statistical relationships involving dependence.
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| Familiar examples of dependent phenomena include the correlation between the physical [[human height|statures]] of parents and their offspring, and the correlation between the [[demand curve|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 mild day based on the correlation between electricity demand and weather. In this example there is a [[causality|causal relationship]], because extreme weather causes people to use more electricity for heating or cooling; however, statistical dependence is not sufficient to demonstrate the presence of such a causal relationship (i.e., [[correlation does not imply causation]]).
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| Formally, ''dependence'' refers to any situation in which random variables do not satisfy a mathematical condition of [[independence (probability theory)|probabilistic independence]]. In loose usage, ''correlation'' can refer to any departure of two or more random variables from independence, but technically it refers to any of several more specialized types of relationship between [[conditional expectation|mean values]]. There are several '''correlation coefficients''', often denoted ''ρ'' or ''r'', measuring the degree of correlation. The most common of these is the [[Pearson product-moment correlation coefficient|Pearson correlation coefficient]], which is sensitive only to a linear relationship between two variables (which may exist even if one is a nonlinear function of the other). Other correlation coefficients have been developed to be more [[robust statistics|robust]] than the Pearson correlation – that is, more sensitive to nonlinear relationships.<ref>Croxton, Frederick Emory; Cowden, Dudley Johnstone; Klein, Sidney (1968) ''Applied General Statistics'', Pitman. ISBN 9780273403159 (page 625)</ref><ref>Dietrich, Cornelius Frank (1991) ''Uncertainty, Calibration and Probability: The Statistics of Scientific and Industrial Measurement'' 2nd Edition, A. Higler. ISBN 9780750300605 (Page 331)</ref><ref>Aitken, Alexander Craig (1957) ''Statistical Mathematics'' 8th Edition. Oliver & Boyd. ISBN 9780050013007 (Page 95)</ref> [[Mutual information]] can also be applied to measure dependence between two variables.
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| [[Image:Correlation examples2.svg|thumb|400px|right|Several sets of (''x'', ''y'') points, with the Pearson correlation coefficient of ''x'' and ''y'' for each set. Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of ''Y'' is zero.]]
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| ==Pearson's product-moment coefficient==
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| {{Main|Pearson product-moment correlation coefficient}}
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| The most familiar measure of dependence between two quantities is the [[Pearson product-moment correlation coefficient]], or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by dividing the [[covariance]] of the two variables by the product of their [[standard deviation]]s. [[Karl Pearson]] developed the coefficient from a similar but slightly different idea by [[Francis Galton]].<ref name="thirteenways">J. L. Rodgers and W. A. Nicewander. [http://www.jstor.org/stable/2685263 Thirteen ways to look at the correlation coefficient]. The American Statistician, 42(1):59–66, February 1988.</ref>
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| The population correlation coefficient ρ<sub>''X,Y''</sub> between two [[random variables]] ''X'' and ''Y'' with [[expected value]]s μ<sub>''X''</sub> and μ<sub>''Y''</sub> and [[standard deviation]]s σ<sub>''X''</sub> and σ<sub>''Y''</sub> is defined as:
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| :<math>\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y},</math>
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| where ''E'' is the [[expected value]] operator, ''cov'' means [[covariance]], and, ''corr'' a widely used alternative notation for the correlation coefficient.
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| The Pearson correlation is defined only if both of the standard deviations are finite and both of them are 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(''X'',''Y'') = corr(''Y'',''X'').
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| The Pearson correlation is +1 in the case of a perfect direct(increasing) linear relationship (correlation), −1 in the case of a perfect decreasing (inverse) linear relationship ('''anticorrelation'''),<ref>Dowdy, S. and Wearden, S. (1983). "Statistics for Research", Wiley. ISBN 0-471-08602-9 pp 230</ref> and some value between −1 and 1 in all other cases, indicating the degree of [[linear dependence]] between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.
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| If the variables are [[statistical independence|independent]], Pearson's correlation coefficient is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. For example, suppose the random variable ''X'' is symmetrically distributed about zero, and ''Y'' = ''X''<sup>2</sup>. Then ''Y'' is completely determined by ''X'', so that ''X'' and ''Y'' are perfectly dependent, but their correlation is zero; they are [[uncorrelated]]. However, in the special case when ''X'' and ''Y'' are [[bivariate Gaussian distribution|jointly normal]], uncorrelatedness is equivalent to independence.
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| If we have a series of ''n'' measurements of ''X'' and ''Y'' written as ''x<sub>i</sub>'' and ''y<sub>i</sub>'' where ''i'' = 1, 2, ..., ''n'', then the ''sample correlation coefficient'' can be used to estimate the population Pearson correlation ''r'' between ''X'' and ''Y''. The sample correlation coefficient is written
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| :<math>
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| r_{xy}=\frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{(n-1) s_x s_y}
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| =\frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}
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| {\sqrt{\sum\limits_{i=1}^n (x_i-\bar{x})^2 \sum\limits_{i=1}^n (y_i-\bar{y})^2}},
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| </math>
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| where <span style="text-decoration: overline">x</span> and <span style="text-decoration: overline">y</span> are the sample [[arithmetic mean|means]] of ''X'' and ''Y'', and ''s''<sub>''x''</sub> and ''s''<sub>''y''</sub> are the [[standard deviation#With sample standard deviation|sample standard deviations]] of ''X'' and ''Y''.
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| This can also be written as:
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| :<math>
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| r_{xy}=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}=\frac{n\sum x_iy_i-\sum x_i\sum y_i}
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| {\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.
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| </math>
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| If ''x'' and ''y'' are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range.<ref>{{cite journal|last=Francis|first=DP|coauthors=Coats AJ, Gibson D|title=How high can a correlation coefficient be?|journal=Int J Cardiol|year=1999|volume=69|pages=185–199|doi=10.1016/S0167-5273(99)00028-5|issue=2}}</ref>
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| For the case of a linear model with a single independent variable, the [[Coefficient of determination|coefficient of determination (R squared)]] is the square of r, Pearson's product-moment coefficient .
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| ==Rank correlation coefficients==
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| {{Main|Spearman's rank correlation coefficient|Kendall tau rank correlation coefficient}}
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| [[Rank correlation]] coefficients, such as [[Spearman's rank correlation coefficient]] and [[Kendall's tau|Kendall's rank correlation coefficient (τ)]] measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship. If, as the one variable increases, the other ''decreases'', the rank correlation coefficients will be negative. It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the [[Pearson product-moment correlation coefficient]], and are best seen as measures of a different type of [[association (statistics)|association]], rather than as alternative measure of the population correlation coefficient.<ref name="Yule and Kendall">Yule, G.U and Kendall, M.G. (1950), "An Introduction to the Theory of Statistics", 14th Edition (5th Impression 1968). Charles Griffin & Co. pp 258–270</ref><ref name="Kendall Rank Correlation Methods">Kendall, M. G. (1955) "Rank Correlation Methods", Charles Griffin & Co.</ref>
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| To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers (''x'', ''y''):
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| :(0, 1), (10, 100), (101, 500), (102, 2000).
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| As we go from each pair to the next pair ''x'' increases, and so does ''y''. This relationship is perfect, in the sense that an increase in ''x'' is ''always'' accompanied by an increase in ''y''. This means that we have a perfect rank correlation, and both Spearman's and Kendall's correlation coefficients are 1, whereas in this example Pearson product-moment correlation coefficient is 0.7544, indicating that the points are far from lying on a straight line. In the same way if ''y'' always ''decreases'' when ''x'' ''increases'', the rank correlation coefficients will be −1, while the Pearson product-moment correlation coefficient may or may not be close to −1, depending on how close the points are to a straight line. Although in the extreme cases of perfect rank correlation the two coefficients are both equal (being both +1 or both −1) this is not in general so, and values of the two coefficients cannot meaningfully be compared.<ref name="Yule and Kendall"/> For example, for the three pairs (1, 1) (2, 3) (3, 2) Spearman's coefficient is 1/2, while Kendall's coefficient is 1/3.
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| ==Other measures of dependence among random variables==
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| The information given by a correlation coefficient is not enough to define the dependence structure between random variables.<ref name="wilmottM.com">{{cite journal|authors=Mahdavi Damghani B.|title=The Non-Misleading Value of Inferred Correlation: An Introduction to the Cointelation Model|journal=Wilmott Magazine|year=2013|doi=10.1002/wilm.10252 }}</ref> The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a [[multivariate normal distribution]]. (See diagram above.) In the case of [[elliptical distribution]]s it characterizes the (hyper-)ellipses of equal density, however, it does not completely characterize the dependence structure (for example, a multivariate t-distribution's degrees of freedom determine the level of tail dependence).
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| [[Distance correlation]] and [[Brownian covariance]] / Brownian correlation <ref>Székely, G. J. Rizzo, M. L. and Bakirov, N. K. (2007). "Measuring and testing independence by correlation of distances", ''[[Annals of Statistics]]'', 35/6, 2769–2794. {{doi| 10.1214/009053607000000505}} [http://personal.bgsu.edu/~mrizzo/energy/AOS0283-reprint.pdf Reprint]
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| </ref><ref>Székely, G. J. and Rizzo, M. L. (2009). "Brownian distance covariance", ''Annals of Applied Statistics'', 3/4, 1233–1303. {{doi| 10.1214/09-AOAS312}} [http://personal.bgsu.edu/~mrizzo/energy/AOAS312.pdf Reprint] | |
| </ref> were introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables; zero distance correlation and zero Brownian correlation imply independence.
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| The [[correlation ratio]] is able to detect almost any functional dependency{{citation needed|date=August 2011}}{{clarify|reason=doesn't seem true for correlation ratio as defined in article of that name|date=August 2011}}, and the [[Entropy (information theory)|entropy]]-based [[mutual information]], [[total correlation]] and [[dual total correlation]] are capable of detecting even more general dependencies. These are sometimes referred to as multi-moment correlation measures{{citation needed |date=August 2011}}, in comparison to those that consider only second moment (pairwise or quadratic) dependence.
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| The [[polychoric correlation]] is another correlation applied to ordinal data that aims to estimate the correlation between theorised latent variables.
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| One way to capture a more complete view of dependence structure is to consider a [[copula (statistics)|copula]] between them.
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| The [[coefficient of determination]] generalizes the correlation coefficient for relationships beyond [[simple linear regression]].
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| ==Sensitivity to the data distribution==
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| The degree of dependence between variables ''X'' and ''Y'' does not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship between ''X'' and ''Y'', most correlation measures are unaffected by transforming ''X'' to ''a'' + ''bX'' and ''Y'' to ''c'' + ''dY'', where ''a'', ''b'', ''c'', and ''d'' are constants. This is true of some correlation statistics as well as their population analogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant to [[monotone function|monotone transformations]] of the marginal distributions of ''X'' and/or ''Y''.
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| [[Image:correlation range dependence.svg|300px|right|thumb|[[Pearson product moment correlation coefficient|Pearson]]/[[Spearman's rank correlation coefficient|Spearman]] correlation coefficients between ''X'' and ''Y'' are shown when the two variables' ranges are unrestricted, and when the range of ''X'' is restricted to the interval (0,1).]]Most correlation measures are sensitive to the manner in which ''X'' and ''Y'' are sampled. Dependencies tend to be stronger if viewed over a wider range of values. Thus, if we consider the correlation coefficient between the heights of fathers and their sons over all adult males, and compare it to the same correlation coefficient calculated when the fathers are selected to be between 165 cm and 170 cm in height, the correlation will be weaker in the latter case. Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations.<ref>{{cite book|last=Thorndike|first=Robert Ladd|title=Research problems and techniques (Report No. 3)|year=1947|publisher=US Govt. print. off.|location=Washington DC}}</ref>
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| Various correlation measures in use may be undefined for certain joint distributions of ''X'' and ''Y''. For example, the Pearson correlation coefficient is defined in terms of [[moment (mathematics)|moments]], and hence will be undefined if the moments are undefined. Measures of dependence based on [[quantile]]s are always defined. Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being [[bias of an estimator|unbiased]], or [[consistent estimator|asymptotically consistent]], based on the spatial structure of the population from which the data were sampled.
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| Sensitivity to the data distribution can be used to an advantage. For example, [[scaled correlation]] is designed to use the sensitivity to the range in order to pick out correlations between fast components of time series.<ref name = "Nikolicetal">Nikolić D, Muresan RC, Feng W, Singer W (2012) Scaled correlation analysis: a better way to compute a cross-correlogram. ''European Journal of Neuroscience'', pp. 1–21, {{DOI|10.1111/j.1460-9568.2011.07987.x }}</ref> By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.
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| ==Correlation matrices==
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| The correlation matrix of ''n'' random variables ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> is the ''n'' × ''n'' matrix whose ''i'',''j'' entry is corr(''X''<sub>''i''</sub>, ''X''<sub>''j''</sub>). If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the [[covariance matrix]] of the [[standardized variable|standardized random variables]] ''X''<sub>''i''</sub> / σ (''X''<sub>''i''</sub>) for ''i'' = 1, ..., ''n''. This applies to both the matrix of population correlations (in which case "σ" is the population standard deviation), and to the matrix of sample correlations (in which case "σ" denotes the sample standard deviation). Consequently, each is necessarily a [[positive-semidefinite matrix]].
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| The correlation matrix is symmetric because the correlation between ''X''<sub>''i''</sub> and ''X''<sub>''j''</sub> is the same as the correlation between ''X''<sub>''j''</sub> and ''X''<sub>''i''</sub>.
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| ==Common misconceptions==
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| ===Correlation and causality===
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| {{Main|Correlation does not imply causation}}
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| The conventional dictum that "[[correlation does not imply causation]]" means that correlation cannot be used to infer a causal relationship between the variables.<ref>{{cite journal | last=Aldrich | first=John | journal=Statistical Science | volume=10 | issue=4 | year=1995 | pages=364–376 | title=Correlations Genuine and Spurious in Pearson and Yule | jstor=2246135 | doi=10.1214/ss/1177009870}}</ref> This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with [[identity (mathematics)|identity]] relations ([[tautology (logic)|tautologies]]), where no causal process exists. Consequently, establishing a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).
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| A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health, or does good health lead to good mood, or both? Or does some other factor underlie both? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.
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| === Correlation and linearity ===
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| [[Image:Anscombe's quartet 3.svg|thumb|325px|right|Four sets of data with the same correlation of 0.816]]
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| The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the [[conditional expectation|conditional mean]] of ''Y'' given ''X'', denoted E(''Y''|''X''), is not linear in ''X'', the correlation coefficient will not fully determine the form of E(''Y''|''X'').
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| The image on the right shows [[scatterplot]]s of [[Anscombe's quartet]], a set of four different pairs of variables created by [[Francis Anscombe]].<ref>{{cite journal | last=Anscombe | first=Francis J. | year=1973 | title=Graphs in statistical analysis | journal=The American Statistician | volume=27 | pages=17–21 | jstor=2682899 | doi=10.2307/2682899}}</ref> The four ''y'' variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (''y'' = 3 + 0.5''x''). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one [[outlier]] which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.
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| These examples indicate that the correlation coefficient, as a summary statistic, cannot replace visual examination of the data. Note that the examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a [[normal distribution]], but this is not correct.<ref name="thirteenways"/>
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| ==Bivariate normal distribution==
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| If a pair (''X'', ''Y'') of random variables follows a [[bivariate normal distribution]], the conditional mean E(''X''|''Y'') is a linear function of ''Y'', and the conditional mean E(''Y''|''X'') is a linear function of ''X''. The correlation coefficient ''r'' between ''X'' and ''Y'', along with the [[Marginal distribution|marginal]] means and variances of ''X'' and ''Y'', determines this linear relationship:
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| :<math>
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| E(Y\mid X) = E(Y) + r\sigma_y\frac{X-E(X)}{\sigma_x},
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| </math>
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| where ''E(X)'' and ''E(Y)'' are the expected values of ''X'' and ''Y'', respectively, and σ<sub>''x''</sub> and σ<sub>''y''</sub> are the standard deviations of ''X'' and ''Y'', respectively.
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| ==Partial correlation==
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| {{Main|Partial correlation}}
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| If a population or data-set is characterized by more than two variables, a [[partial correlation]] coefficient measures the strength of dependence between a pair of variables that is not accounted for by the way in which they both change in response to variations in a selected subset of the other variables.
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| ==See also==
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| {{Commons category|Correlation and dependence}}
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| {{Portal|Statistics}}
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| <div style="-moz-column-count:3; column-count:3;">
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| * [[Association (statistics)]]
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| * [[Autocorrelation]]
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| * [[Canonical correlation]]
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| * [[Coefficient of determination]]
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| * [[Cointegration]]
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| * [[Cointelation]]
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| * [[Concordance correlation coefficient]]
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| * [[Cophenetic correlation]]
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| * [[Copula (probability theory)|Copula]]
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| * [[Correlation function]]
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| * [[Covariance and correlation]]
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| * [[Cross-correlation]]
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| * [[Ecological correlation]]
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| * [[Fraction of variance unexplained]]
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| * [[Genetic correlation]]
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| * [[Goodman and Kruskal's lambda]]
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| * [[Illusory correlation]]
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| * [[Interclass correlation]]
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| * [[Intraclass correlation]]
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| * [[Modifiable areal unit problem]]
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| * [[Multiple correlation]]
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| * [[Point-biserial correlation coefficient]]
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| * [[Quadrant count ratio]]
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| * [[Statistical arbitrage]]
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| * [[Subindependence]]
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| </div>
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| ==References==
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| {{Reflist|colwidth=35em}}
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| ==Further reading==
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| * {{cite book |author=Cohen, J., Cohen P., West, S.G., & Aiken, L.S. |year=2002 |title=Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.) |publisher=Psychology Press |isbn= 0-8058-2223-2 }}
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| * {{springer|title=Correlation (in statistics)|id=p/c026560}}
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| ==External links==
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| {{Wikiversity|Correlation}}
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| {{Wiktionary|correlation}}
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| {{Wiktionary|dependence}}
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| * [http://mathworld.wolfram.com/CorrelationCoefficient.html MathWorld page on (cross-) correlation coefficient(s) of a sample.]
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| * [http://peaks.informatik.uni-erlangen.de/cgi-bin/usignificance.cgi Compute Significance between two correlations] – A useful website if one wants to compare two correlation values.
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| *[http://www.mathworks.com/matlabcentral/fileexchange/20846 A MATLAB Toolbox for computing Weighted Correlation Coefficients]
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| * [http://www.docstoc.com/docs/3530180/Proof-that-the-Sample-Bivariate-Correlation-Coefficient-has-Limits-(Plus-or-Minus)-1 Proof that the Sample Bivariate Correlation Coefficient has Limits ±1]
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| * [http://nagysandor.eu/AsimovTeka/correlation_en/index.html Interactive Flash simulation on the correlation of two normally distributed variables.] Author: Juha Puranen.
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| {{Statistics}}
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| {{DEFAULTSORT:Correlation And Dependence}}
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| [[Category:Covariance and correlation]]
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| [[Category:Statistical dependence]]
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| [[Category:Dimensionless numbers]]
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