Changes

'''Correlation''' describes the relationship of two factors to one another (see [[cause and effect]]). In common usage, it denotes an association of one variable with another in quite general terms; for example, one might say, "success is correlated with hard work". In mathematics, however, and in science and engineering, which make use of mathematical concepts, correlation is a technical term with a precise definition.

Correlation refers to a statistical relationship between two continuous variables. In common speech if two factors are "correlated" it is often incorrectly taken to mean that one factor causes the other (see [[cause and effect]]). Importantly, however, a correlation only means that two factors tend to occur together and is not proof of causation. The co-occurrence of two factors may arise because one of them causes the other, but a correlation can also occur because both are caused by an unknown third factor (see [[correlation is not causation]]). For example, the prices of two unrelated goods might increase during a period of [[inflation]]; the two prices are correlated with each other because the co-occur but were both caused by the same third factor.

Correlation must be distinguished from causation (see article on [[correlation is not causation]]). When one factor changes and another factor changes with it, there is usually a direct relationship between the two factors, observed as a correlation. Alternatively, both changed could be the result of changes in a third factor. For example, the prices of two unrelated goods might increase during a period of [[inflation]]; the two price rises are correlated with each other but neither has caused the other.

Once a correlation has been noted, it is often possible to determine if there is a causal relationship between the variables, though this usually requires additional research by scientists. Research is particularly necessary if there is not a clear ordering of the factors in time. For example, a correlation between the heights of fathers and sons may be produced by fathers directly causing their sons' heights (e.g. through genetics) but is unlikely to be the result of sons causing their fathers' heights. The latter would require a factor (i.e. height of sons) to cause something that occurred earlier in time (i.e. the height of the fathers), and is therefore implausible. In contrast, a correlation between cancer and consumption of alcohol might imply that alcohol causes cancer but could also imply that cancer causes the consumption of alcohol, for example to reduce the pain of the cancer. As there is no logically necessary ordering to these two factors (i.e. the cancer or the consumption could have started first) additional research is necessary to determine causation.

Once a correlation is established, scientists may conduct research to determine causation. Are respiration deaths causing air pollution, or is it the other way around? It is easy to determine that sickness among the elderly does not cause air pollution. Rather, it is chemicals like sulfur dioxide (typically from coal burning [[power plant]]s) which are the culprits. Cities and states measure the amount of pollutants in the air and epidemiologists can use these data, comparing them to the number of people who develop respiratory diseases.

It is sometimes also possible to reject particular causal arguments based on theoretical or substantive knowledge even if there is no obvious ordering of the events in time. For example, are deaths from respiratory disease causing air pollution, or is air pollution causing deaths from respiratory disease? There is no theoretical or substantive reason to think that human deaths from respiratory disease can degrade air quality, but there are both theoretical and substantive reasons to think that chemicals like sulfur dioxide (typically from coal burning [[power plant]]s) can produce respiratory disease. An association between the two is therefore suggestive that air pollution causes respiratory disease though it remains possible that both might be caused by a third factor.

 

Regulations which restrict air pollution are often made on the basis of these correlations, as well as on the cause and effect relationships which the correlations help scientists to discover. However, activists have sometimes selectively used data to create the appearance of a correlation where none exists. [http://www.robinsoncurriculum.com/view/rc/s31p59.htm] Likewise, manipulation of data can be used to obscure a correlation that does exist. It is therefore important that good practices be used in the collection and analysis of data to ensure that reported correlations, or the lack thereof, are reliable

== Formal definition ==

== Formal definition ==

'''t'''= (r)√(n-2)/√(1-r²)

'''t'''= (r)√(n-2)/√(1-r²)

The results of this test should be evaluated against the standard t-distribution with degrees of freedom equal to n-2. As usual in significance tests, a result described as significant (i.e. a low value of ''P'' in the ''t'' test) means that the chance of getting a correlation coefficient at least as large (whichever the direction) as that observed, if there is in fact no correlation, is small. A statistically significant result can therefore arise if the actual correlation is strong, even if the dataset is small, or if the actual correlation is weak but many data are observed.

The results of this test should be evaluated against the standard t-distribution with degrees of freedom equal to n-2. As usual in significance tests, a result described as significant (i.e. typically a probability of occurring naturally of less than 5%) means that the chance of getting a correlation coefficient at least as large (whichever the direction) as that observed, if there is in fact no correlation, is small. A statistically significant result can therefore arise if the actual correlation is strong, even if the dataset is small, or if the actual correlation is weak but many data are observed.

It is important to note that the correlation coefficient should not be calculated when either of the variables is not continuous. That is, when they do not vary continuously and have a meaningful zero. As such, correlating a dichotomous variable (e.g. sex) with a ratio variable (e.g. IQ) is inappropriate and will return uninterpretable results.

It is important to note that the correlation coefficient should not be calculated when either of the variables is not continuous. That is, when they do not vary continuously and have a meaningful zero. As such, correlating a dichotomous variable (e.g. sex) with a ratio variable (e.g. IQ) is inappropriate and will return uninterpretable results.