Type I error (or, error of the first kind) and Type II error (or, error of the second kind) are precise technical terms used in statistics to describe particular flaws in a testing process, where a truenull hypothesis was incorrectly rejected (Type I error) or where one fails to reject a false null hypothesis (Type II error).
The terms are also used in a more general way by social scientists and others to refer to flaws in reasoning. This article is specifically devoted to the statistical meanings of those terms and the technical issues of the statistical errors that those terms describe.

Statistical test theory
In statistical test theory the notion of statistical error is an integral part of hypothesis testing. The test requires an unambiguous statement of a null hypothesis, which usually corresponds to a default "state of nature", for example "this person is healthy", "this accused is not guilty" or "this product is not broken". An alternative hypothesis is the negation of null hypothesis, for example, "this person is not healthy", "this accused is guilty" or "this product is broken". The result of the test may be negative, relative to null hypothesis (not healthy, guilty, broken) or positive (healthy, not guilty, not broken). If the result of the test corresponds with reality, then a correct decision has been made. However, if the result of the test does not correspond with reality, then an error has occurred. Due to the statistical nature of a test, the result is never, except in very rare cases, free of error. Two types of error are distinguished: type I error and type II error.
[edit]Type I error
A type I error, also known as an error of the first kind, is the wrong decision that is made when a test rejects a true null hypothesis (H0). A type I error may be compared with a so called false positive in other test situations. Type I error can be viewed as the error of excessive credulity.[1] In terms of folk tales, an investigator may be "crying wolf" (raising a false alarm) without a wolf in sight (H0: no wolf).
The rate of the type I error is called the size of the test and denoted by the Greek letter α (alpha). It usually equals the significance level of a test. In the case of a simple null hypothesis α is the probability of a type I error. If the null hypothesis is composite, α is the maximum (supremum) of the possible probabilities of a type I error.
[edit]Type II error
A type II error, also known as an error of the second kind, is the wrong decision that is made when a test accepts a false null hypothesis. A type II error may be compared with a so-called false negative in other test situations. Type II error can be viewed as the error of excessive skepticism.[1] In terms of folk tales, an investigator may fail to see the wolf ("failing to raise an alarm"; see Aesop's story of The Boy Who Cried Wolf). Again, H0: no wolf.
The rate of the type II error is denoted by the Greek letter β (beta) and related to the power of a test (which equals 1 − β).
What we actually call type I or type II error depends directly on the null hypothesis. Negation of the null hypothesis causes type I and type II errors to switch roles.
The goal of the test is to determine if the null hypothesis can be rejected. A statistical test can either reject (prove false) or fail to reject (fail to prove false) a null hypothesis, but never prove it true (i.e., failing to reject a null hypothesis does not prove it true).
[edit]Example
As it is conjectured that adding fluoride to toothpaste protects against cavities, the null hypothesis of no effect is tested. When the null hypothesis is true (i.e., there is indeed no effect, but the data give rise to rejection of this hypothesis, falsely suggesting that adding fluoride is effective against cavities), a type I error has occurred.
A type II error occurs when the null hypothesis is false (i.e., adding fluoride is actually effective against cavities, but the data are such that the null hypothesis cannot be rejected, failing to prove the existing effect).
In colloquial usage type I error can be thought of as "convicting an innocent person" and type II error "letting a guilty person go free".
Tabularised relations between truth/falseness of the null hypothesis and outcomes of the test: | Null hypothesis (H0) is true | Null hypothesis (H0) is false | Reject null hypothesis | Type I error
False positive | Correct outcome
True positive | Fail to reject null hypothesis | Correct outcome
True negative | Type II error
False negative |
[edit]Understanding Type I and Type II errors
From the Bayesian point of view, a type I error is one that looks at information that should not substantially change one's prior estimate of probability, but does. A type II error is that one looks at information which should change one's estimate, but does not. (Though the null hypothesis is not quite the same thing as one's prior estimate, it is, rather, one's pro forma prior estimate.)
Hypothesis testing is the art of testing whether a variation between two sample distributions can be explained by chance or not. In many practical applications type I errors are more delicate than type II errors. In these cases, care is usually focused on minimizing the occurrence of this statistical error. Suppose, the probability for a type I error is 1% , then there is a 1% chance that the observed variation is not true. This is called the level of significance, denoted with the Greek letter α (alpha). While 1% might be an acceptable level of significance for one application, a different application can require a very different level. For example, the standard goal of six sigma is to achieve precision to 4.5 standard deviations above or below the mean. This means that only 3.4 parts per million are allowed to be deficient in a normally distributed process
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[edit]Consequences of type I and type II errors
Both types of errors are problems for individuals, corporations, and data analysis. A false positive (with null hypothesis of health) in medicine causes unnecessary worry or treatment, while a false negative gives the patient the dangerous illusion of good health and the patient might not get an available treatment. A false positive in manufacturing quality control (with a null hypothesis of a product being well-made), discards a product, which is actually well-made, while a false negative stamps a broken product as operational. A false positive (with null hypothesis of no effect) in scientific research suggest an effect, which is not actually there, while a false negative fails to detect an effect that is there.
Based on the real-life consequences of an error, one type may be more serious than the other. For example, NASA engineers would prefer to throw out an electronic circuit that is really fine (null hypothesis H0: not broken; reality: not broken; action: thrown out; error: type I, false positive) than to use one on a spacecraft that is actually broken (null hypothesis H0: not broken; reality: broken; action: use it; error: type II, false negative). In that situation a type I error raises the budget, but a type II error would risk the entire mission.
Alternatively, criminal courts set high bar for proof and procedure and sometimes release someone who is guilty (null hypothesis: innocent; reality: guilty; test find: not guilty; action: release; error: type II, false negative) rather than convict someone who is innocent (null hypothesis: innocent; reality: not guilty; test find: guilty; action: convict; error: type I, false positive). Each system makes its own choice regarding where to draw the line.
Minimizing errors of decision is not a simple issue; for any given sample size the effort to reduce one type of error generally results in increasing the other type of error. The only way to minimize both types of error, without just improving the test, is to increase the sample size, and this may or may not be feasible.