Type I and II Errors

In statistics, the terms Type I error (also, α error, or false positive) and type II error (β error, or a false negative) are used to describe possible errors made in a statistical decision process. In 1928, Jerzy Neyman (1894-1981) and Egon Pearson (1895-1980), both eminent statisticians, discussed the problems associated with "deciding whether or not a particular sample may be judged as likely to have been randomly drawn from a certain population" (1928/1967, p.1): and identified "two sources of error", namely:
(α) the error of rejecting a hypothesis that should have been accepted, and( β) the error of accepting a hypothesis that should have been rejected

In 1930, they elaborated on these two sources of error, remarking that "in testing hypotheses two considerations must be kept in view, (1) we must be able to reduce the chance of rejecting a true hypothesis to as low a value as desired; (2) the test must be so devised that it will reject the hypothesis tested when it is likely to be false"
When an observer makes a Type I error, they are observing a statistical difference when in truth there is none (rejecting a null hypothesis when it is actually true). For example, a pregnancy test with a positive result (indicating that the person taking the test is pregnant) has produced a false positive in the case where the person is not pregnant. A Type II error, or a "false negative", is the error of failing to reject a null hypothesis when the alternative hypothesis is the true state of nature. For example, a type II error occurs if a pregnancy test reports negative when the person is, in fact, pregnant.

Scientists recognize two different sorts of error:

• Statistical error: the difference between a computed, estimated, or measured value and the true, specified, or theoretically correct value (see errors and residuals in statistics) that is caused by random, and inherently unpredictable fluctuations in the measurement apparatus or the system being studied.
  
• Systematic error: the difference between a computed, estimated, or measured value and the true, specified, or theoretically correct value that is caused by non-random fluctuations from an unknown source and which, once identified, can usually be eliminated.
  

Statistical error: Type I and Type II

Statisticians speak of two significant sorts of statistical error. The context is that there is a "null hypothesis" which corresponds to a presumed default "state of nature", e.g., that an individual is free of disease, that an accused is innocent, or that a potential login candidate is not authorized. Corresponding to the null hypothesis is an "alternative hypothesis" which corresponds to the opposite situation, that is, that the individual has the disease, that the accused is guilty, or that the login candidate is an authorized user. The goal is to determine accurately if the null hypothesis can be discarded in favor of the alternative. A test of some sort is conducted (a blood test, a legal trial, a login attempt), and data is obtained. The result of the test may be negative (that is, it does not indicate disease, guilt, or authorized identity). On the other hand, it may be positive (that is, it may indicate disease, guilt, or identity). If the result of the test does not correspond with the actual state of nature, then an error has occurred, but if the result of the test correspods with the actual state of nature, then a correct decision has been made.
There are two kinds of error, classified as "Type I error" and "Type II error," depending upon which hypothesis has incorrectly been identified as the true state of nature.

Type I error

Type I error, also known as an "error of the first kind", an α error, or a "false positive": the error of rejecting a null hypothesis when it is actually true. Plainly speaking, it occurs when we are observing a difference when in truth there is none.
A false positive normally means that a test claims something to be positive, when that is not the case. For example, a pregnancy test with a positive result (indicating that the woman taking the test is pregnant) has produced a false positive in the case where the woman is not pregnant.

Type II error

Type II error, also known as an "error of the second kind", a β error, or a "false negative":the error of failing to reject a null hypothesis when the alternative hypothesis is the true state of nature. In other words, this is the error of failing to observe a difference when in truth there is one. This type of error can only occur when the statistician fails to reject the null hypothesis. In the example of a pregnancy test, a type II error occurs if the test reports false when the woman is, in fact, pregnant.

Type I and type II errors

Over time, the notion of these two sources of error has been universally accepted. They are now routinely known as type I errors and type II errors. For obvious reasons, they are very often referred to as false positives and false negatives respectively. The terms are now commonly applied in much wider and far more general sense than Neyman and Pearson's original specific usage, as follows:

• Type I errors (the "false positive"): the error of rejecting the null hypothesis given that it is actually true; e.g., A court finding a person guilty of a crime that they did not actually commit.
  
• Type II errors (the "false negative"): the error of failing to reject the null hypothesis given that the alternative hypothesis is actually true; e.g., A court finding a person not guilty of a crime that they did actually commit.
  

These examples illustrate the ambiguity, which is one of the dangers of this wider use: They assume the speaker is testing for guilt; they could also be used in reverse, as testing for innocence; or two tests could be involved, one for guilt, the other for innocence. (This ambiguity is one reason for the Scottish legal system's third possible verdict:)
The following tables illustrate the conditions.

Condition Present + Positive result = True Positive	Condition absent + Positive result = False Positive Type I error
Condition present + Negative result = False (invalid) Negative Type II error	Condition absent + Negative result = True (accurate) Negative

True Positive	False Positive (i.e. infection reported but not present) Type I error
False Negative (i.e. infection not detected) Type II error	True Negative

Example, testing for guilty/not-guilty:

True Positive	False Positive (i.e. guilt reported unfairly) Type I error
False Negative (i.e. guilt not detected) Type II error	True Negative

True Positive	False Positive (i.e. guilty but not caught) Type I error
False Negative (i.e. innocent but condemned) Type II error	True Negative

Note that, when referring to test results, the terms true and false are used in two different ways: the state of the actual condition (true=present versus false=absent); and the accuracy or inaccuracy of the test result (true positive, false positive, true negative, false negative). This is confusing to some readers. To clarify the examples above, we have used present/absent rather than true/false to refer to the actual condition being tested.