P-value

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In statistical hypothesis testing, the p-value is the probability of obtaining a result at least as "impressive" as that obtained, assuming the truth of the null hypothesis that the finding was the result of chance alone. The fact that p-values are based on this assumption is crucial to their correct interpretation.

More technically, the p-value of an observed value t_observed of some random variable T used as a test statistic is the probability that, given that the null hypothesis is true, T will assume a value as or more unfavorable to the null hypothesis as the observed value t_observed. "More unfavorable to the null hypothesis" can in some cases mean greater than, in some cases less than, and in some cases further away from a specified center.

1 Example
2 Interpretation
3 Frequent misunderstandings
4 External links
5 Additional reading

[edit] Example

For example, say an experiment is performed to determine if a coin flip is fair (50% chance of landing heads or tails), or unfairly biased toward heads (> 50% chance of landing heads). If we choose a two-tailed test then the null hypothesis is that the coin is fair, and that any deviations from the 50% rate can be ascribed to chance alone. Suppose that the experimental results show the coin turning up heads 14 times out of 20 total flips. The p-value of this result would be the chance of a fair coin landing on heads at least 14 times out of 20 flips (as larger values in this case are also less favorable to the null hypothesis of a fair coin) or at most 6 times out of 20 flips. In this case the random variable, T, is a binomial distribution. The probability that 20 flips of a fair coin would result in 14 or more heads is 0.0577. Since this is a two-tailed test, the probability that 20 flips of the coin would result in 14 or more heads or 6 or less heads is 0.115.

Generally, the smaller the P value, the more people there are who would be willing to say that the results came from a biased coin.

[edit] Interpretation

Generally, one rejects the null hypothesis if the p-value is smaller than or equal to the significance level, often represented by the Greek letter $α$ (alpha). If the level is 0.05, then the results are only 5% likely to be as extraordinary as just seen, given that the null hypothesis is true.

In the above example, the calculated p-value exceeds 0.05, and thus the null hypothesis - that the observed result of 14 heads out of 20 flips can be ascribed to chance alone - is not rejected. Such a finding is often stated as being "not statistically significant at the 5 % level".

However, had a single extra head been obtained, the resulting p-value would be 0.02. This time the null hypothesis - that the observed result of 15 heads out of 20 flips can be ascribed to chance alone - is rejected. Such a finding would be described as being "statistically significant at the 5 % level".

There is often an alternative hypothesis, but the construction of the test does not allow for 'supporting' a specific alternative.

Critics of p-values point out that the criterion used to decide "statistical significance" is based on the somewhat arbitrary choice of level (often set at 0.05). A proposed replacement for the p-value is p-rep, which is the probability that an effect can be replicated.

[edit] Frequent misunderstandings

There are several common misunderstandings about p-values.

The p-value is not the probability that the null hypothesis is true, (claimed to justify the "rule" of considering as significant p-values closer to 0 (zero)).

In fact, frequentist statistics does not, and cannot, attach probabilities to hypotheses. Comparison of Bayesian and classical approaches shows that a p-value can be very close to zero while the posterior probability of the null is very close to unity. This is the Jeffreys-Lindley paradox.
The p-value is not the probability that a finding is "merely a fluke" (again, justifying the "rule" of considering small p-values as "significant").

As the calculation of a p-value is based on the assumption that a finding is the product of chance alone, it patently cannot simultaneously be used to gauge the probability of that assumption being true.
The p-value is not the probability of falsely rejecting the null hypothesis. This error is a version of the so-called prosecutor's fallacy.
The p-value is not the probability that a replicating experiment would not yield the same conclusion.
1 − (p-value) is not the probability of the alternative hypothesis being true (see (1)).
The significance level of the test is not determined by the p-value.

The significance level of a test is decided upon before any data are collected, and does not depend on the p-value or any other statistic calculated after the test has been performed.

[edit] External links

Free p-Value Calculator for the Chi-Square test from Daniel Soper's Free Statistics Calculators website. Computes the one-tailed probability value of a chi-square test (i.e., the area under the chi-square distribution from the chi-square value to infinity), given the chi-square value and the degrees of freedom.
Free p-Value Calculator for the Fisher F-test from Daniel Soper's Free Statistics Calculators website. Computes the probability value of an F-test, given the F-value, numerator degrees of freedom, and denominator degrees of freedom.
Free p-Value Calculator for the Student t-test from Daniel Soper's Free Statistics Calculators website. Computes the one-tailed and two-tailed probability values of a t-test, given the t-value and the degrees of freedom.