In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. "A statistically significant difference" simply means there is statistical evidence that there is a difference; it does not mean the difference is necessarily large, important, or significant in the common meaning of the word. This article is about the field of statistics. ... Look up chance in Wiktionary, the free dictionary. ...

The significance level of a test is a traditional frequentist statistical hypothesis testing concept. In simple cases, it is defined as the probability of making a decision to reject the null hypothesis when the null hypothesis is actually true (a decision known as a Type I error, or "false positive determination"). The decision is often made using the p-value: if the p-value is less than the significance level, then the null hypothesis is rejected. The smaller the p-value, the more significant the result is said to be. Statistical regularity has motivated the development of the relative frequency concept of probability. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ... In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ... In statistical hypothesis testing, a Type I error consists of rejecting a null hypothesis that is true, in other words finding a result to have statistical significance when this has in fact happened by chance. ... In statistical hypothesis testing, the p-value of a random variable T used as a test statistic is the probability that T will assume a value at least as extreme as the observed value tobserved, given that a null hypothesis being considered is true. ...

In more complicated, but practically important cases, the significance level of a test is a probability such that the probablility of making a decision to reject the null hypothesis when the null hypothesis is actually true is no more than the stated probability. This allows for those applications where the probability of deciding to reject may be much smaller than the significance level for some sets of assumptions encompassed within the null hypothesis. In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ...

Use in practice

The significance level is usually represented by the Greek symbol, α (alpha). Popular levels of significance are 5%, 1% and 0.1%. If a test of significance gives a p-value lower than the α-level, the null hypothesis is rejected. Such results are informally referred to as 'statistically significant'. For example, if someone argues that "there's only one chance in a thousand this could have happened by coincidence," a 0.1% level of statistical significance is being implied. The lower the significance level, the stronger the evidence. One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ...

In some situations it is convenient to express the statistical significance as 1 − α. The value 1 − α is called the confidence, and the value of α is occasionally called the risk. So 95% confidence level is usually equivalent to a signifcance level of 5%, which can also be called a risk of 5%. In general, when interpreting a stated significance, one must be careful to note what, precisely, is being tested statistically.

Different α-levels have different advantages and disadvantages. Smaller α-levels give greater confidence in the determination of significance, but run greater risks of failing to reject a false null hypothesis (a Type II error, or "false negative determination"), and so have less statistical power. The selection of an α-level inevitably involves a compromise between significance and power, and consequently between the Type I error and the Type II error. In statistical hypothesis testing, a Type II error consists of failing to reject an invalid null hypothesis (i. ... The power of a statistical test is the probability that the test will reject a false null hypothesis (that it will not make a Type II error). ... In statistical hypothesis testing, a Type I error consists of rejecting a null hypothesis that is true, in other words finding a result to have statistical significance when this has in fact happened by chance. ... In statistical hypothesis testing, a Type II error consists of failing to reject an invalid null hypothesis (i. ...

In some fields, for example nuclear and particle physics, it is common to express statistical significance in units of "σ" (sigma), the standard deviation of a Gaussian distribution. A statistical significance of "nσ" can be converted into a value of α via use of the error function: In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... Probability density function of Gaussian distribution (bell curve). ... Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ...

The use of σ is motivated by the ubiquitous emergence of the Gaussian distribution in measurement uncertainties. For example, if a theory predicts a parameter to have a value of, say, 100, and one measures the parameter to be 109 ± 3, then one might report the measurement as a "3σ deviation" from the theoretical prediction. In terms of α, this statement is equivalent to saying that "assuming the theory is true, the likelihood of obtaining the experimental result by coincidence is 0.27%" (since 1 − erf(3/√2) = 0.0027).

Fixed significance levels such as those mentioned above may be regarded as useful in exploratory data analyses. However, modern statistical advice is that, where the outcome of a test is essentially the final outcome of an experiment or other study, the p-value should be quoted explicitly. And, importantly, it should be quoted whether or not the p-value is judged to be significant. This is to allow maximum information to be transfered from a summary of the study into meta-analyses. In statistics, a meta-analysis combines the results of several studies that address a set of related research hypotheses. ...

Pitfalls

A common misconception is that a statistically significant result is always of practical significance, or demonstrates a large effect in the population. Unfortunately, this problem is commonly encountered in scientific writing. Given a sufficiently large sample, extremely small and non-notable differences can be found to be statistically significant, and statistical significance says nothing about the practical significance of a difference.

One of the more common problems in significance testing is the tendency for multiple comparisons to yield spurious significant differences even where the null hypothesis is true. For instance, in a study of twenty comparisons, using an α-level of 5%, one comparison will likely yield a significant result despite the null hypothesis being true. In these cases p-values are adjusted in order to control either the false discovery rate or the familywise error rate. In statistics, the multiple comparisons problem tests null hypotheses stating that the averages of several disjoint populations are equal to each other (homogeneous). ... False discovery rate (FDR) control is a statistical method used in multiple hypothesis testing to correct for multiple comparisons. ... In statistics, familywise error rate (FWER) is the probability of making one or more false discoveries, or type I errors among all the hypotheses when performing multiple pairwise tests[1][2]. // The m specific hypotheses of interest are assumed to be known in advance, but the numbers of true null...

An additional problem is that frequentist analyses of p-values are considered by some to overstate "statistical significance".^[1][2] See Bayes factor for details. Statistical regularity has motivated the development of the relative frequency concept of probability. ... In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing. ...

Yet another common pitfall often happens when a researcher writes the ambiguous statement "we found no statistically significant difference," which is then misquoted by others as "they found that there was no difference." Actually, statistics cannot be used to prove that there is exactly zero difference between two populations. Failing to find evidence that there is a difference does not constitute evidence that there is no difference. This principle is sometimes described by the maxim "Absence of evidence is not evidence of absence."

According to J. Scott Armstrong, attempts to educate researchers on how to avoid pitfalls of using statistical significance have had little success. In the papers "Significance Tests Harm Progress in Forecasting,"^[3] and "Statistical Significance Tests are Unnecessary Even When Properly Done,"^[4] Armstrong makes the case that even when done properly, statistical significance tests are of no value. A number of attempts failed to find empirical evidence supporting the use of significance tests. Tests of statistical significance are harmful to the development of scientific knowledge because they distract researchers from the use of proper methods. Armstrong suggests authors should avoid tests of statistical significance; instead, they should report on effect sizes, confidence intervals, replications/extensions, and meta-analyses. J. Scott Armstrong (born March 26, 1937), Ph. ... In statistics, effect size is a measure of the strength of the relationship between two variables. ... In statistics, a confidence interval (CI) for a population parameter is an interval with an associated probability p that is generated from a random sample of an underlying population such that if the sampling was repeated numerous times and the confidence interval recalculated from each sample according to the same... In statistics, replication is the repetition of the creation of a phenomenon so that the variability associated with the phenomenon can be estimated. ... Look up extension in Wiktionary, the free dictionary. ... In statistics, a meta-analysis combines the results of several studies that address a set of related research hypotheses. ...

Signal–noise ratio conceptualisation of significance

Statistical significance can be considered to be the confidence one has in a given result. In a comparison study, it is dependent on the relative difference between the groups compared, the amount of measurement and the noise associated with the measurement. In other words, the confidence one has in a given result being non-random (i.e. it is not a consequence of chance) depends on the signal-to-noise ratio (SNR) and the sample size. Look up chance in Wiktionary, the free dictionary. ... Signal-to-noise ratio (often abbreviated SNR or S/N) is an electrical engineering concept defined as the ratio of a signal power to the noise power corrupting the signal. ...

Expressed mathematically, the confidence that a result is not by random chance is given by the following formula by Sackett:^[5]

For clarity, the above formula is presented in tabular form below.

Dependence of confidence with noise, signal and sample size (tabular form)

In words, the dependence of confidence is high if the noise is low and/or the sample size is large and/or the effect size (signal) is large. The confidence of a result (and its associated confidence interval) is not dependent on effect size alone. If the sample size is large and the noise is low a small effect size can be measured with great confidence. Whether a small effect size is considered important is dependent on the context of the events compared. In statistics, effect size is a measure of the strength of the relationship between two variables. ... In statistics, a confidence interval (CI) is an interval estimate of a population parameter. ...

In medicine, small effect sizes (reflected by small increases of risk) are often considered clinically relevant and are frequently used to guide treatment decisions (if there is great confidence in them). Whether a given treatment is considered a worthy endeavour is dependent on the risks, benefits and costs.

See also

• A/B testing
• Fisher's method for combining independent tests of significance
• Reasonable doubt

This article does not cite any references or sources. ... In statistics, Fishers method is a data fusion or meta-analysis (analysis after analysis) technique for combining the results from a variety of independent tests bearing upon the same overall hypothesis (H0) as if in a single large test. ... One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ... In the common law, burden of proof is the obligation to prove allegations which are presented in a legal action. ...

References

A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...

External links

• Raymond Hubbard, M.J. Bayarri, P Values are not Error Probabilities. A working paper that explains the difference between Fisher's evidential p-value and the Neyman-Pearson Type I error rate α.
• The Concept of Statistical Significance Testing - Article by Bruce Thompon of the ERIC Clearinghouse on Assessment and Evaluation, Washington, D.C.