# Statistical significance

In statistical hypothesis testing, statistical significance (or a statistically significant result) is attained whenever the observed p-value of a test statistic is less than the significance level defined for the study. The p-value is the probability of obtaining results at least as extreme as those observed, given that the null hypothesis is true. The significance level, α, is the probability of rejecting the null hypothesis, given that it is true. This statistical technique for testing the significance of results was developed in the early 20th century.

In any experiment or observation that involves drawing a sample from a population, there is always the possibility that an observed effect would have occurred due to sampling error alone. But if the p-value of an observed effect is less than the significance level, an investigator may conclude that that effect reflects the characteristics of the whole population, thereby rejecting the null hypothesis. A significance level is chosen before data collection, and typically set to 5% or much lower, depending on the field of study.

The term significance does not imply importance and the term statistical significance is not the same as research, theoretical, or practical significance. For example, the term clinical significance refers to the practical importance of a treatment effect.

## History

Main article: History of statistics

In 1925, Ronald Fisher advanced the idea of statistical hypothesis testing, which he called "tests of significance", in his publication Statistical Methods for Research Workers. Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis. In a 1933 paper, Jerzy Neyman and Egon Pearson called this cutoff the significance level, which they named α. They recommended that α be set ahead of time, prior to any data collection.

Despite his initial suggestion of 0.05 as a significance level, Fisher did not intend this cutoff value to be fixed. For instance, in his 1956 publication Statistical methods and scientific inference he recommended that significant levels be set according to specific circumstances.

The significance level α is the threshhold for p below which the experimenter assumes the null hypothesis is false, and something else is going on. This means α is also the probability of mistakenly rejecting the null hypothesis, if the null hypothesis is true.

Sometimes researchers talk about the confidence level γ = (1 − α) instead. This is the probability of not rejecting the null hypothesis given that it is true.  Confidence levels and confidence intervals were introduced by Neyman in 1937.

## Role in statistical hypothesis testing In a two-tailed test, the rejection region for a significance level of α=0.05 is partitioned to both ends of the sampling distribution and makes up 5% of the area under the curve (white areas).

Statistical significance plays a pivotal role in statistical hypothesis testing. It is used to determine whether the null hypothesis should be rejected or retained. The null hypothesis is the default assumption that nothing happened or changed. For the null hypothesis to be rejected, an observed result has to be statistically significant, i.e. the observed p-value is less than the pre-specified significance level.

To determine whether a result is statistically significant, a researcher calculates a p-value, which is the probability of observing an effect given that the null hypothesis is true. The null hypothesis is rejected if the p-value is less than a predetermined level, α. α is called the significance level, and is the probability of rejecting the null hypothesis given that it is true (a type I error). It is usually set at or below 5%.

For example, when α is set to 5%, the conditional probability of a type I error, given that the null hypothesis is true, is 5%, and a statistically significant result is one where the observed p-value is less than 5%. When drawing data from a sample, this means that the rejection region comprises 5% of the sampling distribution. These 5% can be allocated to one side of the sampling distribution, as in a one-tailed test, or partitioned to both sides of the distribution as in a two-tailed test, with each tail (or rejection region) containing 2.5% of the distribution.

The use of a one-tailed test is dependent on whether the research question or alternative hypothesis specifies a direction such as whether a group of objects is heavier or the performance of students on an assessment is better. A two-tailed test may still be used but it will be less powerful than a one-tailed test because the rejection region for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) of each rejection region for a two-tailed test. As a result, the null hypothesis can be rejected with a less extreme result if a one-tailed test was used. The one-tailed test is only more powerful than a two-tailed test if the specified direction of the alternative hypothesis is correct. If it is wrong, however, then the one-tailed test has no power.

### Stringent significance thresholds in specific fields

In specific fields such as particle physics and manufacturing, statistical significance is often expressed in multiples of the standard deviation or sigma (σ) of a normal distribution, with significance thresholds set at a much stricter level (e.g. 5σ). For instance, the certainty of the Higgs boson particle's existence was based on the 5σ criterion, which corresponds to a p-value of about 1 in 3.5 million.

In other fields of scientific research such as genome-wide association studies significance levels as low as 5×10−8 are not uncommon.

## Limitations

Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive and not replicable. There is also a difference between statistical significance and practical significance. A study that is found to be statistically significant, may not necessarily be practically significant. 

### Effect size

Main article: Effect size

Effect size is a measure of a study's practical significance.  A statistically significant result may have a weak effect. To gauge the research significance of their result, researchers are encouraged to always report an effect size along with p-values. An effect size measure quantifies the strength of an effect, such as the distance between two means in units of standard deviation (cf. Cohen's d), the correlation between two variables or its square, and other measures.

### Reproducibility

Main article: Reproducibility

A statistically significant result may not be easy to reproduce. In particular, some statistically significant results will in fact be false positives. Each failed attempt to reproduce a result increases the belief that the result was a false positive. 

### Controversy around overuse in some journals

Starting in the 2010s, some journals began questioning whether significance testing, and particularly using a threshold of α=5%, was being relied on too heavily as the primary measure of validity of a hypothesis. Some journals encouraged authors to do more detailed analysis than just a statistical significance test. In social psychology, the Journal of Basic and Applied Social Psychology banned the use of significance testing altogether from papers it published, requiring authors to use other measure to evaluate hypotheses and impact.

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