Statistical significance is the low probability of obtaining at least as extreme results given
that the null hypothesis is true. It is an integral part of statistical hypothesis testing
where it helps investigators to decide if a null hypothesis can be rejected. 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 probability of obtaining at least as extreme result, given the null hypothesis
is true, is less than a pre-determined threshold, then an investigator can conclude that the
observed effect actually reflects the characteristics of the population rather than just sampling
error. The present-day concept of statistical significance
originated from Ronald Fisher when he developed statistical hypothesis testing in the early
20th century. These tests are used to determine whether the outcome of a study would lead
to a rejection of the null hypothesis based on a pre-specified low probability threshold
called p-values, which can help an investigator to decide if a result contains sufficient
information to cast doubt on the null hypothesis. P-values are often coupled to a significance
or alpha level, which is also set ahead of time, usually at 0.05. Thus, if a p-value
was found to be less than 0.05, then the result would be considered statistically significant
and the null hypothesis would be rejected. Other significance levels, such as 0.1 or
0.01, are also used, depending on the field of study.
In statistics, statistical significance is not the same as research, theoretical, or
practical significance.
History
The concept of statistical significance was originated by Ronald Fisher when he developed
statistical hypothesis testing, which he described as "tests of significance", in his 1925 publication,
Statistical Methods for Research Workers. Fisher suggested a probability of one in twenty
as a convenient cutoff level to reject the null hypothesis. In their 1933 paper, Jerzy
Neyman and Egon Pearson recommended that the significance level, which they called 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, and in his
1956 publication Statistical methods and scientific inference he recommended that significant
levels be set according to specific circumstances. Role in statistical hypothesis testing
Statistical significance plays a pivotal role in statistical hypothesis testing, where it
is used to determine if a null hypothesis should be rejected or retained. A null hypothesis
is the general or default statement that nothing happened or changed. For a null hypothesis
to be rejected as false, the result has to be identified as being statistically significant,
i.e. unlikely to have occurred by chance alone. To determine if a result is statistically
significant, a researcher would have to calculate 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 the significance or α level. The α level is the probability of
rejecting the null hypothesis given that it is true and is most often set at 0.05. If
the α level is 0.05, then the conditional probability of a type I error, given that
the null hypothesis is true, is 5%. Then a statistically significant result is one in
which the observed p-value is less than 5%, which is formally written as p < 0.05.
If an observed p-value is not lower than the significance level, then rather than simply
accepting the null hypothesis, where feasible it would often be appropriate to increase
the sample size of the study, and see if the significance level is reached.
If the α level is set at 0.05, it means that the rejection region comprises 5% of the sampling
distribution. This 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 containing 2.5% of the distribution. One-tailed tests are more powerful than two-tailed
tests, as a null hypothesis can be rejected with a less extreme result.
Defining significance in terms of sigma
In specific fields such as particle physics and manufacturing, statistical significance
is often expressed in multiples of the standard deviation or sigma of a normal distribution,
with significance thresholds set at a much stricter level. 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. Effect size
Researchers focusing solely on whether their results are statistically significant might
report findings that are not necessarily substantive. To gauge the research significance of their
result, researchers are also encouraged to report the effect size along with p-values:
the effect size quantifies the strength of an effect, such as the distance between two
means or the correlation between two variables. See also
A/B testing ABX test
Effect size Fisher's method for combining independent
tests of significance Look-elsewhere effect
Texas sharpshooter fallacy Reasonable doubt
Statistical hypothesis testing References
Further reading Ziliak, Stephen, and McCloskey, Deirdre,.
The Cult of Statistical Significance: How the Standard Error Costs Us Jobs, Justice,
and Lives. Ann Arbor, University of Michigan Press, 2009.
Thompson, Bruce,. The "significance" crisis in psychology and education. Journal of Socio-Economics,
33, pp. 607–613. Chow, Siu L.,. Statistical Significance: Rationale,
Validity and Utility, Volume 1 of series Introducing Statistical Methods, Sage Publications Ltd,
ISBN 978-0-7619-5205-3 – argues that statistical significance is useful in certain circumstances.
Kline, Rex,. Beyond Significance Testing: Reforming Data Analysis Methods in Behavioral
Research Washington, DC: American Psychological Association.
External links The article "Earliest Known Uses of Some of
the Words of Mathematics" contains an entry on Significance that provides some historical
information. "The Concept of Statistical Significance Testing":
article by Bruce Thompon hosted by the ERIC Clearinghouse on Assessment and Evaluation,
Washington, D.C. "What does it mean for a result to be "statistically
significant"?": an article from the Statistical Assessment Service at George Mason University,
Washington, D.C.