Statistical significance

# Overview
In statistics, a result is called 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 usual sense of the word.
In traditional frequentist statistical hypothesis testing, the significance level of a test is the maximum probability, assuming the null hypothesis, that the statistic would be observed. Hence, the significance level is the probability that the null hypothesis will be rejected in error when it is true (a decision known as a Type I error, or "false positive").  The significance of a result is also called its p-value; the smaller the p-value, the more significant the result is said to be.
Significance is usually represented by the Greek symbol,  (alpha).  Popular levels of significance are 5%, 1% and 0.1%. If a test of significance gives a p-value lower than the -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.
Different -levels have different advantages and disadvantages. A very small -level (say 1%) is less likely to be more extreme than the critical value and so is more significant than high -level values (say 5%). However, smaller -levels run greater risks of failing to reject a false null hypothesis (a Type II error, or "false negative"), and so have less statistical power. The selection of an -level inevitably involves a compromise between significance and power, and consequently between the Type I error and the Type II error.

# 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 -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.
An additional pitfall is that frequentist analyses of p-values overstates statistical significance. See Bayes factor for details.
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.
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," and "Statistical Significance Tests are Unnecessary Even When Properly Done," 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.

# Signalnoise 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.
Expressed mathematically, the confidence that a result is not by random chance is given by the following formula by Sackett:
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 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.