Null hypothesis, from wikipedia

In statistics, a null hypothesis set up to be nullified or refuted. Although it was originally proposed to be any hypothesis, in practice it has come to be identified with the "nil hypothesis", which states that "there is no phenomenon". It is a hypothesis that is presumed true until statistical evidence in the form of a hypothesis test indicates otherwise. For example, if we want to compare the test scores of two random samples of men and women, a null hypothesis would be that the mean score in the male population from which the first sample was drawn was the same as the mean score in the female population from which the second sample was drawn:

H0:μ1 = μ2
where:

H0 = the null hypothesis
μ1 = the mean of population 1, and
μ2 = the mean of population 2.
Alternatively, the null hypothesis can postulate that the two samples are drawn from the same population:

H0:μ1 ? μ2 = 0
Formulation of the null hypothesis is a vital step in statistical significance testing. Having formulated such a hypothesis, we can then proceed to establish the probability of observing the data we have actually obtained, or data more different from the prediction of the null hypothesis, if the null hypothesis is true. That probability is what is commonly called the "significance level" of the results.

In formulating a particular null hypothesis, we are always also formulating an alternative hypothesis, which we will accept if the observed data values are sufficiently improbable under the null hypothesis. The precise formulation of the null hypothesis has implications for the alternative. For example, if the null hypothesis is that sample A is drawn from a population with the same mean as sample B, the alternative hypothesis is that they come from populations with different means (and we shall proceed to a two-tailed test of significance). But if the null hypothesis is that sample A is drawn from a population whose mean is lower than the mean of the population from which sample B is drawn, the alternative hypothesis is that sample A comes from a population with a higher mean than the population from which sample B is drawn, and we will proceed to a one-tailed test.

A null hypothesis is only useful if it is possible to calculate the probability of observing a data set with particular parameters from it. In general it is much harder to be precise about how probable the data would be if the alternative hypothesis is true.

If experimental observations contradict the prediction of the null hypothesis, it means that either the null hypothesis is false, or we have observed an event with very low probability. This gives us high confidence in the falsehood of the null hypothesis, which can be improved by increasing the number of trials. However, accepting the alternative hypothesis only commits us to a difference in observed parameters; it does not prove that the theory or principles that predicted such a difference is true, since it is always possible that the difference could be due to additional factors not recognised by the theory.

For example, rejection of a null hypothesis (that, say, rates of symptom relief in a sample of patients who received a placebo and a sample who received a medicinal drug will be equal) allows us to make a non-null statement (that the rates differed); it does not prove that the drug relieved the symptoms, though it gives us more confidence in that hypothesis.

The formulation, testing, and rejection of null hypotheses is methodologically consistent with the falsificationist model of scientific discovery formulated by Karl Popper and widely believed to apply to most kinds of empirical research. However, concerns regarding the high power of statistical tests to detect differences in large samples have led to suggestions for re-defining the null hypothesis, for example as a hypothesis that an effect falls within a range considered negligible. This is an attempt to address the confusion among non-statisticians between significant and substantial, since large enough samples are likely to be able to indicate differences however minor.