**p-value interpretation, definition, introduction and examples**

**p-value interpretation, definition, introduction and examples**

The p-value also known as observed *level of significance* or exact *level of significance* or the exact probability of committing a *type-I error* (probability of rejecting H_{0}, when it is true), helps to determine the significance of results from hypothesis. The p-value is the probability of obtaining the observed sample results or a more extreme result when the *null hypothesis* (a statement about population) is actually true.

In technical words, one can define p-value as the *lowest level of significance* at which a null hypothesis can be rejected. If p-value is very small or less than the threshold value (chosen *level of significance*), then the observed data is considered as inconsistent with the assumption that the null hypothesis is true and thus null hypothesis must be rejected while the *alternative hypothesis* should be accepted. The p-value is a number between 0 and 1 and in literature it is usually interpreted in the following way:

- A small p-value (<0.05) indicates strong evidence against the null hypothesis
- A large p-value (>0.05) indicates weak evidence against the null hypothesis.
- p-value very close to the cutoff (say 0.05) are considered to be marginal.

Let the p-value of a certain test statistic is 0.002 then it means that the probability of committing a *type-I error *(making a wrong decision) is about 0.2 percent, that is only about 2 in 1,000. For a given sample size, as | *t* | (or any *test statistic*) increases the p-value decreases, so one can reject the *null hypothesis* with increasing confidence.

Fixing the* level of significance* ($\alpha$) (i.e. *type-I error*) equal to the p-value of a *test statistic* then there is no conflict between the two values, in other words, it is better to give up fixing up (*significance level*) arbitrary at some *level of significance* such as (5%, 10% etc.) and simply choose the p-value of the *test statistic*. For example, if the p-value of *test statistic* is about 0.145 then one can reject the null hypothesis at this exact *significance level* as nothing wrong with taking a chance of being wrong 14.5% of the time of someone reject the *null hypothesis*.

p-value addresses only one question: how likely are your data, assuming a true *null hypothesis*? It does not measure support for the *alternative hypothesis*.

Most authors refers to p-value<0.05 as *statistically significant* and p-value<0.001 as highly *statistically significant* (less than one in a thousand chance of being wrong).

p-value is usually incorrectly interpreted as it is usually interpreted as the probability of making a mistake by rejecting a true null hypothesis (a Type-I error). p-value cannot be error rate because:

p-value is calculated based on the assumption that the *null hypothesis* is true and that the difference in the sample by random chances. Consequently, p-value cannot tell about the probability that the *null hypothesis* is true or false because it is 100% true from the perspective of the calculations.

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