# Why is smaller the p-value, larger is the significance? [duplicate]

I am trying to understand the concept of p-value.

Wikipedia mentions:

The p-value is defined as the probability, under the assumption of hypothesis $H$, of obtaining a result equal to or more extreme than what was actually observed.

Though, I found quite vogue the term extreme, I'm taking it as the values beyond the observed value of the test statistic; by that I could somewhat conceived the definition.

Then, like my book, the wiki article writes:

The smaller the p-value, the larger the significance because it tells the investigator that the hypothesis under consideration may not adequately explain the observation. The hypothesis $H$ is rejected if any of these probabilities is less than or equal to a small, fixed but arbitrarily pre-defined threshold value $\alpha,$ which is referred to as the level of significance [...]

Hmmm... I couldn't get that point; why does the smaller p-value means a higher significance viz, the null hypothesis is on the verge of extinction?

Also, as this site summarises:

• High P values: your data are likely with a true null.

• Low P values: your data are unlikely with a true null.

I'm not getting why it is so- why does the lower p-values means the the significance is high?

Wikipedia seems to reason saying this: 'the hypothesis under consideration may not adequately explain the observation'. Can anyone tell me why it is so?

Let's say we have a null hypothesis that the mean of some variable is $0.$ We can draw a random sample from our population and in that sample we find a mean of $1.2\,.$ This could be because our hypothesis is wrong or because we drew a random sample, and the sample will deviate a bit from the population. With (two-sided) statistical testing we compute the probability of drawing a sample with a mean more than or equal to $1.2$ or less than or equal to $-1.2\, .$ If this probability is small then it is unlikely (but not impossible) that the mean of $1.2$ we found in our sample is due to sampling. Since this randomness due to sampling was our only way to "save" the null-hypothesis, we use that small $p$-value as evidence against the null-hypothesis.
• Thanks for answering; you said If this probability is small then it is unlikely (but not impossible) that the mean of 1.2 we found in our sample is due sampling- why is it so? Why does low p-value imply it was not due to sampling fluctuation? – user74724 Mar 23 '16 at 10:06