# P-Value Definition

## What is the P value?

In statistics, the value of p is the probability of obtaining the observed results of a test, assuming that the null hypothesis is correct. This is the level of marginal significance in a statistical hypothesis test representing the probability of a given event occurring. The value of p is used as an alternative to the rejection points to provide the smallest level of significance at which the null hypothesis would be rejected. A smaller p-value means that there is more solid evidence for the alternative hypothesis.

## How is the P value calculated?

The p-values ​​are calculated using tables of p-values ​​or spreadsheets / statistical software. Because different researchers use different levels of significance when considering an issue, a reader can sometimes find it difficult to compare the results of two different tests.

For example, if two studies of the returns of two particular assets were undertaken using two different levels of significance, a reader could not easily compare the probability of returns of the two assets.

To facilitate comparison, researchers often present the p-value in the hypothesis test and allow the reader to interpret the statistical significance themselves. This is called a p-value approach for the hypothesis test.

## Approach to the value of p for hypothesis tests

The p-value approach for the hypothesis test uses the calculated probability to determine if there is evidence to reject the null hypothesis. The null hypothesis, also known as conjecture, is the initial claim regarding a population of statistics.

The alternative hypothesis indicates whether the population parameter differs from the value of the population parameter indicated in the conjecture. In practice, the p-value, or critical value, is indicated in advance to determine how the value required to reject the null hypothesis.

## Type I error

A type I error is the false rejection of the null hypothesis. The probability that a type I error will occur or reject the null hypothesis when it is true is equivalent to the critical value used. Conversely, the probability of accepting the null hypothesis when it is true is 1 minus the critical value.

## Real example of P value

Suppose an investor claims that the performance of his investment portfolio is equivalent to that of the Standard & Poor’s (S&P) 500 index. To determine this, the investor performs a bilateral test. The null hypothesis indicates that the portfolio returns are equivalent to the returns of the S&P 500 over a specified period, while the alternative hypothesis indicates that the portfolio returns and the returns of the S&P 500 are not equivalent. If the investor performed a one-sided test, the alternative hypothesis would indicate that the portfolio returns are lower or higher than the S&P 500 returns.

A commonly used p-value is 0.05. If the investor concludes that the p-value is less than 0.05, there is solid evidence against the null hypothesis. Consequently, the investor would reject the null hypothesis and accept the alternative hypothesis.

Conversely, if the p-value is greater than 0.05, this indicates that there is weak evidence against the conjecture, so that the investor would not reject the null hypothesis. If the investor finds that the p value is 0.001, there is solid evidence against the null hypothesis, and the portfolio returns and the S&P 500 returns may not be equivalent.