P-Value: Definition & Examples

KEY TAKEAWAYS

• The p-value is a statistical probability method that considers statistical significance. 
• It is a gauge between 0 and 1 that a null hypothesis is true or false. 
• A smaller p-value indicates stronger support for the alternative hypothesis.
• The larger the difference is between two observed values, the less likely it is that the difference can be put down to a random chance.

What Is P-Value?

P-value measurement is a statistical probability method. It determines the probability that the results obtained are at least as extreme as the observed results of a hypothesis test. This is all under the assumption that the null hypothesis is correct. 

The p-value serves as an alternative to rejection points. This is to provide the smallest level of statistical significance at which the null hypothesis would be rejected. 

The smaller the p-value is, the stronger evidence there is in favor of the alternative hypothesis. 

The null hypothesis is the initial claim about a population. The alternative hypothesis states whether the population parameter is different from the value of the entire population parameter that is stated in the null hypothesis. 

What Is the P-Value Used For?

Government agencies will often use the p-value to promote the credibility of various studies or reports. For example, the United States Census Bureau states that any analysis that has a p-value greater than 0.10 has to be accompanied by a statement that the difference isn’t statistically different from zero. 

The Census Bureau also has a number of standards that are put in place that stipulate what p-values are acceptable. This is across various different publications.

How Do You Calculate P-Value?

Finding p-value is mainly done by using p-value tables or spreadsheets and statistical software. The calculations associated with the p-value are based on the assumed or known probability distribution. This would be of the specific statistic that is being tested. 

You can calculate p-values from the deviation that is between the observed value, and a chosen reference value. This is with a greater difference between the two values that correspond to a lower p-value. This is given the probability distribution of the statistic. 

In a mathematical sense, the p-value can be calculated using integral calculus. It can be done from the area under the probability distribution curve. This is for all values of statistics that are at the very least as far from the reference value as the observed value. Relative to the total area under the probability distribution curve. 

The calculation that is used to find a p-value can vary. This is based on the type of test that is performed. There are three different hypothesis testing types that can be used to describe the location on the probability distribution curve. They are:

• Lower-tailed test
• Upper-tailed test
• Two-sided test

To put it simply, the larger the difference is between two observed values, the less likely it is that the difference can be put down to a random sample chance. This is then reflected by having a lower p-value.

What Is an Example of P-Value?

Let’s say that an investor is under the impression that their investment portfolio is equivalent to the Standard & Poor’s 500 Index (S&P Index). In order to figure out if this is true, the investor undertakes a single study of a two-tailed test. 

The null hypothesis is that the portfolio’s returns are equal to the S&P 500. This is over a specified period. While the alternative hypothesis says that the returns of the portfolio are in fact not equal to the S&P 500. 

Note: If the investor had instead conducted a one-tailed test, then the alternative hypothesis would state that the returns of the portfolio are either less or greater than the returns of the S&P 500.

The p-value hypothesis test doesn’t always make use of a preselected confidence level. This is the confidence level of the investor that their initial hypothesis that the returns are equivalent is true. Instead, the test provides a measure of how much evidence there is to reject the null hypothesis. 

So the smaller the p-value, the more strong evidence there is against the null hypothesis. Therefore, if the investor eventually finds that there is a 0.001 p-value, then there is a strong case that the null hypothesis is incorrect. This means that the investor can assume with confidence that the portfolio’s returns and the returns of the S&P 500 are not equal. 

It’s important to note that this doesn’t provide an exact threshold as to when the investor should reject the null hypothesis or not. Although it does have a practical advantage. 

The p-value hypothesis test offers a direct way to compare the relative confidence that an investor can have. This is when they have to choose between a wide range of different investment types relative to a benchmark. In this case, the benchmark is the S&P 500. 

To sum up, let’s say there are two portfolios; Portfolio X and Portfolio Y.

Portfolio X differs from the S&P 500 with a p-value of 0.10. While Portfolio Y differs from the S&P 500 with a p-value of 0.01. With this information, the investor can be far more confident that Portfolio Y, which has a lower p-value, will show different results consistently. 

Why is P-Value Important?

The p-value is important because it helps researchers to know if the results they find are replicable or not. 
The size of the p-value can have a significant effect on a variety of factors and industries, including clinical practice, the financial health of the pharmaceutical industry, and the professional and publication success of certain researchers. Although its widespread application and interpretation are still debatable. This is despite the fact that it plays a significant role in discussions about results.

Summary

The p-value approach holds a good degree of statistical significance when looking at probability and statistical tests. 

However, using p-values isn’t an exact science. While it lends itself to showing a high degree of confidence, it is not a binary science. Interpreting p-values should be done at your own risk. This is because there may be a scenario where the p-value points towards a null hypothesis being rejected, whereas it is in fact true. This may point towards artificially low p-values.