P-Value: What It Is, How to Calculate It, and Why It Matters (2024)

What Is P-Value?

In statistics, a p-value is defined as a number that indicates how likely you are to obtain a value that is at least equal to or more than the actual observation if the null hypothesis is correct.

The p-value serves as an alternative to rejection points to provide the smallest level of significance at which thenull hypothesis would be rejected. A smaller p-value means stronger evidence in favor of the alternative hypothesis.

P-value is often used to promote credibility for studies or reports by government agencies. For example, the U.S. Census Bureau stipulates that any analysis with a p-value greater than 0.10 must be accompanied by a statement that the difference is not statistically different from zero. The Census Bureau also has standards in place stipulating which p-values are acceptable for various publications.

Key Takeaways

  • A p-value is a statistical measurement used to validate a hypothesis against observed data.
  • A p-value measures the probability of obtaining the observed results, assuming that the null hypothesis is true.
  • The lower the p-value, the greater the statistical significance of the observed difference.
  • A p-value of 0.05 or lower is generally considered statistically significant.
  • P-value can serve as an alternative to—or in addition to—preselected confidence levels for hypothesis testing.

Understanding P-Value

P-values are usually foundusing p-value tables or spreadsheets/statistical software. These calculations are based on the assumed or known probability distribution of the specific statistic tested. The sample size, which determines the reliability of the observed data, directly influences the accuracy of the p-value calculation. he p-value approach to hypothesis testing uses the calculated he p-value approach to hypothesis testing uses the calculated P-values are calculated from the deviation between the observed value and a chosen reference value, given the probability distribution of the statistic, with a greater difference between the two values corresponding to a lower p-value.

Mathematically, the p-value is calculated using integral calculus from the area under the probability distribution curve for all values of statistics that are at least as far from the reference value as the observed value is, relative to the total area under the probability distribution curve. Standard deviations, which quantify the dispersion of data points from the mean, are instrumental in this calculation.

The calculation for a p-value varies based on the type of test performed. The three test types describe the location on the probability distribution curve: lower-tailed test, upper-tailed test, or two-tailed test. In each case, the degrees of freedom play a crucial role in determining the shape of the distribution and thus, the calculation of the p-value.

In a nutshell, the greater the difference between two observed values, the less likely it is that the difference is due to simple random chance, and this is reflected by a lower p-value.

The P-Value Approach to Hypothesis Testing

The p-value approach to hypothesis testing uses the calculated probability to determine whether there is evidence to reject the null hypothesis. This determination relies heavily on the test statistic, which summarizes the information from the sample relevant to the hypothesis being tested. The null hypothesis, also known as the conjecture, is the initial claim about a population (or data-generating process). The alternative hypothesis states whether the population parameter differs from the value of the population parameter stated in the conjecture.

In practice, the significance level is stated in advance to determine how small the p-value must be to reject the null hypothesis. Because different researchers use different levels of significance when examining a question, a reader may sometimes have difficulty comparing results from two different tests. P-values provide a solution to this problem.

Even a low p-value is not necessarily proof of statistical significance, since there is still a possibility that the observed data are the result of chance. Only repeated experiments or studies can confirm if a relationship is statistically significant.

For example, suppose a study comparing returns from two particularassets was undertaken by different researchers who used the same data but different significance levels. The researchers might come to opposite conclusions regarding whether the assets differ.

If one researcher used a confidence level of 90% and the other required a confidence level of 95% to reject the null hypothesis, and if the p-value of the observed difference between the two returns was 0.08 (corresponding to a confidence level of 92%), then the first researcher would find that the two assets have a difference that is statistically significant, while the second would find no statistically significant difference between the returns.

To avoid this problem, the researchers could report the p-value of the hypothesis test and allow readers to interpret thestatistical significancethemselves. This is called a p-value approach to hypothesis testing. Independent observers could note the p-value and decide for themselves whether that represents a statistically significant difference or not.

Example of P-Value

Aninvestorclaims that their investment portfolio’s performance is equivalent to that of the . To determine this, the investor conducts atwo-tailed test.

The null hypothesis states that the portfolio’s returns are equivalent to the S&P 500’s returns over a specified period, while the alternative hypothesis states that the portfolio’s returns and the S&P 500’s returns are not equivalent—if the investor conducted a one-tailed test, the alternative hypothesis would state that the portfolio’s returns are either less than or greater than the S&P 500’s returns.

The p-value hypothesis test does not necessarily make use of a preselected confidence level at which the investor should reset the null hypothesis that the returns are equivalent. Instead, it provides a measure of how much evidence there is to reject the null hypothesis. The smaller the p-value, the greater the evidence against the null hypothesis.

Thus, if the investor finds that the p-value is 0.001, there is strong evidence against the null hypothesis, and the investor can confidently conclude that the portfolio’s returns and the S&P 500’s returns are not equivalent.

Although this does not provide an exact threshold as to when the investor should accept or reject the null hypothesis, it does have another very practical advantage. P-value hypothesis testing offers a direct way to compare the relative confidence that the investor can have when choosing among multiple different types of investments or portfolios relative to a benchmark such as the S&P 500.

For example, for two portfolios, A and B, whose performance differs from the S&P 500 with p-values of 0.10 and 0.01, respectively, the investor can be much more confident that portfolio B, with a lower p-value, will actually show consistently different results.

Is a 0.05 P-Value Significant?

A p-value less than 0.05 is typically considered to be statistically significant, in which case the null hypothesis should be rejected. A p-value greater than 0.05 means that deviation from the null hypothesis is not statistically significant, and the null hypothesis is not rejected.

What Does a P-Value of 0.001 Mean?

Ap-value of 0.001 indicates that if the null hypothesis tested were indeed true, then there would be a one-in-1,000 chance of observing results at least as extreme. This leads the observer to reject the null hypothesis because either a highly rare data result has been observed or the null hypothesis is incorrect.

How Can You Use P-Value to Compare 2 Different Results of a Hypothesis Test?

If you have two different results, one with a p-value of 0.04 and one with a p-value of 0.06, the result with a p-value of 0.04 will be considered more statistically significant than the p-value of 0.06. Beyond this simplified example, you could compare a 0.04 p-value to a 0.001 p-value. Both are statistically significant, but the 0.001 example provides an even stronger case against the null hypothesis than the 0.04.

The Bottom Line

The p-value is used to measure the significance of observational data. When researchers identify an apparent relationship between two variables, there is always a possibility that this correlation might be a coincidence. A p-value calculation helps determine if the observed relationship could arise as a result of chance.

P-Value: What It Is, How to Calculate It, and Why It Matters (2024)

FAQs

P-Value: What It Is, How to Calculate It, and Why It Matters? ›

Understanding P-Value

What is p-value and how do you calculate it? ›

The p-value is calculated using the sampling distribution of the test statistic under the null hypothesis, the sample data, and the type of test being done (lower-tailed test, upper-tailed test, or two-sided test). The p-value for: a lower-tailed test is specified by: p-value = P(TS ts | H 0 is true) = cdf(ts)

What is the p-value and why is it important? ›

The p value is a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true. P values are used in hypothesis testing to help decide whether to reject the null hypothesis.

Is the p-value of 0.05 significant? ›

If the p-value is less than 0.05, it is judged as “significant,” and if the p-value is greater than 0.05, it is judged as “not significant.” However, since the significance probability is a value set by the researcher according to the circ*mstances of each study, it does not necessarily have to be 0.05.

Why isn't the p-value enough? ›

While a P value can inform the reader whether an effect exists, the P value will not reveal the size of the effect. In reporting and interpreting studies, both the substantive significance (effect size) and statistical significance (P value) are essential results to be reported.

How do you explain p-value to non-technicians? ›

In simple terms the p-value expresses how surprised you are with the data, assuming there is no effect. The lower the p-value, the more incompatible the data seems with your model (i.e. the assumption that there is no effect).

How to interpret p-value in t test? ›

We can work out the chances of the result we have obtained happening by chance. If a p-value reported from a t test is less than 0.05, then that result is said to be statistically significant. If a p-value is greater than 0.05, then the result is insignificant.

Why is it important to correct p-values? ›

That is because the probability of reporting a false positive in a group of independent tests is the sum of the individual p-values. When this is done for hundreds of pathways, we are virtually guaranteed to have some pathways that appear to be significant just by chance.

How to report p-value example? ›

The preferred method of reporting P-values is to use an exact number, with two or three significant decimal places rather than as a range or category (e.g., NS, p > . 05, or p < . 05).

What is p-value and statistical significance for dummies? ›

The end result of a statistical significance test is a p value, which represents the probability that random fluctuations alone could have generated results that differed from the null hypothesis (H0), in the direction of the alternate hypothesis (HAlt), by at least as much as what you observed in your data.

What happens when p-value is equal to significance level? ›

A p-value less than or equal to your significance level (typically ≤ 0.05) is statistically significant. A p-value less than or equal to a predetermined significance level (often 0.05 or 0.01) indicates a statistically significant result, meaning the observed data provide strong evidence against the null hypothesis.

What is the difference between p-value and level of significance? ›

The term significance level (alpha) is used to refer to a pre-chosen probability and the term "P value" is used to indicate a probability that you calculate after a given study.

How to calculate significance level? ›

In most studies, a p-value of 0.05 or less is considered statistically significant — but you can set the threshold higher. A higher p-value of over 0.05 means variation is less likely, while a lower value below 0.05 suggests differences. You can calculate the difference using this formula: (1 - p-value)*100.

Why is p-value important? ›

A p-value measures the probability of obtaining the observed results, assuming that the null hypothesis is true. The lower the p-value, the greater the statistical significance of the observed difference. A p-value of 0.05 or lower is generally considered statistically significant.

What is the biggest problem with using p-value? ›

The p value is sensitive to sample size and variability in the sample. A very large sample size with a very small effect size can yield a significant p value. Such results may offer little inference in scientific studies and are likely to be irreproducible.

Why is p-value so confusing? ›

P Values Don't Provide the Answers that We Really Want

The common misinterpretations are what we really want to learn from hypothesis testing. We'd love to learn the probability that a hypothesis is correct. That would be nice. Unfortunately, hypothesis testing doesn't provide that type of information.

What is the formula for p probability? ›

P(A/B) is known as conditional probability and it means the probability of event A that depends on another event B and is read as "probability of A given B". It says P(A/B) = P(A∩B) / P(B). It is also known as "the probability of A given B".

What is the difference between p-value and significance level? ›

The p-value represents the strength of evidence against the null hypothesis, while the significance level represents the level of evidence required to reject the null hypothesis. If the p-value is less than the significance level, the null hypothesis is rejected, and the alternative hypothesis is accepted.

How do you calculate the p-value in Excel? ›

Use the formula =T. TEST(data_array1, data_array2, tails, type) in the cell where you want to place the P-value result. Enter the data arrays and specify the number of tails and the type of test you want to perform. Press "Enter" to see the result.

What does p 0.001 mean? ›

The p-value indicates how probable the results are due to chance. p=0.05 means that there is a 5% probability that the results are due to random chance. p=0.001 means that the chances are only 1 in a thousand.

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