Picture this: you're diving deep into the world of analytics and statistics, trying to make sense of all those numbers and data points. Suddenly, you stumble upon a little gem called the p-value. It's like a secret code that researchers use to unlock the mysteries of hypothesis testing and significance.

The primary use of the p-value is for decision-making in hypothesis testing. It helps researchers assess if the observed data is enough to reject the null hypothesis for an alternative hypothesis. Researchers also use the p-value to compare groups or test for correlations.

Gather answers using the SurveyMonkey p-value calculator above.

The p-value stands for probability value. It measures the likelihood of a result, assuming the null hypothesis is true. It's a probability gauge showing how likely your result is, assuming no real difference (the null hypothesis).

The p-value quantifies the strength of evidence against the null hypothesis. It is typically compared to a predetermined level of significance, such as 0.05. When the p-value is low, it tells you, "This result probably didn't happen by chance!" This gives you the green light to reject the null hypothesis and consider that your hypothesis might be true. 

The p-value is important because researchers use it to decide whether to accept or reject the null hypothesis. Some examples of research questions that can use the p-value are:

A low p-value suggests there are differences among the groups you tested. It also indicates that real, predictable relationships among variables may exist.

Researchers can then interpret the significance of their findings and communicate the strength of evidence to stakeholders and peers.

To calculate a p-value, first determine the probability of obtaining your data if the null hypothesis were true. Then, compare this probability to your chosen significance level (usually 0.05) to decide if your results are statistically significant.

To calculate a p-value from a z-score, look up the z-score in a standard normal distribution table. Alternatively, use software to find the corresponding probability. This probability represents the likelihood of observing a value as extreme as the z-score under the null hypothesis.

The following formulas give the p-value:

Here’s the step-by-step guide on how to calculate the p-value from a z-score:

To calculate a p-value from a t-score, first, determine the t-score representing the difference between your sample mean and the population mean. Then, use a t-distribution table or software to find the probability of observing that t-score. This indicates the likelihood of obtaining your sample results under the null hypothesis.

The following formula gives the p-value from the t-score.

Where cdft,d represents the cumulative distribution function of the t-Student distribution with d degrees of freedom.

Here’s the step-by-step guide on how to calculate the p-value from a t-score:

To obtain the p-value for a Pearson correlation coefficient, first use the calculated coefficient to derive a t-statistic. Then, you can find its associated p-value using the t-distribution with degrees of freedom (n - 2).

The formula to get the t-statistic from a Pearson correlation coefficient is below:

Where:

After obtaining the z-score, you can calculate the p-value using the cumulative distribution function of the t-distribution. This uses n - 2 degrees of freedom, where n is the sample size.

Here's the general process:

To calculate the p-value from a chi-square score, determine the degrees of freedom associated with the chi-square distribution. Then, use statistical tables or software to find the probability of obtaining a chi-square value as extreme as the observed one.

You can get the p-value with the help of the following formula:

p-value=1− cdfχ² (x; df)

Where:

​You subtract the cumulative probability from 1 because the chi-square distribution is right-skewed, so the tail area to the right of the observed chi-square value corresponds to the p-value.

Here are the step-by-step instructions to calculate the p-value from a chi-square score:

If the p-value is less than or equal to 0.05 (or any chosen significance level), it suggests the result is statistically significant. This means the observed result is significant at the α level.

This means the probability of obtaining an extreme result, assuming the null hypothesis is true, is very low. Typically, this probability is less than 5%.

Therefore, you reject the null hypothesis in favor of the alternative hypothesis. This indicates some evidence to support the claim made by the alternative hypothesis.

If the p-value is greater than 0.05, it suggests that the observed result is not statistically significant at the chosen significance level. In other words, there’s insufficient evidence to reject the null hypothesis. This means we cannot conclude that the observed result differs from what would be expected under the null hypothesis.

Related reading: How to analyze survey data

Some people believe that a p-value of 0.05 means there is a 95% chance the test hypothesis is true and a 5% chance it is false. This is a misinterpretation of the p-value.

P-values indicate the likelihood of observing the data, assuming the null hypothesis is true. They are not direct measures of the probability of hypotheses being true or false.

Treating the p-value as synonymous with effect size or importance is a common misconception. This blurs the line between statistical significance and practical significance.

A small p-value indicates the observed result is unlikely due to random chance. However, it does not convey the effect's magnitude. Additionally, it does not reflect the practical relevance of that effect.

For instance, even tiny deviations from the null hypothesis may yield statistically significant p-values in large datasets despite being practically insignificant. Also, if an experiment yields significant differences multiple times, it’s likely to observe nonsignificant results sometimes because this is based on probability.

Conversely, a large p-value doesn't necessarily imply that the observed effect is trivial. Instead, it suggests that the data do not provide convincing evidence against the null hypothesis. 

To accurately assess the practical importance of findings, it's essential to complement p-values with measures of effect size. Effect size quantifies the magnitude of the observed effect. It helps researchers contextualize results within the broader scope of the research question or application. 

This distinction ensures that statistical significance aligns with meaningful real-world implications. It guides informed decision-making and interpretation of research outcomes.

The multiple testing problem arises when researchers conduct numerous hypothesis tests on the same dataset without appropriately adjusting the significance level. This practice significantly inflates the likelihood of encountering false positives, also known as Type I errors. In these situations, the null hypothesis is incorrectly rejected.

Imagine a scenario where several independent tests are conducted simultaneously. Even if each test maintains a low significance level (e.g., α = 0.05), the cumulative probability of observing at least one significant result by chance alone increases. This occurs as the number of tests increases.

Researchers employ statistical correction techniques such as the Bonferroni correction to make it more difficult to reject the null hypothesis. These solutions will help maintain stringent control over the overall false positive rate. They ensure the probability of false positives across all tests remains below the specified threshold.

Consider the practical implications of your findings within the broader context of your research question or application. Avoid overinterpreting statistically significant results or dismissing nonsignificant results without careful consideration.

Suppose you find a statistically significant improvement in test scores among students taught with a new method. This improvement is compared to those taught using the traditional method.

You should avoid overinterpreting the results. Instead, consider factors such as the size of the effect. Is the score improvement substantial enough to justify implementing the new teaching method on a large scale? Would this finding be replicated in other studies with similar conditions? Are there other factors, such as cost, that need to be considered?

Conversely, nonsignificant results could be due to other factors, such as small sample size or measurement error. 

Therefore, it's important to critically evaluate the study design, data quality, and potential sources of bias before drawing conclusions.

Regardless of their significance, all p-values for all variables in a study should be included. This provides a comprehensive picture of the analysis. It enables readers to assess the robustness of the findings.

By reporting all p-values, researchers convey the complete range of statistical analyses, including those with nonsignificant results. This transparency allows readers to evaluate the consistency and reliability of the findings across different variables and analyses. It also promotes integrity in research by presenting the data in its entirety, without bias or distortion.

Interpreting small p-values requires caution. They can sometimes be misleading indicators of the significance of observed effects. 

Recognizing that small p-values can arise from genuine effects and large sample sizes is crucial. Large sample sizes increase the statistical power to detect trivial deviations from the null hypothesis.

Therefore, small p-values in studies with large sample sizes may not necessarily reflect meaningful or practically significant effects.