## Key points

- A p-value is a probability that
*measures the likelihood*that the data you observed (or more extreme data) occurred by random chance, assuming that the null hypothesis is true. - A common threshold for declaring statistical significance is 0.05.
- Suppose the
**p-value is less than or equal to 0.05**. In that case, you reject the**null hypothesis**and conclude that there is sufficient evidence to support the alternative hypothesis. - Suppose the
**p-value is more significant than 0.05**. In that case,**you fail to reject the null hypothesis**and conclude insufficient evidence supports the alternative hypothesis.

Hi, I'm Zubair Goraya, a data analyst and a writer for Data Analysis, a website that offers tutorials on Rstudio and other data analysis topics.

In this article, I will explain what it means when the p-value is less than 0.05, how to interpret it, and how to use it in Rstudio. I will also answer some frequently asked questions about p-values and statistical significance.

# What is a p-value?

A p-value is a probability that measures the likelihood that the data you observed (or more extreme data) occurred by random chance, assuming that the null hypothesis is true.

The ** null hypothesis (H0)** is that there is no relationship between the variables you are studying or that the effect you are testing is zero.

The ** alternative hypothesis (Ha)** is the opposite of the null hypothesis, stating that there is some relationship or effect.

For example, suppose you want to test whether a new drug lowers blood pressure. You randomly assign 100 patients to receive the drug or a placebo, and measure their blood pressure after four weeks. Your null hypothesis is that the drug does not affect blood pressure, and your alternative hypothesis is that the drug lowers blood pressure.

You can use a statistical test, such as a t-test, to calculate the p-value for your data. The p-value tells you how likely you would observe a difference in blood pressure as significant as (or larger than) the one you observed if the null hypothesis was true and the drug had no effect.

## What does it mean when the p-value is less than 0.05?

A common threshold for declaring statistical significance is 0.05. If the p-value is less than or equal to 0.05, you reject the null hypothesis and conclude that sufficient evidence supports the alternative hypothesis. In other words, you are confident that the difference or effect you observed is not due to chance but rather to some real phenomenon.

For example, suppose your p-value for the blood pressure experiment was 0.01. It means there is only a 1% chance of observing a difference in blood pressure as significant as (or larger than) the one you observed if the null hypothesis was true and the drug had no effect. Since this probability is very low, you reject the null hypothesis and conclude that the drug lowers blood pressure.

On the other hand, if the p-value is less than 0.05, you fail to reject the null hypothesis and conclude that there is not enough evidence to support the alternative hypothesis. In other words, you need to be more confident that the difference or effect you observed is real, possibly due to chance or other factors.

For example, suppose your p-value for the blood pressure experiment was 0.15. This means there is a 15% chance of observing a difference in blood pressure as significant as (or larger than) the one you observed if the null hypothesis was true and the drug had no effect. Since this probability is relatively high, you fail to reject the null hypothesis and conclude that the drug does not lower blood pressure.

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## FAQs

**What is a significance level? **

A significance level (alpha) is a threshold you set before conducting a statistical test. It determines how likely you are willing to accept a false positive result or reject the null hypothesis when it is true. An ordinary significance level is 0.05, meaning you will accept a 5% chance of making a false positive error.

**What is a one-tailed and a two-tailed test? **

A one-tailed test is a test where the alternative hypothesis specifies the direction of the difference or effect.

For example, suppose you are testing whether the drug lowers blood pressure. In that case, your alternative hypothesis is one-tailed because it states that the mean blood pressure of the drug group is lower than that of the placebo group. A two-tailed test is a test where the alternative hypothesis does not specify the direction of the difference or effect. For example, suppose you are testing whether the drug affects blood pressure. In that case, your alternative hypothesis is two-tailed because it states that the mean blood pressure of the drug group is not equal to that of the placebo group.

**How do I choose a significance level?**

The choice of a significance level depends on several factors, such as:

- The importance of the research question
- The consequences of making a false positive or a false negative error
- The prior knowledge or expectations about the phenomenon
- The sample size and variability of the data

In general, you should choose a lower significance level (such as 0.01 or 0.001) if:

- The research question is very important or has serious implications
- You want to minimize the risk of making a false positive error
- You have strong evidence or reason to believe that the null hypothesis is false
- You have a large sample size and low variability in your data
- You should choose a higher significance level (such as 0.1 or 0.2) if:

The research question is not very important or has minor implications

- You want to minimize the risk of making a false negative error
- You have weak evidence or reason to believe that the null hypothesis is false
- You have a small sample size and high variability in your data

**How do I report a p-value? **

When reporting a p-value, you should follow these guidelines:

- Use an equal sign (=) if the p-value is exact, or use a less than sign (<) if
- The p-value is rounded or truncated. For example, p = 0.034 or p < 0.001.
- Use two or three decimal places for the p-value unless it is small or large. For example, p = 0.05 or p = 0.0004, but not p = 0.0500 or p = 0.00000004.
- Do not use zero as a p-value, as it is impossible to observe a p-value of exactly zero. Instead, use a very small number, such as p < 0.001 or p < 2.2e-16 (the smallest number R can represent).
- Do not use asterisks (*) or other symbols to indicate the significance level, as they can be confusing or misleading. Instead, use words or phrases such as "statistically significant" or "not statistically significant".
- Include the degrees of freedom (df) and the test statistic (such as t, F, chi-square, etc.) along with the p-value, if applicable. For example, t(98) = -3.45, p = 0.001.
- Report the effect size (such as Cohen's d, R-squared, etc.), the confidence interval (CI), and the p-value. These measures provide more information about the magnitude and precision of the difference or effect. For example, d = -0.69, 95% CI [-1.03, -0.35], p = 0.001.

**What are some limitations of p-values?**

P-values help test hypotheses and assess statistical significance, but they also have some limitations that you should be aware of:

- P-values depend on the sample size and the variability of the data. A large sample size or a low variability can make a small difference or effect appear statistically significant, even if it is not practically significant. Conversely, a small sample size or a high variability can make a large difference or effect appear not statistically significant, even if it is almost significant.
- P-values do not tell you anything about the difference's direction, magnitude, or effect. It would help if you looked at the descriptive statistics (such as means, medians, etc.) and the effect size (such as Cohen's d, R-squared, etc.) to understand the nature and importance of the difference or effect.
- P-values do not tell you anything about the causality or the mechanism of the difference or effect. It would help if you considered other factors, such as the research design, the confounding variables, the external validity, etc., to draw causal inferences and explain how and why the difference or effect occurred.
- P-values are sensitive to multiple testing and data dredging. Suppose you perform many tests on the same data set or explore many variables without a clear hypothesis. In that case, you increase the chance of finding a statistically significant result by chance alone. This is known as a false discovery or a type I error. You must adjust your significance level or use other methods (such as Bonferroni correction, false discovery rate control, etc.) for multiple testing and data dredging.

**What are some alternatives to p-values? **

P-values are not the only way to test hypotheses and assess statistical significance. Some alternatives to p-values are:

**Bayesian statistics**: This approach uses prior knowledge and beliefs about the phenomenon to update the probability of the hypotheses based on the observed data. Instead of a p-value, it uses a posterior probability or a Bayes factor to compare the hypotheses and measure the strength of the evidence.**Confidence intervals**: This approach uses a range of values containing the true population parameter with a certain confidence level (such as 95% or 99%). Instead of a p-value, it uses a confidence interval to estimate the uncertainty and precision of the difference or effect.**Null hypothesis significance testing (NHST)**: This approach uses a decision rule that specifies when to reject or fail to reject the null hypothesis based on a pre-defined significance level (such as 0.05 or 0.01). Instead of using a p-value, it uses a critical value to compare with the test statistic and make a decision.

## Conclusion

In this article, I explained what it means when the p-value is less than 0.05 and how to interpret it. I hope you found this article helpful and informative. If you have any questions or comments, please get in touch with me at info@rstudiodatalab.com or visit our website at https://www.rstudiodatalab.com/p/order-now.html. We offer professional data analysis services and tutorials for various topics and software. Thank you for reading, and happy analyzing!