What is a Statistically Significant Relationship Between Two Variables?

How do you decide if indeed the relationship between two variables in your study is significant or not? What does the p-value output in statistical software analysis mean? This article explains the concept and provides examples.

What does a researcher mean if he says there is a statistically significant relationship between two variables in his study? What makes the relationship statistically significant?

These questions imply that a test for correlation between two variables was made in that particular study. The specific statistical test could either be the parametric Pearson Product-Moment Correlation or the non-parametric Spearman’s Rho test.

It is now easy to do computations using a popular statistical software like SPSS or Statistica and even using the data analysis function of spreadsheets like the proprietary Microsoft Excel and the open source but less popular Gnumeric. I provide links below on how to use the two spreadsheets.

Once the statistical software has finished processing the data, You will get a range of correlation coefficient values along with their corresponding p-values denoted by the letter p and a decimal number for one-tailed and two-tailed test. The p-value is the one that really matters when trying to judge whether there is a statistically significant relationship between two variables.

The Meaning of p-value

What does the p-value mean? This value never exceeds 1. Why?

The computer generated p-value represents the estimated probability of rejecting the null hypothesis (H0) that the researcher formulated at the beginning of the study. The null hypothesis is stated in such a way that there is “no” difference between the two variables being tested. This means, therefore, that as a researcher, you should be clear about what you want to test in the first place.

For example, your null hypothesis that will lend itself to statistical analysis should be written like this:

H0: There is no relationship between the long quiz score and the number of hours devoted by students in studying their lessons.

If the computed value is exactly 1 (p = 1.0), this means that the relationship is absolutely correlated. There is no doubt that the long quiz score and the number of hours spent by students in studying their lessons are correlated. That means a 100% probability. The greater the number of hours devoted by students in studying their lessons, the higher their long quiz scores.

Conversely, if the p-value is 0, this means there is no correlation at all. Whether the students study or not, their long quiz scores are not affected at all.

In reality however, this is not the case. Many factors or variables influence the long quiz score. Variables like the intelligence quotient of the student, the teacher’s teaching skill, difficulty of the quiz, among others affect the score.

Now, this means that the p-value should not be 1 or numbers greater than that. If you get a p-value of more than 1 in your computation, that’s nonsense. Your p-value, I repeat once again, should range between 1 and 0.

To illustrate, if the p-value you obtained during the computation is equal to 0.5, this means that there is a 50% chance that one variable is correlated to the other variable. In our example, we can say that there is a 50% probability that the long quiz score is correlated to the number of hours spent by students in studying their lessons.

Deciding Whether the Relationship is Significant

If the probability in the example given above is p = 0.05, is it good enough to say that indeed there is a statistically significant relationship between long quiz score and the number of hours spent by students in studying their lessons? The answer is NO. Why?

In today’s standard rule or convention in the world of statistics, statisticians adopt a significance level denoted by alpha (α) as a pre-chosen probability for significance. This is usually set at either 0.05 (statistically significant) or  0.01 (statistically highly significant). These numbers represent 5% and 1% probability, respectively.

Comparing the computed p-value with the pre-chosen probabilities of 5% and 1% will help you decide whether the relationship between two variables is significant or not. So, if say the p-values you obtained in your computation are 0.5, 0.4, or 0.06; you should accept the null hypothesis. That is, if you set alpha at 0.05 (α = 0.05). If the value you got is below 0.05 or p < 0.05, then you should accept your alternative hypothesis.

In the above example, the alternative hypothesis that should be accepted when the p-value is less than 0.05 will be:

H1There is a relationship between the long quiz score and the number of hours devoted by students in studying their lessons.

The strength of the relationship is indicated by the correlation coefficient or r values. Guilford (1956) suggested the following categories as guide:

< 0.20slight; almost negligible relationship
0.20 – 0.40low correlation; definite but small relationship
0.40 – 0.70moderate correlation; substantial relationship
0.70 – 0.90high correlation; marked relationship
> 0.90very high correlation; very dependable relationship

You may read the following articles to see example computer outputs and how these are interpreted.

How to Use Gnumeric in Comparing Two Groups of Data

Heart Rate Analysis: Example of t-test using MS Excel Analysis ToolPak


Guilford, J. P., 1956. Fundamental statistics in psychology and education. New York: McGraw-Hill. p. 145.

© 2014 May 29 P. A. Regoniel

4 thoughts on “What is a Statistically Significant Relationship Between Two Variables?”

  1. The article is really very useful to me as it helps me analyze the meaning of significant relationship between two variables. However, I have yet to go over the article, read it several times to finally come with a better grasp of the idea behind relationship. Thanks anyway.

  2. Thanks for this article,It’s really helpfull.I have a question regarding my project.It is “Relationship between ocupational stress and Marital Adjustment among working women in defence”. Here,Is it necessary to do t-test as I have already calculated correlation and the sample size is 30? Please help me out from this.

  3. Dear Mr. Regoniel
    Thanks so much for this wonderful article . Pls. can U explain why the significance (alpha) level of 0.01 is higher than that of 0.05, Also in reference to the illustration which U gave in your article, as regards the alpha level of 0.05 does it mean that if I get a computed value of 0.04, then I should accept the alternative hypothesis and reject the null hypothesis

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