Pearson Correlation Coefficient Formula: How to Calculate and Read It in SPSS

What is the correlation coefficient formula

The correlation coefficient formula measures the strength and direction of the linear relationship between two quantitative variables. In a thesis, the index you will most often see is the Pearson correlation coefficient, denoted by r. The value of r ranges from -1 to +1.

The Pearson formula is:

r = [nΣXY - (ΣX)(ΣY)] / √{[nΣX² - (ΣX)²][nΣY² - (ΣY)²]}

Here, X and Y are the two variables being compared, n is the number of observations, and ΣXY is the sum of the products of each pair of X and Y values. When you run the analysis in SPSS, the software calculates this formula from each respondent's data. You do not need to calculate it manually with a calculator, but you do need to understand what r is telling you before drawing a conclusion.

When r is positive, the two variables tend to increase in the same direction. When r is negative, one variable increases while the other tends to decrease. When r is close to 0, the linear relationship between the two variables is weak or nearly absent. A correlation coefficient describes an association, but it does not prove that X causes Y.

You can also review related topics in the Statistics category if you are working through several SPSS output tables in the same Chapter 4.

Why the correlation coefficient formula matters in quantitative research

In quantitative research, correlation is commonly used at three points. First, you check the preliminary relationships between the independent variables and the dependent variable. This result helps you see whether the data move in a direction that is consistent with the theoretical model.

Next, correlation can reveal a risk that the independent variables are too closely related to one another. This may lead to multicollinearity when you run the regression. A correlation matrix does not replace checking VIF in the Coefficients or Collinearity Statistics table.

Finally, some studies use Pearson correlation as the main analysis for testing the relationship between two variables. In that situation, you need to read three pieces of information together: the direction of r, the absolute size of r, and the p-value in the Sig. (2-tailed) column.

For example, r = 0.52 and Sig. = 0.003 indicate a positive linear relationship, a moderate relationship under the commonly used interpretation, and statistical significance at the 5% level. You still need to connect this conclusion to your hypothesis and research design. Do not write that X certainly increases Y simply because the Pearson result is significant.

Correlation can also be affected by an outlier. A few data points that lie far from the rest can substantially increase or decrease r. Check outliers in your data before interpreting an unusual correlation coefficient.

What correlation coefficient value is acceptable

There is no single number that makes a correlation acceptable for every study. Separate two questions: whether the coefficient is statistically significant and how strong the relationship is. Statistical significance is usually assessed with the p-value, while strength is assessed using the absolute value of r.

The table below provides a reference framework for interpretation. The categories may be presented differently across fields, so use a consistent approach throughout your thesis.

The strength ranges in the table are practical interpretations, not rigid thresholds for every field. When reporting the result, give the exact value of r instead of writing only “high correlation”. For significance testing, you can use the 0.05 level if that is the level selected in your research method.

A correlation with r = 0.18 can still be statistically significant when the sample size is large. Conversely, r = 0.40 may not reach significance when the sample size is small or the data vary substantially. Do not make the decision by looking at r alone and ignoring Sig. (2-tailed).

How to read the correlation coefficient formula in the output

In SPSS, go to Analyze > Correlate > Bivariate. Move the two variables you want to test into the Variables box, select Pearson, select Two-tailed, and click OK. If your hypothesis was directional from the beginning, you may consider One-tailed depending on your research requirements, but explain the choice clearly in the methods chapter.

SPSS produces a table called Correlations. The diagonal always contains Pearson Correlation equal to 1.000 because each variable is being compared with itself. Read the cell where the two variables intersect, such as the cell between SAT and LOY, instead of reading the diagonal.

The table below is illustrative output, not the result of an actual study.

In this example, r between SAT and LOY is 0.526. The positive sign indicates that the two variables have a positive relationship. Sig. (2-tailed) equals 0.003, which is below 0.05, so the relationship is statistically significant at the 5% level. In SPSS, the ** symbol usually indicates significance at the 0.01 level, but check the note below the table.

The N column shows that 124 observations were used in the calculation. If N differs between pairs of variables, SPSS may be excluding rows with missing data using Pairwise or Listwise deletion, depending on your selection. Record this because the actual sample size for each coefficient may differ.

A safe reading order is: identify the correct pair of variables, take Pearson Correlation, check the sign of r, check Sig. (2-tailed), and then check N. If you are using the table to prepare for regression, also read VIF afterward. Do not conclude that there is multicollinearity from one pair of correlations alone.

What to do when the correlation coefficient result does not meet the criterion

If Sig. is greater than or equal to 0.05, you do not have enough statistical evidence to reject H0 concerning the absence of a linear correlation in the sample. A suitable sentence is “no statistically significant linear correlation was identified”, rather than “the two variables are completely unrelated”.

If r is close to 0 but the scatterplot has a curved pattern, Pearson may be unsuitable because it measures a linear relationship. Review the scatterplot and the theoretical logic before choosing another analysis. Do not turn a result that does not meet the criterion into an acceptable result by deleting data rows arbitrarily.

If r is very high between two independent variables, check whether the variables are measuring almost the same content. Review the coding, variable names, and questionnaire. When you run the regression, also report VIF using the criterion that VIF should be below 5, as proposed by Hair et al. (Hair et al., 2019).

If you suspect an outlier, inspect each case using the original data table, boxplot, and scatterplot. Remove an observation only when there is a clear data-related reason, such as an incorrectly entered value or failure to meet the sample conditions. Keep the case code, the reason for the decision, and the results before and after processing so you can explain the decision to your supervisor.

If your data use a Likert scale, check whether the items have been coded in the same direction. A reverse-coded item that has not been recoded can sharply reduce the correlation between variables or change its sign. Before running the analysis again, review what a scale is and check the scale-coding procedure used in your study.

Distinguishing the correlation coefficient formula from easily confused measures

Pearson r and the p-value are different pieces of information. r shows the direction and size of the relationship, while the p-value shows the statistical evidence against the null hypothesis of no population correlation. A large r with a p-value that is not yet small can occur when the sample size is limited.

Pearson correlation is also different from the regression coefficient. Pearson does not clearly distinguish a predictor variable from an outcome variable. Regression assigns variables specific roles and estimates the expected change in Y when X changes while the other variables in the model are held constant.

Correlation does not mean causation. If service quality and loyalty have a positive correlation, you still need an appropriate model, hypothesis, and research design before discussing an effect. A third variable may influence both variables being observed.

Finally, do not confuse a mediator with a moderator when interpreting the model. A mediator explains the mechanism through which an effect is transmitted, while a moderator changes the strength or direction of a relationship. You can also review mediator, moderator, and moderating if the correlation table is a preparation step for a mediation or moderation model.

Common mistakes

The first mistake is reading the diagonal 1.000 as the result between two variables. Always identify the row and column names for the pair of variables you need to report.

The second mistake is writing “there is a correlation” simply because r is different from 0. You must check Sig. (2-tailed) and state the selected significance level. If the p-value does not meet the criterion, use cautious wording.

The third mistake is drawing a causal conclusion from a correlation. The sentence “SAT affects LOY” requires more appropriate analysis and reasoning, such as regression or a structural model. The Correlations table alone cannot support it.

The fourth mistake is deleting rows to make the result look better. Missing data, outliers, and reverse-coded items must be handled according to the rules stated in the methods section. Keep an original data file, a cleaned data file, and a change log.

The final mistake is reporting r without N and the p-value. A minimum results table should tell the reader the coefficient, its statistical significance, and the number of observations used.

Read more: outliers, moderating, moderator, what a scale is, mediator.

Frequently asked questions

What is the Pearson correlation coefficient formula used for?

The formula measures the strength and direction of the linear relationship between two quantitative variables. In SPSS, you usually read the result in the Correlations table through three main columns or rows: Pearson Correlation, Sig. (2-tailed), and N.

What correlation coefficient value is acceptable?

Do not use one threshold to decide whether a result is acceptable. Use the absolute value of r to describe strength, use the p-value to assess statistical significance, and then compare the result with your hypothesis and research context.

Why is Pearson Correlation negative?

A negative value indicates that the two variables tend to move in opposite directions in a linear relationship. For example, as X increases, Y tends to decrease. A negative sign does not mean that the result is wrong, but you should check reverse-coded items and the coding procedure before interpreting it.

What should I do when Sig. in the Correlations table is greater than 0.05?

You do not have enough evidence to conclude that the two variables have a statistically significant linear correlation at the 5% level. Check the scatterplot, outliers, missing data, and coding. Do not automatically remove a variable or alter the data simply to make the p-value smaller than 0.05.

Can Pearson be used for Likert data?

In many theses, composite scores or mean scores from Likert scales are entered into Pearson analysis. Check how the composite variable was constructed, the reliability of the scale, and the assumptions of the analysis according to the methods guidance for your study.

Does Pearson correlation prove a causal relationship?

No. Pearson shows how much two variables vary together in a sample. To discuss an effect or causal relationship, you need to rely on the theoretical model, research design, and a more suitable analysis.

How to write this in your thesis

Open your SPSS file, run Analyze > Correlate > Bivariate, compare each variable pair with the illustrative output, and record r, Sig., and N in your results table. If you need to run and explain the analysis directly on your .sav or .csv file, use M4 data analysis from DoThesis.