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A correlational study measures two or more variables as they naturally occur and estimates how strongly, and in which direction, they are related, without manipulating any of them. The result is usually a correlation coefficient, and it describes association only: it cannot by itself show that one variable causes the other. This page defines the design, gives examples, works one calculation through by hand, and shows how to report and compute the result.
What is a correlational study?
In a correlational study (also called correlational research or a correlation study), the researcher records variables as they are and examines whether they vary together. OpenStax Psychology 2e describes the central statistic as a correlation coefficient, a number from -1 to +1: the size of the number gives the strength of the relationship and the sign gives its direction, with values near 0 indicating a weak or absent relationship. The same chapter states that correlational research shows how strong a relationship is and in what direction it runs, but that a relationship does not necessarily imply cause and effect, because a third, confounding variable may influence both.
Textbooks classify the design differently: some treat it as a subtype of descriptive research, others as a separate category. Either way, no variable is assigned or manipulated by the researcher, which is what separates it from an experiment. For the three directions a relationship can take, see types of correlation.
Correlational study examples
The examples below are illustrative designs written for this page. They are not reports of real studies.
Example of a correlational study: three common data-collection routes
| Route | Illustrative research question | Variables (both measured, neither assigned) |
|---|---|---|
| Survey | Is weekly study time related to course grade among first-year students? | Self-reported hours; final grade |
| Archival records | Is clinic waiting time related to patient-reported satisfaction across sites? | Recorded minutes; satisfaction score |
| Naturalistic observation | Is the number of people at a study area related to the noise level there? | Counted occupants; measured decibels |
Each of these can be turned into a causal question only by changing the design, for example by assigning participants to different conditions, as in an experiment. The correlational version answers a narrower question: do the two measurements move together?
Sample correlational study: the worked example below
A sample correlational study of eight students pairs hours studied in a week (x) with exam score (y). Section three calculates the coefficient from these numbers.
Formula: Pearson’s correlation coefficient
For paired measurements, Pearson’s r is
r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √[ Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)² ]
- xᵢ, yᵢ: the i-th pair of observations.
- x̄, ȳ: the means of x and y.
- Σ: sum over all n pairs.
- The numerator is the sum of cross-products of deviations; the denominator rescales it so r falls between -1 and +1.
Worked example with the arithmetic shown
Illustrative data, n = 8. Means: x̄ = 36 / 8 = 4.5 and ȳ = 517 / 8 = 64.625.
| x (hours) | y (score) | x − x̄ | y − ȳ | (x − x̄)(y − ȳ) |
|---|---|---|---|---|
| 1 | 52 | -3.5 | -12.625 | 44.19 |
| 2 | 55 | -2.5 | -9.625 | 24.06 |
| 3 | 61 | -1.5 | -3.625 | 5.44 |
| 4 | 60 | -0.5 | -4.625 | 2.31 |
| 5 | 68 | 0.5 | 3.375 | 1.69 |
| 6 | 72 | 1.5 | 7.375 | 11.06 |
| 7 | 70 | 2.5 | 5.375 | 13.44 |
| 8 | 79 | 3.5 | 14.375 | 50.31 |
- Σ(x − x̄)(y − ȳ) = 152.5
- Σ(x − x̄)² = 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 = 42
- Σ(y − ȳ)² = 587.875
- r = 152.5 / √(42 × 587.875) = 152.5 / √24,690.75 = 152.5 / 157.13 = 0.971
- r² = 0.942, so about 94% of the variance in scores is shared linearly with hours in this tiny sample.
- Significance test: t = r√(n − 2) / √(1 − r²) = 0.971 × √6 / √0.058 = 9.86 on 6 degrees of freedom, p < .001 (two-tailed).
- 95% confidence interval (Fisher z transformation, standard error 1/√(n − 3)): [0.84, 0.99].
Eight points is far too few for a real study; the numbers are chosen so the arithmetic can be followed by hand.
Interpretation: what the result means
| What you see | How to read it |
|---|---|
| Sign of r is positive | Higher values of one variable tend to go with higher values of the other. |
| Sign of r is negative | Higher values of one variable tend to go with lower values of the other. |
| |r| close to 1 | A strong relationship: points lie close to a straight line. |
| r close to 0 | A weak or absent linear relationship (a curved pattern can still exist). |
| Any value of r | Association only; it does not by itself establish cause and effect. |
The first four rows follow the OpenStax description of the coefficient. Always plot the data: the NIST/SEMATECH e-Handbook lists what a scatter plot reveals, namely whether X and Y are related, whether the relation is linear or non-linear, whether the spread of Y changes with X, and whether outliers are present. See scatterplot interpretation for patterns. For how large an effect is in practice, see effect size in research.
Assumptions and common mistakes
- Reading correlation as causation. A third variable may drive both measures, and the direction of influence cannot be read from r. See correlation vs causation.
- Assuming linearity. Pearson’s r summarises a straight-line pattern. A quadratic or other curved pattern, which a scatter plot shows, can give r near 0 despite a clear relationship.
- Ignoring outliers. A single extreme point can raise or lower r; the scatter plot is where outliers show up.
- Using Pearson’s r for the wrong data. For ranked or non-normal data, rank-based coefficients are used instead; see Kendall’s tau. Check the type of each variable first.
- Restricted range (editorial note). If a sample covers only a narrow slice of one variable, r is usually smaller than it would be over the full range.
If the aim is prediction rather than description, the next step is regression analysis.
How to report a correlational study (APA style)
APA style reports the coefficient with its degrees of freedom in parentheses, the value without a leading zero (because r cannot exceed 1), and the exact p value or p < .001. For the worked example:
Hours studied and exam score were strongly positively correlated, r(6) = .97, 95% CI [.84, .99], p < .001.
State the sample size, how each variable was measured, and that the design is correlational, so readers do not infer causation.
How to calculate it in Excel, R or Python
With hours in A2:A9 and scores in B2:B9, Excel:
=CORREL(A2:A9, B2:B9)
R, which also returns the t statistic, degrees of freedom, p value and confidence interval:
x <- 1:8y <- c(52, 55, 61, 60, 68, 72, 70, 79)cor.test(x, y)
Python (SciPy):
from scipy import statsx = [1, 2, 3, 4, 5, 6, 7, 8]y = [52, 55, 61, 60, 68, 72, 70, 79]r, p = stats.pearsonr(x, y)print(r, p)
All three give r = 0.971 for these data. A longer walkthrough is in how to find the correlation coefficient in Excel.
FAQ
What is a correlational study in simple terms?
It is a study that measures two or more variables as they occur and checks whether they change together, without the researcher changing anything.
What is an example of a correlational study?
Surveying students about weekly study hours and comparing those hours with their grades is one illustrative example; neither variable is assigned by the researcher.
Can a correlational study show cause and effect?
No. It shows strength and direction of a relationship only; a confounding variable may influence both measures.
What is the difference between a correlational study and an experiment?
In an experiment the researcher manipulates a variable and assigns participants to conditions; in a correlational study every variable is only measured.
What does a correlation of 0 mean?
It means no linear relationship was found. A non-linear pattern may still exist, which is why the data should be plotted.
Sources
- OpenStax, Psychology 2e, “Analyzing Findings”: openstax.org/books/psychology-2e/pages/2-3-analyzing-findings (read 2026-11-08).
- NIST/SEMATECH e-Handbook of Statistical Methods, “Scatter Plot”: itl.nist.gov/div898/handbook/eda/section3/scatterp.htm (read 2026-11-08).
- Worked example: illustrative data computed for this page and checked against SciPy
pearsonr.
