Pearson Correlation: Interpretation and APA Reporting

The Basics

When to Use It

Use it when you've measured two continuous variables on the same people and want to know whether they rise and fall together along a straight line, and how strongly. Neither variable has to be the “outcome.” For example:

Do students who study longer score higher on the exam?
Is daily screen time related to depression symptom scores in adolescents?
Do people with larger working-memory spans respond faster on the Stroop task?

Vocabulary You Need

Definition Covariance (

\text{cov}_{xy}

)

The average product of two variables' deviations from their means. A sample's covariance divides the sum of those products by n − 1 rather than n, as the sample variance does, to estimate the population's value without bias. Positive when above-average scores on one go with above-average scores on the other, negative when they go with below-average scores. Its units (such as hours × points) make its size hard to read, which is why it's usually standardized into a correlation. Glossary →

Definition Pearson's r (

r

)

Covariance divided by the product of the two variables' standard deviations, so it runs from −1 (a perfect downhill line) through 0 (no straight-line relationship) to 1 (a perfect uphill line). It has no units, so it is its own effect size. It only measures linear relationships. Glossary →

Definition Coefficient of Determination (

r^2

)

With two variables it's the square of their correlation, r²: the share of the sample variation in one variable accounted for by its best-fitting straight-line prediction from the other, the same whichever variable is predicted. It doesn't split the variables into shared and separate causes. In a regression with several predictors it's written R²: the share of the outcome's variation that lines up with the model's predictions, 1 − SS_residual ÷ SS_total, which equals the squared correlation between the outcome and the predictions. It's often called variance explained, where “explained” means predicted statistically; it says nothing about cause. It can't go down when a predictor is added, which is why adjusted R² exists. It's never larger than |r|, and much smaller for modest correlations (r = .30 gives r² = .09), a useful check on how big a correlation really is. Glossary →

Assumptions and Inference

1. Paired Numeric Scores Each person has a score on both variables, both scores are numbers on an equal-interval scale, and both vary.

What It Means: r pairs each person's two scores, so both must come from the same person, and it adds and multiplies deviations from the means, so it treats a one-point gap as the same amount anywhere on each scale. Times, counts and totals of many items usually qualify; a single rating item, or ranks, may not. If everyone has the same score on either variable, r can't be computed, and if one variable barely varies in your sample, r says little about a wider range.
Applies To: Variables
How to Check: By reasoning about the measures, not by a test. Check that both columns on a row belong to the same person, ask whether one more unit means the same amount everywhere on each scale, and look at each variable's SD and range.
How Much It Matters: It decides whether r fits. Study time in hours and exam points both qualify, and both spread widely (SDs of 3.00 hours and 10.00 points).

What to Do

Alternative Test Spearman Correlation. For ranks, or ordered scores whose steps may not be equal, correlate the ranks instead.

2. Linear Relationship (for What r Means) r measures the strength and direction of a straight-line association; it can be computed for a curve, but then it may be small or misleading even when the variables are strongly related.

What It Means: As one variable goes up, the other tends to change by about the same amount across its whole range. A curved relationship, one that levels off or a U shape, breaks this.
Applies To: Relationship
How to Check: Look at a scatterplot of the two variables: a roughly oval cloud tilted one way is fine; a clear bend, a ceiling or a U is not.
How Much It Matters: A lot, for what r means. r only measures the straight-line part of a relationship, so a curve that levels off makes r understate it, and a U shape can give an r near zero. The exam data follow a roughly straight trend.

What to Do

Alternative Test If the relationship bends but always rises (or always falls), Spearman correlation measures that consistent one-direction trend without needing a straight line.
Transformation Log-transform a variable whose relationship levels off, such as reaction time, then correlate. The result describes the transformed scale.
Proceed with Caveat If the curve itself is the finding, don't summarize it with r. Show the scatterplot and model the curve with regression.

3. No Extreme Outliers No single case is far enough from the rest to drive the result.

What It Means: A few points far from the rest can pull a statistic around on their own, so it describes those few points more than everyone else. In a test, that can create an effect that isn't there or hide one that is.
Applies To: Observations
How to Check: Plot the two variables and look for points far from the rest, including points that are ordinary on each variable but unusual in combination. A squared Mahalanobis distance across the two flags those: with two variables, values above 13.8 are rare (p < .001). Check each flagged point against the raw records, and compare r with and without it.
How Much It Matters: A lot, especially in small samples, and for every result, r included. r is built from products of deviations, so one point far from both means carries a lot of weight. The points that matter most can be unusual only in combination, such as lots of study and a low score, so check the pair, not each variable alone. In the exam data, no student sits far from the cloud (largest squared Mahalanobis distance 6.5, against a cutoff of 13.8), and dropping any one student keeps r between .28 and .34.

What to Do

Correction Check each extreme point against the raw data, and fix or remove data-entry errors.
Proceed with Caveat If the outlier is a real person, report r with and without it and say which you trust.
Alternative Test Spearman Correlation. Ranks cap how far any one point can pull the result.

4. Independent Observations (for the p-Value and Interval) The people (or other units) in your study don't influence each other.

What It Means: Each person's data comes from a separate, unconnected source, so knowing one person's scores tells you nothing about anyone else's. In designs that measure people more than once, their own scores are meant to be related; it's different people who must be independent. Methods built for clustered data, such as multilevel models and GEE, also allow members of the same group to be related, as long as the analysis names the grouping; then it's the groups that must be independent.
Applies To: Observations
How to Check: By design, not by a test. Did each person contribute their own data, with no clusters such as roommates, couples, therapy groups or classrooms whose members influence each other? If there are clusters, does the analysis model every one of them?
How Much It Matters: A lot. Pairs of scores that depend on each other, such as study partners or students from the same small section, make the standard error of r too small, so p comes out too small.

What to Do

Design Change Collect one pair of scores per independent person.
Alternative Test If people come in clusters (classrooms, couples, repeated days), a multilevel model can estimate the relationship while accounting for the clustering.

5. Bivariate Normality (for the p-Value and Interval) The pairs of scores form an oval cloud, with each variable roughly bell-shaped.

What It Means: Each variable is roughly normal and the scatterplot forms an oval cloud; two bell-shaped variables alone don't guarantee it. The p-value and confidence interval rely on it; r itself, as a description, doesn't. Large samples protect the p-value of a test against zero fairly well. They don't protect a test against any other value, a comparison of two correlations, or the Fisher-z interval, whose coverage can get worse with more data when the scores are strongly nonnormal.
Applies To: Joint distribution
How to Check: Look at a histogram of each variable and at the scatterplot for strong skew, heavy tails or a funnel shape. Don't rely on a normality test: with small samples it misses real problems, and with large ones it flags trivial ones.
How Much It Matters: Usually not much, and only for p and the interval, not r itself. Pearson's test holds its false-positive rate well with non-normal data, except in very small samples or with very heavy tails or strong skew (Bishara & Hittner, 2012). Heavy tails and skew can also cost power. The Fisher-z interval is less forgiving: with nonnormal scores its coverage can fall short even in large samples, especially when either variable has a skewness beyond ±1 or a kurtosis of 2 or more (Bishara & Hittner, 2017). With 70 students, skewness of 0.19 for study time and 0.06 for exam scores, and kurtosis of −0.03 and −0.70, it's fine here.

What to Do

Alternative Test Spearman Correlation. It works on ranks, so skew and heavy tails matter much less. It asks a slightly different question (a consistent one-direction trend, not a straight line).
Transformation Transform skewed variables toward a normal shape before correlating, especially with a rank-based inverse normal (RIN) transformation, which turns each variable's ranks into normal scores. In simulations, this kept error rates lowest for samples of 20 or more (Bishara & Hittner, 2012), and its interval, like Spearman's, kept close to 95% coverage (Bishara & Hittner, 2017). The correlation then describes the transformed scores, not the raw ones.
Robust Estimation Get p from a permutation test, which shuffles one variable to see what chance produces and assumes no distribution shape. It tests whether the variables are independent, not just uncorrelated, and Bishara and Hittner (2012) found it most useful for very small, very nonnormal samples (10 or fewer).

Limitations

This Is Not Your Test If…

Your scores are ranks, or badly skewed Spearman Correlation →
You want to predict one variable from the other Simple Linear Regression →
You need to control for a third variable Partial Correlation →
You're comparing correlations between groups Comparing Correlation Coefficients →
You're checking whether raters or retests agree Agreement and Reliability →
Your variables are categories, not scores Categorical Data and Proportions →

What It Doesn't Tell You

Why. Study time and scores move together, but nothing here shows that studying raised scores; motivation or prior knowledge could drive both. Holding a third variable such as prior grades constant takes a partial correlation (r between the two after removing the third variable's straight-line part from each) or a regression.
Whether it matters. Significance doesn't make the relationship useful: knowing a student's study time narrows down their exam score only a little.
Who else. The result describes students like these 70, over the range of study times they reported. Restricting the range of study times, say to heavy studiers only, can weaken or otherwise distort the correlation, so the result may not carry over to a wider range of students.
That there's no relationship, if p had been above .05. A non-significant r would have meant the data were inconclusive about a straight-line trend, not that study time doesn't matter. A curved relationship can even have an r near zero.

Common Mistakes

Don'tSwitch to a one-sided p because the result went the way you hoped.

InsteadUse a one-sided test only if you predicted the direction before collecting data, and say so.
Don'tDescribe a correlation as one variable causing, raising or improving the other.

InsteadSay the variables are associated. Causal claims need an experiment or a design that rules out third variables.
Don'tCompute r without looking at a scatterplot.

InsteadPlot first. A curve, an outlier or two separate clusters can make r misleading.
Don'tConclude two variables are unrelated because r is near zero.

InsteadSay there's no straight-line relationship, and check the scatterplot for a curve r can't see.
Don'tRead r = .30 as 30% of the variation explained.

InsteadSquare it: r² = .09, so a straight line through study time accounts for 9% of the variation in exam scores, statistically, not causally.
Don'tCompare correlations from samples that cover very different ranges of scores.

InsteadCheck each sample's spread. A restricted range usually shrinks r, and can distort it in other ways, even when the relationship is the same.
Don'tCompare correlations from measures of very different reliability as if they were equal.

InsteadRemember that independent, random measurement error generally pulls r toward zero, while correlated or systematic errors can distort it in other ways, and report each measure's reliability.
Don'tRead a p-value as the chance that the null hypothesis is true.

InsteadRead it as: if the null were true, how often data at least this extreme would turn up.

Result Decoder

Significance and Effect Size Guidelines

Benchmarks Significance (p Value)

p < .001
Very strong evidence against the null hypothesis.
.001 ≤ p < .01
Strong evidence against the null hypothesis.
.01 ≤ p < .05
Statistically significant at α = .05. Moderate evidence against the null hypothesis.
.05 ≤ p < .10
Not significant. Weak evidence; sometimes called a “trend.” Treat it as inconclusive.
p ≥ .10
Not significant. Little evidence against the null, which is not the same as evidence of no effect.

These evidence labels are one common convention for describing p in graded language rather than as significant or not; the boundaries are rough guides, not rules, and other authors draw them differently. Muff et al. (2022) argue for this kind of language of evidence over a single cutoff. α is chosen before the analysis, and p never says how big or important an effect is.

More about statistical significance ↓

Benchmarks Effect Size (Pearson's r)

|r| < .10 Below Small
Smaller than the conventional small benchmark. Whether an effect this size matters depends on the setting.
.10 ≤ |r| < .30 Small
A weak straight-line trend, often hard to see in a scatterplot without computing it.
.30 ≤ |r| < .50 Medium
Cohen described a medium effect as likely visible to the naked eye of a careful observer; whether it shows in a given scatterplot depends on the data and the sample size.
|r| ≥ .50 Large
A strong straight-line trend, usually clear in a scatterplot unless outliers or a small sample muddy it.

The sign shows direction; the benchmarks apply to its size. They are conventions (Cohen, 1988, 1992), not measures of practical or clinical importance.

More about effect size ↓

How It's Reported

Equation Form

Example

r(68) = .30, 95% CI [.07, .50], t(68) = 2.59, p = .012

Full Text (APA Style)

Example

Students studied for an average of 8.00 hours (SD = 3.00) and scored an average of 70.00 points (SD = 10.00). Students who studied longer tended to score higher, r(68) = .30, 95% CI [.07, .50], a medium correlation by convention (r² = .09). The correlation was statistically significant, t(68) = 2.59, p = .012.

Put the degrees of freedom, n − 2, in parentheses: r(68).
Drop the leading zero for r and p: neither can go past 1.
Write “tended to,” “was associated with” or “went with,” not “led to” or “improved.” The design can't support a causal claim.
Give the confidence interval. With 70 people it runs from a below-small correlation to one at the edge of large, which the single .30 hides.

Reading Each Piece

The equation form above, piece by piece, in this order: how big, how certain, then how surprising.

r(68) = .30	Students who studied longer tended to score higher. A correlation of .30 sits right at the conventional medium benchmark. The 68 in parentheses is the degrees of freedom, n − 2. Benchmarks ↑
95% CI [.07, .50]	The true correlation is plausibly anywhere from below small (.07) to the edge of large (.50). The whole range is above zero, but 70 students can't pin the strength down precisely.
t(68) = 2.59	t = 2.59 on 68 degrees of freedom is the test statistic for the null hypothesis that the population correlation is zero, and it's where p comes from. SPSS doesn't print t; R does.
p = .012	If study time and exam scores had no straight-line relationship among all students like these, a sample correlation at least this far from zero, in either direction, would turn up about 1.2% of the time. That's below .05, so the correlation is statistically significant. Benchmarks ↑

Calculations

The Running Example

Example

The exam study: 70 undergraduates, each with hours spent studying and a final exam score, asking whether the two go together.

Null Hypothesis:: $\rho = 0$
Alternative (Two-Sided):: $\rho \neq 0$

$\rho$ (“rho”) is the population correlation: the straight-line correlation between study time and exam score among all students like these, which the sample's r = .30 estimates.

What Your Data Should Look Like

One row per person and one numeric column for each variable. Both scores on a row must come from the same person.

Example

participant	study_hours	exam_score
1	10.1	65
2	11.1	70
3	7.5	54
4	6.8	53
5	7.2	71
6	11.7	78
… 64 more rows

	n	M	SD
Study Hours	70	8.00	3.00
Exam Score	70	70.00	10.00

Pearson correlation between the two: r = .30

Simulated data.

Download all 70 rows (data.csv)

Napkin Math

The whole test, one idea at a time. Under each heading, a line says why the step matters, and the green note reads the equation in plain English with the exam study's numbers. Collapse all to read just the headings: they tell the whole story on their own.

1. Measure How the Two Scores Move Together

When students who study more than average also score above average, both of their deviations have the same sign, and the product is positive. Averaging those products shows which way the two scores move together.

$\text{cov}_{xy} = \dfrac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n - 1}$ = $\dfrac{\sum (x_i - 8.00)(y_i - 70.00)}{70 - 1}$

= 9.01 hour-points

$x_i$ is each student's study time and $y_i$ their exam score. For each student, multiply how far they sit from 8.00 hours by how far they sit from 70.00 points, add up all 70 products, and divide by $n - 1 = 69$ (one less than $n$ , because the deviations are measured from the sample's own means). That's 9.01 hour-points: positive, so more study time tends to go with higher scores.
2. Figure Out How Much Each Score Varies on Its Own

The covariance is in hour-points, a unit nobody can picture. To get rid of the units, we need each variable's own typical spread.

$s = \sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n - 1}}$

$s_x$ = 3.00 hours
$s_y$ = 10.00 points

The same formula, run once per variable: $s_x$ = 3.00 hours of study time and $s_y$ = 10.00 exam points.
3. Put the Covariance on a Common Scale

Dividing by both spreads strips out the units and leaves a number between −1 and 1 with no units, so it can be compared across scales. It still depends on how varied the sample is and how reliably each variable was measured. This is the correlation, and it's also the effect size.

$r = \dfrac{\text{cov}_{xy}}{s_x \, s_y}$ = $\dfrac{9.01}{3.00 \times 10.00}$

= .30

$\text{cov}_{xy}$ is our 9.01 hour-points, $s_x$ is 3.00 hours and $s_y$ is 10.00 points. 9.01 ÷ (3.00 × 10.00) = 9.01 ÷ 30.0 = .30: a medium uphill trend, right at the benchmark.
4. See How Much Variation a Straight Line Accounts For

Squaring the correlation gives the share of the sample variation in one variable accounted for by its best-fitting straight-line prediction from the other, and it's the same whichever variable you predict. It's a statistical share, not a split into shared causes, and a reminder that a medium correlation still leaves most of the variation unmatched.

$r^2 = r \times r$ = $0.30 \times 0.30$

= .09

$r$ is .30, so $r^2$ = .30 × .30 = .09. A straight line through study time accounts for 9% of the variation in exam scores (and a line through exam scores for 9% of the variation in study time); the other 91% isn't linearly tied to it.
5. Account for How Much Information We Had

Testing a correlation is the same as testing the slope of a straight line through the points, and a line has two estimated parts, where it starts (its intercept) and how steeply it climbs (its slope), so two pieces of information are used up. The degrees of freedom tell us which t distribution to judge the result against.

$df = n - 2$ = $70 - 2$

= 68

$n$ is our 70 students. 70 − 2 = 68.
6. Turn the Correlation into a Test Statistic

Even when two variables are uncorrelated in the population, a sample correlation drifts away from zero by chance, and less so in bigger samples. t rescales r by the sample size so it can be compared with the t distribution, the spread of results you'd expect if the population correlation were zero.

$t = \dfrac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}$ = $\dfrac{0.30\sqrt{68}}{\sqrt{1 - 0.09}}$

= 2.59

$r$ is our .30 and $n - 2$ is 68. .30 × √68 (8.25) = 2.47, and dividing by √(1 − .09) (0.95) gives 2.59. This t is for testing a correlation of zero only. Dividing .30 by 2.59 doesn't give the standard error behind the confidence interval, which comes from the different scale in steps 7 and 8.
7. Stretch the Correlation onto a Scale Where Uncertainty Is Even

A correlation can't go past 1, so near the edges its chance variation gets squashed and lopsided. Fisher's transformation stretches the scale so the uncertainty is the same size everywhere, which makes a fair interval possible.

$z_r = \tfrac{1}{2}\ln\!\left(\dfrac{1 + r}{1 - r}\right)$ = $\tfrac{1}{2}\ln\!\left(\dfrac{1.30}{0.70}\right)$

= 0.31

$\ln$ is the natural log. 1.30 ÷ 0.70 = 1.86, its log is 0.62, and half of that is 0.31. Near zero, $z_r$ and $r$ are almost the same; they only part ways as $r$ nears ±1.
8. Give a Range of Plausible Values for the True Correlation

Build the range on the stretched scale, where it's symmetric, then convert both ends back into correlations. The result is lopsided around .30, as it should be with a ceiling at 1.

$\tanh\!\left(z_r \pm 1.96 \times \dfrac{1}{\sqrt{n - 3}}\right)$ = $\tanh\!\left(0.31 \pm 1.96 \times \dfrac{1}{\sqrt{67}}\right)$

= [.07, .50]

$1/\sqrt{n - 3}$ is the standard error on the stretched scale: 1 ÷ √67 = 0.122. 1.96 is the cutoff that holds the middle 95% of a normal curve, so 1.96 × 0.122 = 0.24 and the range is 0.31 ± 0.24, from 0.07 to 0.55. $\tanh$ undoes step 7's stretch (on a calculator, $(e^{2z} - 1)/(e^{2z} + 1)$ ): tanh(0.07) = .07 and tanh(0.55) = .50.
9. Ask How Surprising the Result Would Be If There Were No Straight-Line Relationship

Finally, ask how often a correlation at least this far from zero, in either direction, would turn up if there were no straight-line relationship in the population and the assumptions held.

$p = 2 \times P(T_{df} \le -|t|)$ = $2 \times P(T_{68} \le -2.59)$

= .012

$P(T_{68} \le -2.59)$ is the chance that a t with 68 df lands at or below −2.59. Doubling it counts both directions: .012, about 1 in 86.

Run It in Software

The code below reads the running example's data from data.csv; save it in the same folder you run the code from.

SPSS (version 30, as documented by IBM)

Analyze → Correlate → Bivariate.
Move study_hours and exam_score into Variables.
Leave Pearson and Two-tailed checked.
Click Confidence interval, ask for a 95% interval, and click Continue, then OK.

Or paste this syntax:

CORRELATIONS
  /VARIABLES=study_hours exam_score
  /PRINT=TWOTAIL NOSIG FULL
  /CI CILEVEL(95)
  /MISSING=PAIRWISE.

Reading the Output

SPSS prints the correlation as a matrix: each variable against every other, with 1s down the diagonal. Read the off-diagonal cell for Pearson Correlation and Sig. (2-tailed), and the second table for the 95% Confidence Interval. Report them in the order r, interval, p. SPSS doesn't print t.

Correlations

		study_hours	exam_score
study_hours	Pearson Correlation	1	.300*
	Sig. (2-tailed)		.012
	N	70	70
exam_score	Pearson Correlation	.300*	1
	Sig. (2-tailed)	.012
	N	70	70

* Correlation is significant at the 0.05 level (2-tailed).

Confidence Intervals

	Pearson Correlation	Sig. (2-tailed)	95% CI Lower	Upper
study_hours – exam_score	.300	.012	.070	.500

The interval uses Fisher's r-to-z transformation, as in the Napkin Math. The two halves of the matrix are the same result; read it once.

Output laid out from this page's computed results, not captured from a live run; labels and layout may differ slightly in your version.

R

study <- read.csv("data.csv")
cor.test(study$study_hours, study$exam_score)

	Pearson's product-moment correlation

data:  study$study_hours and study$exam_score
t = 2.5936, df = 68, p-value = 0.01162
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.06998938 0.49976992
sample estimates:
      cor
0.3000292

R leads with the test and puts r (cor) last. Read it in the other order: cor, then the interval, then t and p-value.

Output from a real run: R 4.6.1, run 2026-10-03.

Python

import pandas as pd
from scipy import stats

study = pd.read_csv("data.csv")
result = stats.pearsonr(study["study_hours"], study["exam_score"])
result.statistic              # 0.3000
result.pvalue                 # 0.01162
result.confidence_interval()  # (0.0700, 0.4998)

Output from a real run: Python 3.9.6 (scipy 1.12.0, pandas 2.3.3), run 2026-10-03; results shown as comments.

Impact

Effect Size

Pearson's r is its own effect size. It has no units and runs from −1 to 1, so it can be compared across scales, but it still depends on the design, how varied the sample is, and how reliably each variable was measured: independent, random measurement error generally pulls r toward zero, while correlated or systematic errors can distort it in other ways. Squaring it, r², gives the share of the sample variation in one variable accounted for by its best-fitting straight-line prediction from the other (the same whichever you predict). r² makes modest correlations look tiny, so many methodologists prefer r itself as the measure of size.

r = \dfrac{\text{cov}_{xy}}{s_x \, s_y} = \dfrac{9.01}{3.00 \times 10.00} = .30 \qquad r^2 = .09

Students who studied longer tended to score higher, a correlation of .30: right at the conventional medium benchmark. A straight line through study time accounts for 9% of the differences in exam scores; the other 91% isn't linearly tied to it. That's a statistical share, not a measure of shared causes.

In the variables' own units, students who studied an hour longer scored about 1 point higher on average (the slope of the best-fitting straight line for predicting exam score from study time). That compares different students; it doesn't say what an extra hour would do for any one of them. Benchmarks are conventions, and whether a 1-point difference matters is a question about the exam, not statistics.

Benchmarks Pearson's r at a Glance

Below Small

|r| < .10

Small

.10 ≤ |r| < .30

Medium

.30 ≤ |r| < .50

Large

|r| ≥ .50

Full benchmarks and caveats ↑

Statistical Significance

p answers one narrow question: if the two variables had no straight-line relationship in the population (ρ = 0), how often would a sample correlation land at least this far from zero, in either direction? Below the conventional α = .05, the correlation is called statistically significant.

If study time and exam scores had no straight-line relationship among all students like these, a sample of 70 would show a correlation at least this far from zero, in either direction, about 1.2% of the time, roughly 1 in 86. That's below .05, so it's significant.

p is not the probability that there's no relationship, and it says nothing about how strong the relationship is: with 1,000 people, an r of .07 would be significant too. That's what r and its interval are for.

Benchmarks p at a Glance

Very Strong

p < .001

Strong

.001 ≤ p < .01

Significant

.01 ≤ p < .05

Weak

.05 ≤ p < .10

Little

p ≥ .10

Strength of evidence against the null hypothesis.

Full benchmarks and caveats ↑

Power and Sample Size

Power is the chance that a study detects an effect that's really there. A real relationship doesn't guarantee a significant result: with too few people, it can easily come out non-significant. Power depends on four things: how strong the true correlation is, how many people you have, α, and whether the test is one- or two-sided.

Suppose, before collecting any data, we expected study time and exam scores to correlate about .30. With 85 students, the study would come out significant about 80% of the time. Our 70 students gave about 72% power: a real .30 would be missed more than a quarter of the time.

Researchers usually aim for at least 80% power. Plan it before collecting data, using the smallest correlation that would matter. Small correlations need big samples: 80% power for r = .10 takes about 783 people. These figures, and the planner's, use Fisher's z approximation for a test against a correlation of zero, with independent pairs and a two-sided α = .05 unless you change it; Cohen's (1992) table gives the same 85 and 783. Try it:

Other Info

Follow-Up and Post Hoc Tests

None needed. This test answers a single question, so there are no planned contrasts or post hoc tests to run.

Related Methods

General CaseSimple Linear Regression: the same model with a direction added. Standardize both variables and the regression slope equals r, and the t-test for the slope is identical to the t-test for r. It's the bridge from correlation to the general linear model.
Related DesignComparing Correlation Coefficients: the next step up. It takes two Pearson correlations from separate groups of people and tests whether they differ from each other, on the same Fisher's z scale this page uses for its interval.
Related DesignConvergent Validity: an application to measurement. A new measure's Pearson correlations with established measures of the same construct, and with a measure of a different one, are compared, often after correcting them for measurement error.
Special CaseItem Analysis (Classical Test Theory): Item analysis's corrected item-total correlation is a point-biserial correlation: Pearson's r between an item scored 0 or 1 and the total of the other items.
Related DesignPartial Correlation: the same correlation with one or more other variables held constant: each variable's straight-line overlap with them is removed from both before correlating. Each control variable costs one degree of freedom.
Special CaseSpearman Correlation: the rank-based version. It is Pearson's r computed on ranks instead of raw scores, so it asks about a consistent direction in the scores' order rather than a straight line, and skew and stray extreme points matter much less. Its p can come from exact, permutation or approximate methods; Pearson's t-test is only an approximation for ranks.

References

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