Correlation Coefficient Calculator

StatisticsLast updated: August 17, 2026

Correlation is a statistical measure of the tendency of two numeric variables to vary together. The Pearson correlation coefficient (r) summarizes both the direction and the strength of that relationship in a single number, and it always falls between −1 and +1: the closer it is to +1 the stronger the positive linear relationship, the closer to −1 the stronger the negative one, and values near 0 indicate that no linear relationship is present.

From your X and Y data, this calculator computes Pearson r, the explained variance (r²), the t statistic, and the two-tailed p-value; it also reports the nonparametric Spearman rho coefficient for ranked data and situations with outliers. For students, academics, and researchers writing theses and papers, it produces a ready-made interpretation sentence in APA format.

Separate values with commas, spaces, or line breaks. Both decimal points and decimal commas are accepted.
Each value must belong to the observation (the same participant) in the same position in the X field.

Thresholds for Interpreting a Correlation Coefficient

The sign of a correlation coefficient gives the direction of the relationship and its absolute value gives the strength. The bounds below are the thresholds this calculator uses to produce its "direction and strength of the relationship" label; they apply in the same way to Pearson r and to Spearman rho.

Negligible relationship
|r| < 0.10
In practice there is no linear co-variation between the two variables
Weak relationship
0.10 – 0.30
The shared variance is between 1% and 9%; it can come out significant in large samples, but its practical value is limited
Moderate relationship
0.30 – 0.50
A level frequently met in the social sciences; the shared variance is between 9% and 25%
Strong relationship
0.50 – 0.70
Between 25% and 49% of the variance in one variable is shared with the other
Very strong relationship
|r| ≥ 0.70
The shared variance is above 49%; among scale items this level can also point to item redundancy
Significance threshold
p < 0.05
Whether the coefficient differs from zero in the population is tested two-tailed against a t distribution with n − 2 degrees of freedom

The calculator labels the r value it computes according to these boundsKaynak: Widely used correlation interpretation bounds based on the effect size criteria of Cohen (1988)

Formula

r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √[ Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)² ]
t = r · √(n − 2) / √(1 − r²)  ,  df = n − 2
Spearman ρ: the values are converted to ranks and the same formula is applied

Whether the computed r differs from zero in the population is tested two-tailed against a t distribution with n − 2 degrees of freedom; if p < 0.05 the relationship is taken to be significant.

How to Calculate

  1. Pair up your data so that every participant (observation) has both an X and a Y value; leave out observations with a missing measurement.
  2. Paste the X and Y values into the two fields in the same order — separate them with commas, spaces, or line breaks.
  3. In the results section, read r, r², t, the degrees of freedom, the p-value, and Spearman rho.
  4. Copy the APA-formatted sentence from the interpretation line (e.g. r(10) = 0.64; p = 0.026) into your report, and interpret direction, strength, and significance together.

Worked Examples

Study hours and exam score

Across 12 students, weekly study hours (X) and exam score (Y) show a positive, strong, and significant relationship: r = 0.637; r(10) = 0.64; p = 0.026. Because r² = 0.406, about 40.6% of the variance in scores is shared with study time.

Pearson correlation coefficient (r): 0.637 · Direction and strength of the relationship: positive, strong · r² (explained variance): 0.406 (40.6%)

Screen time and sleep (a negative relationship)

For 10 people, daily screen time (in hours) and sleep duration show a negative, very strong relationship: r = -0.979; p < 0.001. As screen time rises, sleep duration falls markedly; Spearman rho supports the same conclusion at -0.982.

Pearson correlation coefficient (r): -0.979 · Direction and strength of the relationship: negative, very strong · r² (explained variance): 0.959 (95.9%)

A non-significant relationship

In this set of 8 observations, r = 0.043 is very close to zero and the relationship is not significant: r(6) = 0.04; p = 0.919. There is no evidence of a linear relationship between the two variables.

Pearson correlation coefficient (r): 0.043 · Direction and strength of the relationship: positive, negligible · r² (explained variance): 0.002 (0.2%)

Frequently Asked Questions

How large does a correlation coefficient have to be to count as high?
Values of r whose absolute value exceeds 0.50 are generally treated as strong, and those above 0.70 as very strong. The 0.30 to 0.50 range indicates a moderate relationship and 0.10 to 0.30 a weak one. These bounds are widely accepted guidelines rather than hard rules; the best practice is to interpret your result against the literature in your own field.
What does a negative correlation mean?
A negative correlation means that as one variable increases the other tends to decrease; a negative relationship is expected between screen time and sleep duration, for instance. The sign of r gives only the direction of the relationship; its strength is read from the absolute value. An r of −0.60 and an r of +0.60 describe relationships of equal strength.
If r comes out near zero, does that mean there is no relationship at all?
No. An r near zero only shows that there is no linear relationship. The variables may still have a strong non-linear relationship, such as a U shape, which Pearson r cannot capture. For that reason it is always advisable to examine a scatter plot of the data before interpreting the coefficient.
When should Spearman be used instead of Pearson?
Spearman rho (ρ) is preferred for ordinal measurements such as Likert-type scales and ranking data, or when there are pronounced outliers. Because Spearman is computed from ranks rather than raw values, it requires no normality assumption and is far less affected by extreme values. This calculator reports both coefficients together.
How much data does a correlation analysis need?
Technically 3 data pairs are enough; with a sample that small, however, the result is extremely unstable. For a stable estimate of a correlation, the literature generally recommends at least 30 observations. The smaller the sample, the wider the confidence interval, and a single outlier can distort r severely.
Is correlation evidence of causation?
No. A correlation only shows that two variables vary together; it cannot say which one causes the other, or whether the relationship arises from a third variable. The classic example: ice cream sales correlate with drowning incidents, but the real driver is air temperature. Establishing causation requires an experimental design.