Types of Correlation: Direction, Form, Variables

Q: What are the 3 types of correlation?

Positive, negative and zero correlation, which classify the direction of a relationship. Other schemes classify by form (linear, nonlinear) and by number of variables (simple, partial, multiple).

Q: What are the types of correlation in psychology?

Psychology reports the same types: positive, negative and zero, usually shown on a scatter plot and summarised with Pearson’s r , or Spearman’s coefficient for ranked measures.

Q: What do the types of correlation look like on a scatter diagram?

Positive: points slope upward. Negative: points slope downward. Zero: no consistent slope. Linear: points hug a straight line. Nonlinear: points follow a curve. See negative correlation scatter plots .

Q: Can a correlation be zero when variables are related?

Yes. A curved relationship can give r near 0 because r measures only linear association.

Correlation is not one thing. A correlation can be classified by direction (positive, negative or zero), by form (linear or nonlinear, monotonic or not) and by how many variables are involved (simple, partial or multiple). Each classification points to a different coefficient, so naming the type first tells you which statistic to compute. This guide defines each type, shows which coefficient fits it, and works one example with the arithmetic shown.

Types of correlation at a glance

The three classifications answer different questions, and one relationship can sit in all three at once (for example, a positive, monotonic, simple correlation).

Classification	Types	Question it answers
Direction	Positive, negative, zero	Do the variables move together, in opposite directions, or not systematically?
Form	Linear, nonlinear (monotonic or non-monotonic)	Do the points follow a straight line, a curve that always rises or falls, or a curve that changes direction?
Number of variables	Simple (bivariate), partial, multiple	Two variables only, two variables with others held constant, or one variable against several?

Types of correlation by direction

Positive correlation

In a positive correlation, high values of one variable tend to go with high values of the other, and low with low. In OpenStax’s words, a positive r means that y tends to rise as x rises. On a scatter plot the cloud of points slopes upward from left to right. See strong positive correlation for examples.

Negative correlation

In a negative correlation, high values of one variable tend to go with low values of the other. A negative r means y tends to fall as x rises, and the cloud slopes downward from left to right.

Zero correlation (no correlation)

With zero correlation, the scatter plot shows no systematic upward or downward drift. OpenStax cautions that if r = 0 there is likely no linear correlation; a curved pattern can also produce r near 0, which is why you should always look at the plot. Worked cases are in no-correlation examples.

Types of correlation by form: linear, monotonic and nonlinear

Linear correlation means the points cluster around a straight line. OpenStax describes the linear relationship as strong when the points are close to a straight line. Pearson’s r measures the strength and direction of exactly this linear association.

Monotonic correlation means one variable never decreases (or never increases) as the other rises, but the path need not be straight. The SciPy documentation describes Spearman’s coefficient as a nonparametric measure of the monotonicity of a relationship, with −1 or +1 implying an exact monotonic relationship. See monotonic relationships.

Nonlinear, non-monotonic correlation includes U-shaped and other curves that change direction. Neither Pearson’s nor Spearman’s coefficient summarises these well, and r can be close to 0 even when the variables are strongly related. See nonlinear relationships.

Types of correlation by number of variables

Simple (bivariate) correlation relates two variables and ignores everything else. It is what most people mean by “the correlation.”

Partial correlation measures the association between two variables after the linear effect of one or more other variables (covariates) has been removed from both. For a single control variable z the standard formula is:

r_xy·z = (r_xy − r_xzr_yz) / √[(1 − r_xz²)(1 − r_yz²)]

where r_xy, r_xz and r_yz are the simple Pearson correlations between each pair. Partial correlation is one way to check whether an apparent association survives control for a third variable; it does not by itself establish causation (see spurious correlation).

Multiple correlation (R) is the correlation between one outcome and the best linear combination of several predictors; R² is the familiar share of variance explained in multiple regression.

Which coefficient fits which type?

Situation	Coefficient	Reason
Two numeric variables, roughly straight-line pattern	Pearson r	Measures strength and direction of linear association (OpenStax 12.3)
Curved but consistently rising or falling pattern, or ordinal data	Spearman ρ (r_s)	Measures monotonicity (SciPy documentation)
Ranked data with many ties or small samples	Kendall τ	Rank-based; see Kendall’s tau
Association with a third variable removed	Partial correlation	Formula above
One outcome, several predictors	Multiple R	Correlation of outcome with fitted values from regression

The full comparison of Pearson, Spearman, Kendall and point-biserial is in the correlation coefficient guide.

Worked example: monotonic but not linear

Take six pairs where y = x²: x = 1, 2, 3, 4, 5, 6 and y = 1, 4, 9, 16, 25, 36. The relationship is perfectly monotonic but curved, so the two coefficients disagree.

Pearson. Means: x̄ = 21/6 = 3.5 and ȳ̄ = 91/6 = 15.1667. Sums of squares and products about the means: S_xx = 17.5, S_yy = 894.833, S_xy = 441 − 6 × 3.5 × 15.1667 = 122.5. Then r = 122.5 / √(17.5 × 894.833) = 122.5 / 125.14 = 0.979.

Spearman. The ranks of x and y are identical (1 to 6), so the rank correlation is exactly 1.000.

Pearson is below 1 because the points do not lie on a straight line; Spearman is exactly 1 because y never falls as x rises. Both values were computed in Python with SciPy (pearsonr gives p = .0007).

Interpretation: reading the result

Result	What it says
r positive, near +1	Strong linear tendency for y to rise with x
r negative, near −1	Strong linear tendency for y to fall as x rises
r near 0	Likely no linear correlation; check the scatter plot for a curve or a horizontal pattern
Spearman near ±1 but Pearson lower	Relationship is monotonic but curved

The coefficient always lies between −1 and +1 (OpenStax 12.3), and a strong correlation does not imply causation.

Assumptions and common mistakes

Reporting Pearson’s r without plotting the data; curved patterns and outliers can hide or fake a linear relationship.
Reading r = 0 as “no relationship” rather than “no linear relationship.”
Using Pearson for ordinal data; Spearman or Kendall fits ranked scales (see levels of measurement).
Treating a correlation as evidence of cause.

How to report a correlation (APA style)

For the worked example: Pearson’s r(4) = .98, p < .001, with degrees of freedom n − 2 = 4. Report the sample size, the coefficient used, and the direction in words. For a rank correlation, state that Spearman’s was used and give r_s. See APA statistics reporting for the general conventions.

How to calculate each type in Excel, R or Python

Excel. Pearson: =CORREL(A2:A7,B2:B7). Spearman (Excel 365, ranks computed inline): =CORREL(RANK.AVG(A2:A7,A2:A7,1),RANK.AVG(B2:B7,B2:B7,1)). More in finding the correlation coefficient in Excel.

x <- 1:6; y <- x^2cor(x, y)                       # Pearson, 0.979cor(x, y, method = "spearman")  # 1cor.test(x, y)                  # r, df, p-value

Python.

import numpy as npfrom scipy import statsx = np.arange(1, 7); y = x**2print(stats.pearsonr(x, y))     # statistic 0.979, p 0.0007print(stats.spearmanr(x, y))    # statistic 1.0print(stats.kendalltau(x, y))# partial correlation of x and y controlling for zdef partial_r(x, y, z):    rxy = np.corrcoef(x, y)[0, 1]    rxz = np.corrcoef(x, z)[0, 1]    ryz = np.corrcoef(y, z)[0, 1]    return (rxy - rxz * ryz) / np.sqrt((1 - rxz**2) * (1 - ryz**2))

The pingouin library also provides partial_corr(data, x, y, covar, method).

Frequently asked questions

What are the 3 types of correlation?