---
title: "Partial and multiple correlation definitions, properties"
subject: "Probability and Statistics"
chapter: 5
key: "semester_5/probability_and_statistics:partial-multiple-correlation-define"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/partial-multiple-correlation-define
site: BCT Engineering
---

# Partial and multiple correlation definitions, properties

Probability and Statistics · Chapter 5: Correlation and regression

- Asked in: 2073 Chaitra, 5 marks; 2072 Chaitra, 5 marks; 2071 Chaitra, 5 marks; 2071 Shrawan, 5 marks; 2070 Chaitra, 6 marks (asked 5 times)
- Video: [Multiple and Partial Correlation Coefficients and Their Basic Properties | ONE SHOT | GATE STATIS...](https://www.youtube.com/watch?v=yXIqx8hs4cc) — Statistics by Punam
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/partial-multiple-correlation-define
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5

## Question

Define partial and multiple correlation with suitable examples. Write down the properties of partial and multiple correlation.

## Short answer

Partial correlation (e.g., r₁₂.₃) measures linear association between two variables controlling for others; it lies between -1 and 1. Multiple correlation (e.g., R₁.₂₃) measures association between one variable and a set of predictors; it lies between 0 and 1, and its square is the proportion of variance explained.

## Answer

#### Partial correlation

Partial correlation measures the linear relationship between two variables when the linear effect of one or more other variables is removed (held constant). For three variables, the partial correlation of X₁ and X₂ keeping X₃ constant is

r₁₂.₃ = (r₁₂ − r₁₃r₂₃) / √[(1 − r₁₃²)(1 − r₂₃²)]

Example: the correlation between a student's marks in Mathematics (X₁) and Physics (X₂) is partly due to their general intelligence/study hours (X₃). r₁₂.₃ gives the correlation between Mathematics and Physics marks after removing the effect of study hours.

#### Multiple correlation

Multiple correlation measures the degree of linear relationship between one variable (dependent) and the combined effect of two or more other variables (independent). For three variables,

R₁.₂₃ = √[(r₁₂² + r₁₃² − 2r₁₂r₁₃r₂₃) / (1 − r₂₃²)]

Example: the correlation between crop yield (X₁) and the combined effect of rainfall (X₂) and fertilizer (X₃).

#### Properties of partial correlation

1. It lies between −1 and +1: −1 ≤ r₁₂.₃ ≤ 1.
2. It is symmetric: r₁₂.₃ = r₂₁.₃.
3. Its order equals the number of variables held constant (r₁₂.₃ is of first order; r₁₂ is of order zero).
4. It is a simple correlation of the residuals of X₁ and X₂ after regressing each on X₃.
5. Its sign depends on the numerator r₁₂ − r₁₃r₂₃.
6. If X₃ is uncorrelated with both X₁ and X₂ (r₁₃ = r₂₃ = 0), then r₁₂.₃ = r₁₂.

#### Properties of multiple correlation

1. It is never negative: 0 ≤ R₁.₂₃ ≤ 1.
2. R₁.₂₃ ≥ |r₁₂| and R₁.₂₃ ≥ |r₁₃|; it is at least as large as any simple correlation of X₁ with X₂ or X₃.
3. R₁.₂₃ = 1 means X₁ is an exact linear function of X₂ and X₃; R₁.₂₃ = 0 means X₁ is uncorrelated with both.
4. R₁.₂₃² is the coefficient of multiple determination: the proportion of variation of X₁ explained by X₂ and X₃.
5. It is the simple correlation between X₁ and its estimate X̂₁ from the multiple regression of X₁ on X₂, X₃.
6. R₁.₂₃ ≠ R₂.₁₃ in general (it depends on which variable is dependent).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Partial correlation r12.3 measures:
   A. The linear relation of X1 with X2 and X3 combined
   B. The correlation of X1 and X2 after removing the linear effect of X3
   C. The correlation of X3 with the average of X1 and X2
   D. The nonlinear relation between X1 and X2
   Answer: B. It is the correlation of two variables with the linear effect of the other variable(s) held constant. (Key point: Partial correlation)

2. Multiple correlation R1.23 measures:
   A. Correlation of X1 and X2 with X3 held constant
   B. The average of r12 and r13
   C. The relationship between X2 and X3
   D. The linear relationship between dependent X1 and the combined effect of X2 and X3
   Answer: D. R1.23 is the correlation between X1 and the combined effect of X2 and X3. (Key point: Multiple correlation)

3. Which is true of r12.3?
   A. r12.3 = r21.3 and it lies between -1 and 1
   B. It always lies between 0 and 1
   C. r12.3 differs from r21.3
   D. It is of order 2
   Answer: A. Partial correlation is symmetric, lies in [−1, 1], and r12.3 is of first order (one variable held constant). (Key point: Properties of partial correlation)

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