Quick Answer: Partial Correlation Result

The main SPSS, Python, R and Excel report tested the association between G1 first-period grade and G3 final grade while controlling G2 second-period grade. The sample contained N = 649 complete cases. Before controlling G2, the zero-order correlation between G1 and G3 was very strong, r = 0.8264.

After the shared influence of G2 was removed, the partial correlation between G1 and G3 dropped to r = 0.1606. The test statistic was t(646) = 4.1359, with a two-tailed p = 0.000040. The relationship remained statistically significant, but the practical interpretation changed from a very strong raw relationship to a small adjusted relationship.

What Is Partial Correlation?

Partial correlation is a correlation coefficient that measures the relationship between two variables after one or more other variables have been statistically controlled. It is useful when a simple correlation may be inflated because both variables are connected to a third variable.

In the worked example, G1 and G3 are strongly correlated before any adjustment. That makes sense because a student who performs well early in a course often performs well at the final stage too. However, G2 sits between them as a very strong related grade variable. Partial correlation removes the part of G1 and G3 that is predictable from G2, then correlates the remaining residual parts.

Because partial correlation is based on adjusted residual relationships, it is often used in education, psychology, health research, economics and social science. It helps the researcher ask whether an association is still present after a specific control variable is held constant.

Zero-Order vs Partial vs Semi-Partial Correlation

Students often confuse zero-order correlation, partial correlation and semi-partial correlation. The difference is based on how the control variable is removed.

Partial Correlation Formula

When there is one control variable, the partial correlation formula uses three ordinary Pearson correlations: the correlation between X and Y, the correlation between X and Z, and the correlation between Y and Z.

The significance test for a partial correlation uses this t statistic:

For the main result, this gives t(646) = 4.1359 and p = 0.000040. The result is statistically significant, but the adjusted coefficient is small compared with the original zero-order correlation.

Null and Alternative Hypotheses for Partial Correlation

The hypothesis statement should name the two main variables and the control variable. In this guide, the main question is whether G1 and G3 remain related after controlling G2.

Dataset and Variables Used

The worked example uses the student performance dataset. The main pair is G1 and G3, and the control variable is G2. The logic is educationally meaningful: G1 is the first-period grade, G2 is the second-period grade and G3 is the final grade.

Before interpreting the partial correlation, the dataset should be checked with descriptive statistics, correlation matrix, correlation heatmap, correlation assumptions, p-value, confidence interval and effect size guides.

SPSS Output Interpretation for Partial Correlation

The SPSS output should be read in two stages. First, the ordinary correlation matrix shows the raw relationship between G1 and G3. Second, the partial correlation table shows the adjusted relationship after G2 is controlled. The important change is the drop from r = 0.8264 to partial r = 0.1606.

Open the SPSS Partial Correlation output PDF

SPSS Interpretation Summary

The SPSS result means that G1 still has a statistically significant positive relationship with G3 after G2 is controlled. However, the effect is no longer large. This is the main substantive finding: the raw G1-G3 correlation is very strong, but most of it is explained by the grade progression captured by G2.

Python Chart-by-Chart Interpretation

The Python charts show the full story visually: the strong zero-order relationship, the weaker residual relationship after controlling G2, the coefficient drop, the adjusted matrix, the strongest remaining adjusted pairs and the p-value summary.

Python Chart 1: Zero-Order Scatterplot

The zero-order scatterplot shows a strong positive relationship between G1 and G3. Students with higher G1 scores generally have higher G3 scores. This raw pattern explains why the zero-order correlation is very large at r = 0.8264.

This chart is important because it shows the relationship before statistical control. If the analysis stopped here, the report would conclude that G1 and G3 are strongly related. Partial correlation asks whether this relationship remains after G2 is removed.

Python Chart 2: Partial Residual Scatterplot

The residual scatterplot shows the relationship between the part of G1 not predicted by G2 and the part of G3 not predicted by G2. The positive trend is still visible, but it is much weaker than the raw scatterplot. This is why the partial correlation is only r = 0.1606.

The chart supports the final interpretation: G1 has a small unique relationship with G3 after G2 is held constant. The association is statistically significant because the sample is large, but the residual pattern is not strong.

Python Chart 3: Zero-Order vs Partial Correlation

This chart makes the main result easy to understand. The zero-order correlation is 0.8264, while the partial correlation is 0.1606. The drop is large because G2 accounts for much of the shared relationship between G1 and G3.

The decision is not simply that the relationship “disappeared.” The adjusted relationship remains significant. The correct conclusion is that the original strong relationship becomes small once G2 is controlled.

Python Chart 4: Partial Correlation Heatmap

The heatmap displays adjusted correlations among several variables after controlling G2. The main G1-G3 cell is positive but modest at about 0.161. Other relationships remain stronger, especially the adjusted relationship between mother’s education and father’s education at about 0.626.

This chart shows that controlling G2 does not affect every relationship in the same way. Grade relationships involving G1 and G3 shrink strongly, while some background variables remain related after the same control variable is removed.

Python Chart 5: Largest Pairwise Partial Correlations

The ranking chart highlights the largest adjusted associations in the matrix. The strongest remaining relationship is Medu vs Fedu, around 0.626. Other notable adjusted relationships include age with failures, G1 with age, G1 with G3 and age with absences.

This chart provides context for the main result. The adjusted G1-G3 relationship is significant, but it is not the strongest relationship in the controlled matrix. It is better described as a small unique relationship rather than a dominant association.

Python Chart 6: Partial Correlation P-Values

The p-value chart separates statistical significance from effect size. The G1-G3 partial correlation has p = 0.000040, so it is statistically significant. Some other adjusted relationships are not significant, such as G3 with age, which has p around 0.600.

This chart supports careful reporting. A small coefficient can be statistically significant when the sample is large, and a visually weak residual relationship can still pass the p-value test.

R Chart-by-Chart Validation

The R charts validate the same analysis using a separate workflow and colorful chart style. They repeat the zero-order view, residual view, coefficient comparison, heatmap, largest adjusted pairs and p-value display.

R Chart 1: Colorful Zero-Order Scatterplot

The R zero-order scatterplot confirms the strong raw G1-G3 pattern. The points follow a clear upward trend, matching the high zero-order correlation of 0.8264. This confirms that the raw association is not a Python-only result.

The chart provides the baseline against which the partial residual chart should be compared. The raw correlation is strong because G1, G2 and G3 are all grade measures that move together.

R Chart 2: Colorful Partial Residual Scatterplot

The R residual scatterplot confirms the weaker adjusted relationship. Once G2 is removed from both G1 and G3, the residual dots are much more dispersed. The remaining positive slope is small, matching the partial correlation of 0.1606.

This chart is the visual proof for the main interpretation. G1 still matters after G2 is controlled, but it no longer dominates the final grade relationship.

R Chart 3: Colorful Zero-Order vs Partial Correlation

The R comparison chart repeats the same coefficient drop from 0.8264 to 0.1606. This is the most direct chart for explaining why partial correlation changes the conclusion.

The chart should be described as a reduction in the unique relationship, not as a contradiction. The raw relationship includes shared G2 information; the partial relationship removes that shared component.

R Chart 4: Colorful Partial Correlation Heatmap

The R heatmap validates the Python heatmap. The G1-G3 adjusted cell remains positive and small, while Medu-Fedu remains one of the strongest adjusted relationships. This reinforces that partial correlation is pair-specific and context-specific.

Use this chart when readers need the broader matrix rather than only the main G1-G3 result. It shows that controlling G2 affects grade-related relationships especially strongly.

R Chart 5: Colorful Largest Pairwise Partial Correlations

The R ranking chart confirms which adjusted pairs remain strongest. Medu and Fedu stand out, followed by other moderate or small adjusted relationships. The main G1-G3 partial correlation is present but not large.

This chart helps prevent overstatement. Statistical significance is not the same as dominance. In the controlled matrix, the adjusted G1-G3 link is meaningful but small.

R Chart 6: Colorful Partial Correlation P-Values

The R p-value chart confirms that the main partial correlation is statistically significant. It also shows that not every adjusted coefficient reaches significance, which is why both coefficient size and p-value must be reported together.

The final R interpretation is the same as the Python interpretation: the G1-G3 relationship remains significant after controlling G2, but its adjusted magnitude is small.

Excel Results Explained

The Excel workbook gives the clearest formula-based view of the analysis. It contains setup values, descriptive statistics, the zero-order correlation matrix, the partial correlation matrix, p-values, a worked example and a dashboard. The worked example focuses on G1 and G3 controlling G2.

Excel Worked Example: G1 and G3 Controlling G2

The Excel result shows exactly why partial correlation is useful. G1 and G3 look very strongly related before adjustment. But G2 is very strongly related to both of them, so it absorbs most of the shared grade-performance information. After this shared G2 component is removed, the remaining G1-G3 association is only 0.1606.

The workbook also includes a wider partial correlation matrix controlling G2. Important values include G1 with age = -0.1637, G3 with failures = -0.1068, age with failures = 0.3038, and Medu with Fedu = 0.6257. The corresponding p-value table shows that the main G1-G3 partial relationship is significant, while some other adjusted relationships are not.

Excel Formula Steps Used in the Workbook

SPSS, R, Python and Excel Workflows for Partial Correlation

The same analysis can be reproduced in all four tools. The workflow always requires the same logic: choose X and Y, choose the control variable, compute the raw correlation, remove the control variable and test the adjusted relationship.

Code Blocks for Partial Correlation

SPSS Syntax for Partial Correlation

* Partial Correlation in SPSS.
* Example: G1 and G3 controlling G2.

OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Partial_Correlation_Output.

GET DATA
  /TYPE=TXT
  /FILE='dataset.csv'
  /DELCASE=LINE
  /DELIMITERS="," 
  /QUALIFIER='"'
  /ARRANGEMENT=DELIMITED
  /FIRSTCASE=2
  /VARIABLES=
    school A10 sex A5 age F8.0 Medu F8.0 Fedu F8.0 studytime F8.0 failures F8.0 absences F8.0 G1 F8.2 G2 F8.2 G3 F8.2.
CACHE.
EXECUTE.

DESCRIPTIVES VARIABLES=G1 G3 G2 age studytime failures absences Medu Fedu
  /STATISTICS=MEAN STDDEV VARIANCE MIN MAX.

CORRELATIONS
  /VARIABLES=G1 G3 G2 age studytime failures absences Medu Fedu
  /PRINT=TWOTAIL
  /MISSING=LISTWISE.

PARTIAL CORR
  /VARIABLES=G1 G3 BY G2
  /SIGNIFICANCE=TWOTAIL
  /MISSING=LISTWISE.

PARTIAL CORR
  /VARIABLES=G1 G3 age studytime failures absences Medu Fedu BY G2
  /SIGNIFICANCE=TWOTAIL
  /MISSING=LISTWISE.

OUTPUT SAVE OUTFILE='Partial-Correlation-SPSS-Output.spv'.
OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE='Partial-Correlation-SPSS-Output.pdf'.

Python Code for Partial Correlation

import pandas as pd
import numpy as np
from scipy import stats

# Load data
raw = pd.read_csv("dataset.csv")
vars_needed = ["G1", "G3", "G2", "age", "studytime", "failures", "absences", "Medu", "Fedu"]
df = raw[vars_needed].apply(pd.to_numeric, errors="coerce").dropna()

# Main variables
x = df["G1"].to_numpy(dtype=float)
y = df["G3"].to_numpy(dtype=float)
z = df[["G2"]].to_numpy(dtype=float)

# Zero-order correlation
r_xy, p_xy = stats.pearsonr(x, y)

# Residualize x and y on the control variable G2
Z = np.column_stack([np.ones(len(df)), z])
bx = np.linalg.lstsq(Z, x, rcond=None)[0]
by = np.linalg.lstsq(Z, y, rcond=None)[0]
x_resid = x - Z @ bx
y_resid = y - Z @ by

# Partial correlation is correlation between residuals
r_partial, p_partial = stats.pearsonr(x_resid, y_resid)
n = len(df)
k = 1
df_error = n - k - 2
t_value = r_partial * np.sqrt(df_error / (1 - r_partial**2))
p_from_t = stats.t.sf(abs(t_value), df_error) * 2

print("N:", n)
print("Zero-order r(G1,G3):", round(r_xy, 4))
print("Partial r(G1,G3 | G2):", round(r_partial, 4))
print("t statistic:", round(t_value, 4))
print("df:", df_error)
print("p-value:", p_from_t)

R Code for Partial Correlation

# Partial Correlation in R: G1 and G3 controlling G2

df <- read.csv("dataset.csv")
vars <- c("G1", "G3", "G2", "age", "studytime", "failures", "absences", "Medu", "Fedu")
work <- na.omit(df[vars])

# Zero-order correlation
zero_order <- cor(work$G1, work$G3)

# Formula using three Pearson correlations
r_xy <- cor(work$G1, work$G3)
r_xz <- cor(work$G1, work$G2)
r_yz <- cor(work$G3, work$G2)
partial_r <- (r_xy - r_xz * r_yz) / sqrt((1 - r_xz^2) * (1 - r_yz^2))

n <- nrow(work)
k <- 1
df_error <- n - k - 2
t_value <- partial_r * sqrt(df_error / (1 - partial_r^2))
p_value <- 2 * pt(abs(t_value), df = df_error, lower.tail = FALSE)

cat("N =", n, "\n")
cat("Zero-order r(G1,G3) =", round(zero_order, 4), "\n")
cat("Partial r(G1,G3 | G2) =", round(partial_r, 4), "\n")
cat("t =", round(t_value, 4), "df =", df_error, "p =", p_value, "\n")

# Residual method validation
x_resid <- residuals(lm(G1 ~ G2, data = work))
y_resid <- residuals(lm(G3 ~ G2, data = work))
cor.test(x_resid, y_resid)

Excel Formulas for Partial Correlation

Assume:
X = G1 range
Y = G3 range
Z = G2 control range

Zero-order correlations:
=CORREL(X_range,Y_range)     // rxy
=CORREL(X_range,Z_range)     // rxz
=CORREL(Y_range,Z_range)     // ryz

Partial correlation controlling one variable:
=(rxy-rxz*ryz)/SQRT((1-rxz^2)*(1-ryz^2))

Sample size:
=COUNT(X_range)

Degrees of freedom with one control:
=n-1-2

Test statistic:
=partial_r*SQRT(df/(1-partial_r^2))

Two-tailed p-value:
=T.DIST.2T(ABS(t),df)

Decision:
=IF(p_value<0.05,"Significant partial relationship","Not significant partial relationship")

APA Reporting Wording for Partial Correlation

When reporting a Partial Correlation, mention the two main variables, the control variable, sample size, degrees of freedom, zero-order correlation, partial correlation, p-value and practical interpretation.

APA-Style Full Report

Short APA-Style Version

Excel Worked-Example Wording

Common Mistakes in Partial Correlation Interpretation

When to Use Partial Correlation

Use partial correlation when you want to measure the relationship between two continuous variables while holding one or more additional variables constant. It is especially useful when a third variable may explain, inflate or hide the raw relationship.

Partial correlation is useful in education when earlier grades, attendance or background factors may affect later grades. It is useful in psychology when age, baseline ability or scale scores must be controlled. It is useful in health research when age, body mass index or baseline values must be held constant. If the goal is prediction rather than adjusted association, consider regression instead.

Downloads and Resources for Partial Correlation

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