Quick Answer: ANCOVA Result

The worked ANCOVA example asks whether G3 final grade differs by school after adjusting for G2 prior grade. This is different from ordinary ANOVA. ANOVA compares raw group means, while ANCOVA compares adjusted group means after removing the predictable effect of a covariate.

The Python adjusted-means chart shows GP with an adjusted G3 mean of 11.999 and MS with an adjusted G3 mean of 11.732. The raw means were farther apart, but the adjustment for G2 made the school difference much smaller. The school-effect F curve shows an observed F value of 5.99, which is above the critical F value of 3.86, so the adjusted school effect is statistically meaningful in that output.

What Is ANCOVA?

ANCOVA stands for Analysis of Covariance. It is used when you want to compare group means while statistically controlling for a continuous variable that is related to the outcome. The controlled variable is called a covariate.

In this guide, school is the group factor, G3 is the final grade outcome, and G2 is the covariate. The goal is to compare the G3 means of school groups after placing the groups on the same G2 level. This produces adjusted means, also called estimated marginal means.

The key difference between ANCOVA vs ANOVA is adjustment. ANOVA tests whether raw group means differ. ANCOVA tests whether group means differ after removing variation explained by the covariate. If the covariate is strongly related to the dependent variable, ANCOVA gives a cleaner comparison than a raw mean comparison.

ANCOVA is connected to many assumption and diagnostic topics, including Descriptive Statistics, Variance, Standard Error, Confidence Interval, P Value, Effect Size, Null and Alternative Hypothesis, and Parametric vs Nonparametric Tests.

ANCOVA Formula

A one-way ANCOVA model with one covariate can be written as:

In this formula, Y_ij is the dependent variable value, μ is the grand mean, τ_i is the group effect, β is the covariate slope, X_ij is the covariate value, X̄ is the covariate mean, and ε_ij is the residual error.

Adjusted Mean Formula

The adjusted mean estimates what each group mean would be if all groups had the same average covariate value. This is the core idea behind ANCOVA. It changes the comparison from raw means to covariate-controlled means.

ANCOVA F Statistic

The ANCOVA F statistic tests whether adjusted group differences are large compared with adjusted residual error. If the F statistic is large and the p value is below .05, the adjusted group effect is statistically significant.

Diagnostic Null and Alternative Hypothesis for ANCOVA

ANCOVA has a main group hypothesis, a covariate hypothesis and an assumption-check hypothesis for homogeneous regression slopes. These should be separated clearly in reporting.

Dataset and ANCOVA Variables Used

The worked example uses the student performance dataset. The dependent variable is G3 final grade. The grouping factor is school. The covariate is G2 prior grade. Studytime is used in supporting context charts, but the core ANCOVA model compares school groups after controlling G2.

Before final reporting, ANCOVA assumptions should be checked using residual plots, Q-Q plots, variance checks and influence diagnostics. Useful supporting guides include Shapiro-Wilk Test, Jarque-Bera Test, Anderson-Darling Test, Q-Q Plot Normality Check, P-P Plot Normality Check, Skewness and Kurtosis Normality Check, Studentized Residuals, Cook’s Distance, and Mahalanobis Distance.

Python Chart-by-Chart Interpretation

The Python charts below show the ANCOVA workflow for school and G2. They explain the adjusted regression, adjusted means, residual behavior, F statistic, raw-versus-adjusted comparison, covariate distribution and studytime context.

Python Chart 1: G2 vs G3 by School Adjusted Regression

This chart compares G2 prior grade with G3 final grade for GP and MS schools. Both fitted regression lines rise strongly from left to right, which shows that higher G2 is associated with higher G3. The two school lines run close to each other and appear almost parallel across most of the G2 range.

The parallel pattern supports the ordinary ANCOVA assumption that the covariate slope is similar across school groups. The plot also shows a few low-score cases near the bottom of the graph, especially at moderate G2 values, which should be checked as possible unusual prediction errors or influential observations.

In the article report, this chart should be used to say that G2 is a strong covariate for G3 and that the school comparison should be interpreted after checking the school × G2 interaction. The chart supports using G2 as an adjustment variable before comparing school means.

Python Chart 2: Adjusted Means by School

This chart shows the adjusted mean G3 for each school at the mean level of G2. GP has an adjusted mean of 11.999, while MS has an adjusted mean of 11.732. The adjusted difference is small, about 0.267 grade points.

The chart is important because it shows the final ANCOVA-style group comparison, not the raw group comparison. The adjusted school means are close, which means that much of the raw school difference is reduced after controlling for G2 prior grade.

In reporting, these adjusted means should be written before the school-effect F test. A correct sentence is: “After adjusting for G2, the estimated mean G3 score was 11.999 for GP and 11.732 for MS.” The p value and F statistic should then be used to decide whether this adjusted difference is statistically significant.

Python Chart 3: Residuals vs Fitted Values

This chart compares ANCOVA residuals with fitted G3 values. Most residuals are scattered around the zero line across fitted values from about 5 to 19, which suggests that the model is generally capturing the main G3 pattern.

The plot also shows several large negative residuals, including values below -6 and near -9. These cases represent students whose observed G3 scores were much lower than the ANCOVA model predicted. One positive residual above 5 is also visible. These points do not create a clear curved pattern, but they do show that unusual prediction errors exist.

The chart supports a cautious residual interpretation. The model does not show a strong funnel or systematic curve, but the large negative residuals should be mentioned as possible unusual cases. In reporting, say that residuals were mostly centered around zero, with several large negative residuals reviewed through outlier or influence diagnostics.

Python Chart 4: School Effect F Distribution Curve

This chart shows the school-effect F statistic after adjusting for G2. The dashed critical line is at F = 3.86, while the observed school-effect line is at F = 5.99. The observed value is to the right of the critical value.

Because the observed F statistic exceeds the critical value, the Python output supports rejecting the null hypothesis for the adjusted school effect at the .05 level. This means the adjusted school difference is statistically meaningful even though the adjusted means are close.

In reporting, this chart should be connected directly to the ANCOVA table. The result can be written as: “The adjusted school effect exceeded the critical F value, indicating a statistically significant school effect after controlling for G2.” The final report should also include degrees of freedom, p value and effect size from the table.

Python Chart 5: Raw vs Adjusted Means by School

This chart compares raw G3 means with adjusted G3 means for GP and MS. The raw GP mean is much higher than the raw MS mean. After adjustment, GP decreases to about 12.0, while MS increases to about 11.7.

The chart shows that G2 adjustment greatly reduces the school difference. This is the main practical reason for using ANCOVA in this example. The raw mean comparison makes the school gap look larger, but the adjusted comparison shows that prior grade explains a large part of that gap.

In reporting, this chart should be used to explain why the final conclusion must be based on adjusted means. The article should not report the raw mean gap as the ANCOVA result. It should say that covariate adjustment narrowed the school difference substantially.

Python Chart 6: G2 Boxplot by School

This chart shows the distribution of the G2 covariate in the two school groups. GP has a higher median G2, around 12, while MS has a lower median around 10. The GP box is centered higher, and MS includes a very low outlying G2 value near zero.

The chart explains why covariate adjustment is needed. Since the groups do not begin at the same G2 level, a raw G3 comparison may mix school differences with prior-grade differences. ANCOVA controls this prior-grade imbalance before comparing G3 means.

In reporting, say that G2 distributions differed by school and that G2 was therefore a meaningful covariate to control. The low MS G2 outlier should be reviewed with Outlier Detection, Z Score, Mahalanobis Distance, and Studentized Residuals.

Python Chart 7: Mean G3 by Studytime Within School

This chart shows mean G3 by studytime category within GP and MS schools. GP is higher than MS in all four studytime categories. Both groups increase from lower studytime levels to studytime category 3, and then the mean is slightly lower or flatter at category 4.

The chart provides context for student performance but does not replace the ANCOVA school-and-G2 model. It shows that studytime has a visible relationship with G3 and that the school pattern remains visible inside studytime categories.

In reporting, this chart should be described as a supporting descriptive figure. The main ANCOVA conclusion remains the adjusted school comparison after controlling for G2, not the studytime comparison unless studytime is added to the model.

R Chart-by-Chart Validation

The R charts validate the analysis through group means, spread, variation decomposition, F distribution, residual normality and effect size. These outputs are useful because they show the same student performance pattern from a second software workflow.

R Chart 1: Group Means with Confidence Intervals

This R chart shows mean G3 across studytime groups. The means are 10.844 for group 1, 12.092 for group 2, 13.227 for group 3, and 13.057 for group 4. The mean rises from group 1 to group 3 and remains high in group 4.

The confidence intervals are narrow enough to show a clear upward pattern across studytime categories. Studytime group 1 has the lowest mean, while groups 3 and 4 have the highest means.

In reporting, this chart supports the descriptive statement that higher studytime groups show higher G3 performance. If used in the ANCOVA post, it should be presented as supporting model context rather than as the adjusted school effect.

R Chart 2: Boxplot by Group

This boxplot shows G3 distributions across studytime groups. Groups 3 and 4 have higher medians than groups 1 and 2. Groups 1 and 2 include low outliers near zero, and group 1 also has a high outlier near 18.

The chart shows that the group mean differences are not only caused by one bar chart result. The distribution centers are visibly higher for the higher studytime groups, but the low outliers in groups 1 and 2 should be reviewed because they can affect means and residual normality.

In reporting, use this chart to support the statement that group distributions differ and that outliers were checked. If outliers influence the model, follow-up diagnostics such as Cook’s Distance and Influence Diagnostics should be mentioned.

R Chart 3: Sum of Squares Decomposition

This chart decomposes variation into between-group and within-group components. The between-group sum of squares is 465.078, while the within-group sum of squares is 6298.189. Most variation is still inside groups, which is normal in student performance data.

Even though within-group variation is much larger, the between-group component is large enough relative to the error term to produce a significant F statistic in the R output. This means group differences are statistically meaningful despite substantial individual variation.

In reporting, this chart should be used to explain the source of the F statistic. The group effect is not judged by the raw size of between-group sum of squares alone; it is judged by comparing between-group mean square with within-group mean square.

R Chart 4: F Statistic Distribution Curve

This chart shows the observed F statistic on the F distribution. The observed F value is 15.876, with df1 = 3 and df2 = 645. The critical F value is 2.62.

The observed F value is far to the right of the critical value, so the null hypothesis of equal group means is rejected in this companion F-test output. This result matches the summary table where the p value is extremely small.

In reporting, this chart supports the sentence: “The group effect was statistically significant, F(3, 645) = 15.876, p < .001.” In the ANCOVA article, this should be described as supporting F-test evidence from the validation workflow.

R Chart 5: Residual Histogram

This residual histogram is centered near zero, with the tallest bar around the center of the distribution. Most residuals fall between about -5 and +5, but the left tail extends farther, with several negative residuals below -10.

The chart shows that the residual distribution is broadly centered correctly, but the lower tail is heavier than the upper tail. This same issue appears again in the Q-Q plot, where the lower-left residuals depart from the reference line.

In reporting, write that residuals were mostly centered near zero, but the lower tail showed several extreme negative residuals. If this departure is important for the research question, use Log Transformation, Square Root Transformation, Box-Cox Transformation, or Reciprocal Transformation as sensitivity checks.

R Chart 6: Residual Q-Q Plot

The Q-Q plot shows that the central residuals follow the diagonal line fairly well. This means the middle of the residual distribution is approximately normal.

The tails show stronger departures, especially in the lower-left corner where several negative residuals fall far below the reference line. This confirms the residual histogram pattern: the model has a noticeable group of unusually low observed G3 values compared with predicted values.

In reporting, say that residual normality was acceptable in the center but showed tail departures. Formal checks such as Shapiro-Wilk Test, Anderson-Darling Test, Jarque-Bera Test, Lilliefors Test, and Kolmogorov-Smirnov Test can be used if a formal residual normality statement is needed.

R Chart 7: Effect Size Summary

This effect-size chart reports Cohen’s f as 0.2717, epsilon squared as 0.0644, eta squared as 0.0688, omega squared as 0.0643, and partial eta squared as 0.0688.

The values show that the group effect is statistically significant but not large in practical terms. An eta squared around .069 means the grouping variable explains about 6.9% of the outcome variation in this companion F-test output.

In reporting, include effect size with the F statistic and p value. A complete sentence is: “The group effect was significant, F(3, 645) = 15.876, p < .001, η² = .069.”

SPSS Output Interpretation for ANCOVA

The SPSS output section verifies the model workflow through output-style figures, residual diagnostics, variance checks, a summary table and the downloadable PDF. The supplied SPSS-style charts show the companion F-test summary by studytime group and diagnostic evidence that supports assumption reporting.

Download ANCOVA SPSS Output PDF

SPSS Output Items to Read

SPSS Chart 1: Group Mean CI Bar

This chart shows mean G3 for studytime groups with 95% confidence intervals. Group 1 has the lowest mean, group 2 is higher, and groups 3 and 4 have the highest means.

The pattern shows that G3 performance increases from the lowest studytime group to the higher studytime groups. The confidence intervals are shown around each mean, and the group 4 interval is wider because that group has fewer cases.

In reporting, use this chart as descriptive evidence for the group pattern. It supports the summary-table conclusion that group means are not equal, but the final statistical decision should come from the F-test table.

SPSS Chart 2: G3 Distribution by Studytime

This chart shows the distribution of G3 scores across studytime groups using boxplots. The median and mean markers are lowest in group 1 and higher in groups 3 and 4. Low outliers appear in groups 1 and 2, including cases near zero.

The distribution pattern supports the same conclusion as the mean chart: higher studytime groups generally have higher G3 scores. The outliers also explain why residual diagnostics show tail departures in the histogram and Q-Q plot.

In reporting, mention that group distributions showed higher central performance for studytime groups 3 and 4, with low-score outliers mainly in groups 1 and 2. These outliers should be reviewed before deleting or changing any case.

SPSS Chart 3: F Distribution Curve

This chart shows the observed F statistic on the F distribution. The observed value is F = 15.876, with df1 = 3 and df2 = 645. The p value shown on the chart is 5.706e-10.

The observed F statistic is far into the right tail of the distribution, so the null hypothesis of equal group means is rejected. This is a very strong statistical result for the companion F-test output.

In reporting, write the result as F(3, 645) = 15.876, p < .001. If this is used as part of the ANCOVA post, clearly describe it as the supporting F-test output and not as the adjusted school mean chart.

SPSS Chart 4: Residuals vs Fitted Values

This chart plots residuals against fitted values. The points appear in vertical bands because fitted values are based on group means in this companion model. Most residuals are around zero, but several negative residuals fall below -10.

The vertical pattern is expected when fitted values take only a few group-mean levels. The important diagnostic issue is the presence of extreme negative residuals, which show cases where the observed G3 score was far below the fitted group value.

In reporting, write that residuals were centered around zero but contained several large negative errors. These unusual cases should be reviewed with Studentized Residuals, Cook’s Distance, or Influence Diagnostics.

SPSS Chart 5: Residual Q-Q Plot

This Q-Q plot shows that the middle residuals lie close to the reference line, so the central part of the residual distribution is approximately normal.

The lower-left tail departs strongly from the line, with several large negative residuals. The upper tail also shows some departure, but the lower tail is the stronger issue. This matches the residual histogram and residuals-versus-fitted plot.

In reporting, say that residual normality was reasonable in the center but showed tail departures caused by extreme low residuals. A careful report should mention that residual diagnostics were reviewed rather than simply saying that normality was perfect.

SPSS Chart 6: Variance Check by Studytime Group

This chart compares group standard deviations across studytime groups. Groups 1, 2 and 4 have standard deviations close to about 3, while group 3 has a lower standard deviation around 2.5.

The chart also reports a Fligner p value of 0.2621. Because this p value is greater than .05, the variance check supports similar group variances in this output.

In reporting, write that the variance assumption was supported by the Fligner test, p = .262. This can be discussed alongside Levene Test, Brown-Forsythe Test, Bartlett’s Test, Hartley F Max Test, or Cochran C Test.

SPSS Chart 7: Summary Table

This summary table reports F(3, 645) = 15.876, p = 5.706e-10, eta squared = 0.069, and the decision to reject H0. It also reports group summaries: group 1 has n = 212 and mean = 10.84434; group 2 has n = 305 and mean = 12.09180; group 3 has n = 97 and mean = 13.22680; group 4 has n = 35 and mean = 13.05714.

The table confirms that group 1 has the lowest mean, groups 3 and 4 have the highest means, and the overall group effect is statistically significant. The effect size is about .069, so the result is statistically strong but explains a modest portion of the outcome variation.

In reporting, use this table for the final F-test sentence: “The group effect was significant, F(3, 645) = 15.876, p < .001, η² = .069.” The group means and confidence intervals should be reported before or after this sentence to show the direction of the effect.

SPSS Output PDF: Final Verification

SPSS, R, Python and Excel Workflows for ANCOVA

The same ANCOVA model can be reproduced in SPSS, R, Python and Excel. The software changes, but the model question stays the same: does G3 differ by school after adjusting for G2?

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for ANCOVA

SPSS Syntax for ANCOVA

* ANCOVA in SPSS.
* Dependent variable: G3.
* Fixed factor: school.
* Covariate: G2.

TITLE "ANCOVA: G3 by School Adjusted for G2".

UNIANOVA G3 BY school WITH G2
  /METHOD=SSTYPE(3)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ PARAMETER HOMOGENEITY
  /EMMEANS=TABLES(school) WITH(G2=MEAN) COMPARE ADJ(LSD)
  /CRITERIA=ALPHA(.05)
  /DESIGN=G2 school.

* Homogeneity of regression slopes check.
UNIANOVA G3 BY school WITH G2
  /METHOD=SSTYPE(3)
  /PRINT=DESCRIPTIVE ETASQ PARAMETER
  /CRITERIA=ALPHA(.05)
  /DESIGN=G2 school G2*school.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="ANCOVA-SPSS-Output.pdf".

Python Code for ANCOVA

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols

df = pd.read_csv("dataset.csv")

df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df["G2"] = pd.to_numeric(df["G2"], errors="coerce")
df = df.dropna(subset=["G3", "G2", "school"])

# Main ANCOVA model
model = ols("G3 ~ G2 + C(school)", data=df).fit()
ancova_table = sm.stats.anova_lm(model, typ=2)

print(ancova_table)
print(model.summary())

# Homogeneity of regression slopes check
interaction_model = ols("G3 ~ G2 * C(school)", data=df).fit()
interaction_table = sm.stats.anova_lm(interaction_model, typ=2)

print(interaction_table)

# Residual diagnostics
df["fitted"] = model.fittedvalues
df["residual"] = model.resid
df["studentized_residual"] = model.get_influence().resid_studentized_internal

print(df[["G3", "G2", "school", "fitted", "residual", "studentized_residual"]].head())

R Code for ANCOVA

df <- read.csv("dataset.csv")

df$G3 <- as.numeric(df$G3)
df$G2 <- as.numeric(df$G2)
df$school <- as.factor(df$school)

df_model <- na.omit(df[, c("G3", "G2", "school", "studytime")])

# Main ANCOVA model
ancova_model <- aov(G3 ~ G2 + school, data = df_model)
summary(ancova_model)

# Homogeneity of regression slopes check
slope_model <- aov(G3 ~ G2 * school, data = df_model)
summary(slope_model)

# Adjusted means
library(emmeans)
emmeans(ancova_model, ~ school)

# Diagnostics
par(mfrow = c(2, 2))
plot(ancova_model)

Excel Formula Notes for ANCOVA

Excel does not have a direct ANCOVA button.
Use regression with dummy coding.

Model:
G3 = b0 + b1*G2 + b2*school_dummy + error

school_dummy:
GP = 0
MS = 1

Run:
Data > Data Analysis > Regression

Y Range:
G3 column

X Range:
G2 column + school_dummy column

Interpret:
G2 p value = covariate effect
school_dummy p value = adjusted school effect
school_dummy coefficient = adjusted group difference

Residual:
=Observed_G3 - Predicted_G3

Decision:
If school_dummy p value < .05, adjusted school means differ.

APA Reporting Wording

When reporting ANCOVA, include the dependent variable, factor, covariate, adjusted means, F statistic, degrees of freedom, p value, effect size and assumption checks. A good APA paragraph also says whether the homogeneity of regression slopes assumption was tested.

Common Mistakes

When to Use ANCOVA

Use ANCOVA when you want to compare group means and you also have a meaningful continuous covariate that is related to the dependent variable. The covariate should improve precision or remove a pre-existing difference between groups.

For related assumption and correction topics, see Mauchly’s Test of Sphericity, Greenhouse-Geisser Correction, Huynh-Feldt Correction, Brown-Forsythe Test, and Levene Test.

Downloads and Resources for ANCOVA

The SPSS output PDF below verifies the ANCOVA model, adjusted means, assumption checks, residual diagnostics and result interpretation used in this guide. Replace placeholder script links with final uploaded files after the dataset, Python script, R script, SPSS syntax and Excel workbook are uploaded to WordPress Media Library.

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