Quick Answer: T Test for Unequal Variances Result

The worked T Test for Unequal Variances compares mean G3 final grade between GP and MS. GP has n = 423, mean 12.5768, standard deviation 2.6256, and variance 6.8940. MS has n = 226, mean 10.6504, standard deviation 3.8340, and variance 14.6995. Since MS has a larger variance, the unequal-variance Welch method is appropriate.

The observed mean difference is 1.9264 in favor of GP. The Welch standard error is approximately 0.2852. The Welch t statistic is approximately 6.7545, with adjusted degrees of freedom approximately 340.49, and the p value is p < .001. The 95% confidence interval for the mean difference is approximately 1.3654 to 2.4874. Since the interval is fully above zero, GP students have significantly higher average G3 scores than MS students.

What Is a T Test for Unequal Variances?

A T Test for Unequal Variances is an independent samples t test used when the two groups do not have similar variances. It is usually called Welch’s t test. In SPSS-style output, it appears as the Equal variances not assumed row.

The purpose of this test is the same as the ordinary independent samples t test: compare two group means. The difference is how the standard error and degrees of freedom are calculated. Instead of pooling the two variances into one shared estimate, Welch’s method keeps the variance components separate.

In this example, GP and MS are independent school groups. GP has a smaller variance, while MS has a larger variance. Because the spread is not the same across groups, Welch’s method is safer than the equal-variances-assumed Student’s t test.

T Test for Unequal Variances should be interpreted with group means, group variances, variance ratio, Welch standard error, Welch degrees of freedom, t statistic, confidence interval, p value and effect size. Related guides include T Test Assumptions, Levene Test, Standard Deviation, Standard Error, Confidence Interval, P Value, and Effect Size.

T Test for Unequal Variances Formula

The Welch T Test for Unequal Variances compares the difference between two group means with a standard error that keeps group variances separate:

The observed mean difference in this example is:

The Welch standard error is:

Welch degrees of freedom are adjusted using the Welch-Satterthwaite equation:

Null and Alternative Hypothesis for T Test for Unequal Variances

The T Test for Unequal Variances tests whether two independent population means are equal while allowing the population variances to be unequal.

Dataset and Group Variables Used

The worked example uses a student performance dataset. The outcome variable is G3 final grade. The grouping variable is school, with two independent groups: GP and MS. The unequal-variance workflow compares group variances, group means, Welch standard error components, confidence intervals, t distributions and SPSS-style output tables.

Before interpreting a T Test for Unequal Variances, examine group statistics, variance comparison, standard deviation comparison, distribution by group and the difference between equal-variance and Welch results. Useful supporting guides include T Test Assumptions, Levene Test, Standard Deviation, Standard Error, and Confidence Interval.

SPSS Output Interpretation for T Test for Unequal Variances

In SPSS, the T Test for Unequal Variances is interpreted from the Equal variances not assumed row of the Independent Samples Test table. This is the Welch-style row. It is preferred when Levene’s test, variance ratio, standard deviation comparison or visual spread checks show that the equal-variance assumption is questionable.

SPSS Group Statistics

SPSS Unequal Variance Summary

SPSS Equal Variances Not Assumed T Test

Python Chart-by-Chart Interpretation

The Python charts below show the complete T Test for Unequal Variances workflow. They include group variance comparison, group means, Welch standard error components, Welch confidence interval, Welch t distribution, group distributions, Welch versus equal-variance p value comparison and SPSS-style tables.

Python Chart 1: Group Variance Comparison for Unequal Variances

This chart compares the two group variances. GP has variance 6.8940, while MS has variance 14.6995. MS has the larger spread.

The chart explains why the unequal-variance t test is appropriate. When group variances differ, Welch’s method avoids forcing the groups into one pooled variance estimate.

Python Chart 2: Group Means for Unequal Variances

This chart shows the group means being compared. GP has mean 12.5768, while MS has mean 10.6504. The observed difference is positive and favors GP.

The chart is useful because the t test begins with the group mean difference. Welch’s method then asks whether that difference is large relative to the unequal-variance standard error.

Python Chart 3: Welch Standard Error Components

This chart explains the key calculation behind Welch’s test. The GP standard error component is approximately 0.0163, while the MS component is approximately 0.0650. MS contributes more uncertainty because it has a larger variance and smaller sample size.

This chart is central to the unequal-variance method. It shows that Welch’s test does not hide variance differences inside a pooled estimate.

Python Chart 4: Welch Confidence Interval for Mean Difference

This chart shows the confidence interval for the mean difference under Welch’s method. The interval is approximately 1.3654 to 2.4874, and it is fully above zero.

Because zero is not inside the interval, the result is statistically significant. The positive interval also shows that the direction favors GP.

Python Chart 5: Welch T Distribution for Unequal Variances

This chart places the Welch t statistic in the t-test decision framework. The observed statistic is approximately 6.7545, with adjusted degrees of freedom approximately 340.49.

The observed t statistic is far from zero, which explains why the p value is below .001. The decimal degrees of freedom are normal in Welch’s test.

Python Chart 6: Distribution by Group for Unequal Variances

This distribution chart shows the raw score pattern behind the Welch test. GP is centered higher, while MS has a wider spread. This matches both the mean comparison and the variance comparison.

The chart supports the unequal-variance interpretation because the group distributions do not show identical spread.

Python Chart 7: Welch vs Equal Variance P Value Comparison

This chart compares the p value from Welch’s unequal-variance method with the p value from the equal-variance method. In this dataset, both methods lead to a significant conclusion.

The value of Welch’s test is not that it always changes the conclusion. Its value is that it protects the analysis when the equal variance assumption is questionable.

Python Chart 8: Group Statistics Table

This table provides the descriptive statistics behind the Welch test. GP has higher mean and lower variance, while MS has lower mean and higher variance.

The table is important because it explains both parts of the result: the mean difference and the unequal variance condition.

Python Chart 9: Unequal Variance Summary Table

This summary table explains why the unequal-variance route is selected. It identifies the larger variance group, variance ratio and Welch method recommendation.

This table is useful for reporting because it connects assumption checking with the final choice of t-test row.

Python Chart 10: Independent Samples Unequal Variances T Test Table

This is the main unequal-variance t-test table. It reports the Welch mean difference, standard error, t statistic, adjusted degrees of freedom, p value and confidence interval.

The table supports the final conclusion that GP has a significantly higher mean G3 score than MS under the equal variances not assumed row.

R Chart-by-Chart Validation

The R charts validate the Python and SPSS-style conclusions using a separate workflow. The same pattern appears: MS has larger variance, GP has higher mean, Welch standard error uses separate components, the confidence interval excludes zero, the t statistic is significant, and the unequal-variance tables support the Welch interpretation.

R Chart 1: Group Variance Comparison for Unequal Variances

The R variance chart confirms the Python result. MS has the larger variance, which supports using Welch’s unequal-variance t test.

This validation shows that the variance pattern is reproducible across software.

R Chart 2: Group Means for Unequal Variances

The R group means chart confirms that GP has the higher average G3 score. The observed mean difference is positive and matches the Python output.

This chart validates the direction of the final hypothesis-test result.

R Chart 3: Welch Standard Error Components

The R standard-error chart confirms the separate variance-component logic of Welch’s test. The MS component is larger because MS has greater variance and fewer observations.

This chart explains why Welch’s test is safer than pooling the variances.

R Chart 4: Welch Confidence Interval for Mean Difference

The R confidence interval chart confirms that the Welch confidence interval is fully above zero. The group difference is statistically significant and positive.

This chart supports the final conclusion that GP has the higher mean G3 score.

R Chart 5: Welch T Distribution for Unequal Variances

The R t-distribution chart validates the inferential result. The observed Welch t statistic is far from zero, so the p value is very small.

The chart helps readers understand why the null hypothesis of equal means is rejected.

R Chart 6: Distribution by Group for Unequal Variances

The R distribution chart confirms the group-level pattern. GP is centered higher, while MS has a wider distribution.

This visual supports both the mean-difference conclusion and the unequal-variance assumption decision.

R Chart 7: Welch vs Equal Variance P Value Comparison

The R comparison chart confirms that both Welch and equal-variance approaches produce significant p values in this dataset.

However, Welch remains the better reporting choice when the variance assumption is questionable because it does not require equal variances.

R Chart 8: Group Statistics Table

The R group statistics table repeats the descriptive summary. GP has the higher mean and smaller variance, while MS has the lower mean and larger variance.

This table confirms the numerical basis for using Welch’s test.

R Chart 9: Unequal Variance Summary Table

The R unequal-variance summary table confirms that the variance condition supports the Welch approach. It connects the assumption check with the final statistical method.

This table is useful for transparent reporting because it explains why the unequal-variance row is used.

R Chart 10: Independent Samples Unequal Variances T Test Table

The R unequal-variance t-test table confirms the main inference. The Welch result is significant, the confidence interval is above zero, and the group mean difference favors GP.

This validates the final conclusion using a separate workflow.

Additional Output 1: Group Variance Comparison for Unequal Variances

This additional output preserves the complete chart set and confirms that MS has the larger variance.

The repeated visual strengthens the assumption explanation for using Welch’s t test.

Additional Output 2: Group Means for Unequal Variances

This additional group means chart confirms that GP has the higher average G3 score.

It supports the direction of the final Welch t-test result.

Additional Output 3: Welch Standard Error Components

This additional standard-error chart repeats the main Welch calculation idea. The standard error is built from separate variance components.

This is the key difference between Welch’s t test and the equal-variances-assumed Student’s t test.

Additional Output 4: Welch Confidence Interval for Mean Difference

This additional confidence interval output confirms that the interval remains fully above zero.

The chart supports the conclusion that the group mean difference is statistically significant and positive.

Additional Output 5: Welch T Distribution for Unequal Variances

This additional t-distribution chart confirms the statistical decision. The observed Welch t statistic is far from zero.

The chart provides visual support for the small p value.

Additional Output 6: Distribution by Group for Unequal Variances

This additional distribution chart confirms the visible spread difference between GP and MS.

It supports the unequal-variance decision because group distributions do not have identical spread.

Additional Output 7: Welch vs Equal Variance P Value Comparison

This additional p-value comparison chart confirms that the conclusion remains significant under both approaches.

The reason to prefer Welch is not only the p value; it is the safer assumption structure under unequal variances.

Additional Output 8: Group Statistics Table

This additional statistics table repeats the group sample size, mean, standard deviation, variance and standard error information.

It provides the numerical foundation for the Welch t-test result.

Additional Output 9: Unequal Variance Summary Table

This additional summary table confirms the unequal-variance context and the recommendation to use Welch’s method.

The table connects assumption evidence with the final test selection.

Additional Output 10: Independent Samples Unequal Variances T Test Table

This additional t-test table repeats the final Welch result. It confirms the adjusted degrees of freedom, mean difference, confidence interval and p value.

The output provides final evidence that GP has a significantly higher average G3 score than MS under the unequal-variance method.

SPSS, R, Python and Excel Workflows for T Test for Unequal Variances

The same T Test for Unequal Variances workflow can be reproduced in SPSS, R, Python and Excel. SPSS reports the unequal-variance result in the Equal variances not assumed row. R uses t.test(..., var.equal = FALSE). Python uses scipy.stats.ttest_ind(..., equal_var=False). Excel uses the Data Analysis ToolPak option t-Test: Two-Sample Assuming Unequal Variances.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for T Test for Unequal Variances

SPSS Syntax for T Test for Unequal Variances

* T Test for Unequal Variances in SPSS.
* Dependent variable: G3.
* Grouping variable: school.
* Groups: GP and MS.
* Interpret the Equal variances not assumed row when variances differ.

TITLE "T Test for Unequal Variances: Welch Test for G3 by School".

T-TEST GROUPS=school('GP' 'MS')
  /VARIABLES=G3
  /MISSING=ANALYSIS
  /CRITERIA=CI(.95).

MEANS TABLES=G3 BY school
  /CELLS MEAN COUNT STDDEV VARIANCE SEMEAN.

EXAMINE VARIABLES=G3 BY school
  /PLOT BOXPLOT HISTOGRAM
  /STATISTICS DESCRIPTIVES
  /MISSING LISTWISE.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="T-Test-for-Unequal-Variances-SPSS-Output.pdf".

Python Code for T Test for Unequal Variances

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

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

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

gp = data.loc[data["school"] == "GP", "G3"]
ms = data.loc[data["school"] == "MS", "G3"]

n1, n2 = len(gp), len(ms)
m1, m2 = gp.mean(), ms.mean()
s1, s2 = gp.std(ddof=1), ms.std(ddof=1)
v1, v2 = gp.var(ddof=1), ms.var(ddof=1)

mean_difference = m1 - m2

variance_ratio = max(v1, v2) / min(v1, v2)

# Welch t test
t_stat, p_value = stats.ttest_ind(gp, ms, equal_var=False)

# Welch standard error components
se1_component = v1 / n1
se2_component = v2 / n2
welch_se = np.sqrt(se1_component + se2_component)

# Welch-Satterthwaite degrees of freedom
welch_df = ((se1_component + se2_component) ** 2) / (
    (se1_component ** 2) / (n1 - 1) +
    (se2_component ** 2) / (n2 - 1)
)

critical_t = stats.t.ppf(0.975, welch_df)
ci_low = mean_difference - critical_t * welch_se
ci_high = mean_difference + critical_t * welch_se

# Optional comparison with equal-variance t test
t_equal, p_equal = stats.ttest_ind(gp, ms, equal_var=True)

print("T Test for Unequal Variances: Welch")
print("GP n =", n1, "mean =", m1, "sd =", s1, "variance =", v1)
print("MS n =", n2, "mean =", m2, "sd =", s2, "variance =", v2)
print("Variance ratio =", variance_ratio)
print("Mean difference GP - MS =", mean_difference)
print("Welch SE =", welch_se)
print("Welch t =", t_stat)
print("Welch df =", welch_df)
print("Welch p =", p_value)
print("95% CI =", (ci_low, ci_high))
print("SE component GP =", se1_component)
print("SE component MS =", se2_component)
print("Equal-variance t =", t_equal)
print("Equal-variance p =", p_equal)

R Code for T Test for Unequal Variances

# T Test for Unequal Variances in R
# Welch independent samples t test

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

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

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

gp <- df_model$G3[df_model$school == "GP"]
ms <- df_model$G3[df_model$school == "MS"]

n1 <- length(gp)
n2 <- length(ms)
m1 <- mean(gp)
m2 <- mean(ms)
s1 <- sd(gp)
s2 <- sd(ms)
v1 <- var(gp)
v2 <- var(ms)

variance_ratio <- max(v1, v2) / min(v1, v2)
mean_difference <- m1 - m2

# Welch t test
result_welch <- t.test(gp, ms, var.equal = FALSE)
print(result_welch)

# Manual Welch calculations
se1_component <- v1 / n1
se2_component <- v2 / n2
welch_se <- sqrt(se1_component + se2_component)

welch_df <- (se1_component + se2_component)^2 /
  ((se1_component^2 / (n1 - 1)) + (se2_component^2 / (n2 - 1)))

cat("GP n =", n1, "mean =", m1, "sd =", s1, "variance =", v1, "\n")
cat("MS n =", n2, "mean =", m2, "sd =", s2, "variance =", v2, "\n")
cat("Variance ratio =", variance_ratio, "\n")
cat("Mean difference GP - MS =", mean_difference, "\n")
cat("Welch SE =", welch_se, "\n")
cat("Welch df =", welch_df, "\n")
cat("SE component GP =", se1_component, "\n")
cat("SE component MS =", se2_component, "\n")

# Comparison with equal-variance t test
result_equal <- t.test(gp, ms, var.equal = TRUE)
print(result_equal)

Excel Formulas for T Test for Unequal Variances

Step 1:
Place GP G3 values in one column and MS G3 values in another column.

Step 2:
Calculate group sample sizes:
=COUNT(GP_range)
=COUNT(MS_range)

Step 3:
Calculate group means:
=AVERAGE(GP_range)
=AVERAGE(MS_range)

Step 4:
Calculate group standard deviations:
=STDEV.S(GP_range)
=STDEV.S(MS_range)

Step 5:
Calculate group variances:
=VAR.S(GP_range)
=VAR.S(MS_range)

Step 6:
Calculate variance ratio:
=MAX(GP_variance,MS_variance)/MIN(GP_variance,MS_variance)

Step 7:
Calculate observed mean difference:
=GP_mean-MS_mean

Step 8:
Calculate Welch standard error:
=SQRT((GP_variance/GP_n)+(MS_variance/MS_n))

Step 9:
Calculate Welch t statistic:
=mean_difference/Welch_standard_error

Step 10:
Calculate Welch degrees of freedom:
=((GP_variance/GP_n)+(MS_variance/MS_n))^2 /
(((GP_variance/GP_n)^2/(GP_n-1))+((MS_variance/MS_n)^2/(MS_n-1)))

Step 11:
Calculate two-tailed p value:
=T.DIST.2T(ABS(t_cell),welch_df_cell)

Step 12:
Calculate 95% CI lower:
=mean_difference-T.INV.2T(0.05,welch_df_cell)*Welch_standard_error

Step 13:
Calculate 95% CI upper:
=mean_difference+T.INV.2T(0.05,welch_df_cell)*Welch_standard_error

Step 14:
Run Excel ToolPak:
Data Analysis > t-Test: Two-Sample Assuming Unequal Variances

APA Reporting Wording for T Test for Unequal Variances

The T Test for Unequal Variances should be reported with group means, standard deviations, group sample sizes, variance evidence, Welch t statistic, adjusted degrees of freedom, p value, mean difference and confidence interval. The wording should clearly state that Welch’s unequal-variance method was used.

Common Mistakes in T Test for Unequal Variances

When to Use T Test for Unequal Variances

Use a T Test for Unequal Variances when comparing two independent group means and the equal-variance assumption is questionable. It is especially useful when one group has a much larger variance, when sample sizes are unequal, or when Levene’s test suggests that variances differ.

Downloads and Resources

Use the following downloadable resources to reproduce the T Test for Unequal Variances workflow in SPSS, Python, R and Excel. Replace the placeholder links with the final hosted file URLs after uploading your scripts and templates to WordPress Media Library.

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