Quick Answer: T Test for Equal Variances Result

The worked T Test for Equal Variances uses G3 final grade as the outcome and school as the grouping variable. The two independent groups are GP and MS. The group statistics show GP n = 423, M = 12.5768, SD = 2.6256, variance = 6.8940 and MS n = 226, M = 10.6504, SD = 3.8340, variance = 14.6995.

The variance ratio is 2.1322, with MS having the larger variance. The equal variance output marks the equal variance status as Supported by the practical variance-ratio decision rule, while the formal F variance ratio line shows Sig. = 0.0000. This means the practical and strict formal interpretations should be reported carefully. For the equal-variances-assumed t-test row, the result is t(647) = 7.5426, p = 0.0000, with a mean difference of 1.9264 and a 95% confidence interval from 1.4249 to 2.4279.

What Is a T Test for Equal Variances?

A T Test for Equal Variances refers to the independent samples t test when the two groups are treated as having equal population variances. This is also called the equal variances assumed t test or the classic Student’s independent samples t test. The key idea is that both groups are assumed to estimate the same underlying variance, so the analysis combines group variances into one pooled variance estimate.

In this example, the two independent groups are GP and MS. The outcome is G3 final grade. The question is whether the average G3 score differs between the two school groups when the equal-variance t-test row is used.

The equal variance assumption matters because it controls how the standard error is calculated. If the assumption is reasonable, the pooled variance method is efficient. If the assumption is not reasonable, the Welch unequal-variance t test is safer because it uses separate variance components and adjusted degrees of freedom.

T Test for Equal Variances should be interpreted with group variances, group standard deviations, variance ratio, pooled variance, equal-variance test output, t statistic, confidence interval 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 Equal Variances Formula

The equal-variances independent samples t test uses pooled variance. The test statistic compares the group mean difference with the pooled standard error:

The pooled variance is calculated as:

The variance ratio used for equal variance screening is:

Null and Alternative Hypothesis for T Test for Equal Variances

The topic involves two related hypothesis ideas. First, the equal variance check asks whether the two group variances can be treated as similar. Second, the equal-variances-assumed t test asks whether the two group means are different.

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 groups: GP and MS. The equal variance workflow compares group variances, group standard deviations, absolute deviations, F ratio output, pooled variance components, and the equal-variances-assumed independent samples t test.

Before interpreting a T Test for Equal Variances, examine the group spread using variance bars, standard deviation bars, boxplots, absolute deviations, and formal equality-of-variance output. Useful supporting resources include T Test Assumptions, Levene Test, Standard Deviation, Box Plot Interpretation, and Parametric vs Nonparametric Tests.

SPSS Output Interpretation for T Test for Equal Variances

In SPSS, equal variance information appears in the Independent Samples Test output. The analyst normally checks the equality-of-variance result, then decides whether to interpret Equal variances assumed or Equal variances not assumed. The output images supplied for this post include group statistics, an equal-variance test table, and an independent samples equal-variances-assumed t-test table.

SPSS Group Statistics

SPSS Test for Equality of Variances

SPSS Equal Variances Assumed T Test

Python Chart-by-Chart Interpretation

The Python charts below show the complete T Test for Equal Variances workflow. They include variance comparison, standard deviation comparison, boxplot spread, absolute deviations, F variance ratio, equal-variances-assumed t distribution, pooled variance components, and SPSS-style output tables.

Python Chart 1: Group Variance Comparison for Equal Variances

This chart compares the group variances directly. The GP variance is 6.8940, while the MS variance is 14.6995. MS has the larger spread in G3 scores.

The visual makes the equal variance question clear. If one variance is much larger than the other, the analyst should check whether the equal-variance t-test row is appropriate or whether Welch’s t test should be compared.

Python Chart 2: Group Standard Deviation Comparison

This chart compares standard deviations instead of variances. GP has SD = 2.6256, while MS has SD = 3.8340. Standard deviations are easier to interpret because they are in the same unit as G3 scores.

The MS standard deviation is larger, meaning MS scores are more spread out around their mean. This supports the need to discuss equal variance carefully.

Python Chart 3: Boxplot Spread by Group

The boxplot gives a visual assumption check. It shows the center, spread and outliers for each group. MS appears more spread out than GP, which matches the variance and standard deviation charts.

Boxplots are important because equal variance is not only a numerical issue. A visual spread check helps identify outliers, skewness and spread differences that may influence the t-test assumption.

Python Chart 4: Absolute Deviations from Group Median

This chart shows absolute deviations from each group median. It is related to the logic behind robust variance tests such as Levene’s test or Brown-Forsythe style checks. Larger deviations mean greater within-group spread.

The chart helps explain why equal variance is tested through deviations from group centers. If one group consistently has larger deviations, the equal-variance assumption becomes more questionable.

Python Chart 5: F Variance Ratio Distribution

This chart places the observed variance ratio on an F distribution. The observed ratio is 2.1322, with the larger variance belonging to MS. The formal significance value is 0.0000.

This is the strict formal test view of variance equality. Because the p value is very small, a strict F-test interpretation suggests that the variances are statistically different. However, practical variance-ratio rules may still classify the spread difference as manageable for teaching or exploratory reporting.

Python Chart 6: Equal Variances Assumed T Distribution

This chart shows the equal-variances-assumed t statistic. The observed value is t = 7.5426 with df = 647. The value is far from zero, so the group mean difference is statistically significant.

This chart connects the variance assumption with the final mean comparison. Once the equal-variance row is used, the pooled-variance standard error leads to a strong t statistic and a p value of 0.0000.

Python Chart 7: Pooled Variance Components for Equal Variances

This chart explains the pooled variance calculation. The equal-variances-assumed t test combines the two group variances using their degrees of freedom. The larger GP sample contributes heavily because it has more observations.

Pooled variance is the key feature of the equal-variance t test. It produces a shared estimate of variability, which is then used to calculate the standard error of the mean difference.

Python Chart 8: Group Statistics Table

This table summarizes the group-level statistics used in the analysis. GP has n = 423, M = 12.5768, SD = 2.6256, and variance = 6.8940. MS has n = 226, M = 10.6504, SD = 3.8340, and variance = 14.6995.

The table is important because it provides the raw numerical evidence behind the variance comparison and equal-variances-assumed t test.

Python Chart 9: Equal Variances Test Table

This table reports the equality-of-variance check. The variance ratio is 2.1322, with MS as the larger variance group. The formal significance is 0.0000, while the practical equal-variance status in the table is marked as Supported.

The correct interpretation is careful: the variance ratio rule may support continuing with the equal-variance demonstration, but the formal p value suggests that Welch’s test should also be considered for strict reporting.

Python Chart 10: Independent Samples Equal Variances T Test Table

This is the main equal-variances-assumed t-test table. It reports t = 7.5426, df = 647, Sig. (2-tailed) = 0.0000, mean difference = 1.9264, standard error difference = 0.2554, and 95% CI [1.4249, 2.4279].

The result supports rejecting the null hypothesis of equal means. GP students have a significantly higher average G3 score than MS students under the equal-variances-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 and standard deviation, spread differences are visible in boxplots and deviations, the F variance ratio is 2.1322, the pooled-variance t test is significant, and the equal-variances-assumed test table rejects the null hypothesis of equal means.

R Chart 1: Group Variance Comparison for Equal Variances

The R variance chart confirms the Python result. MS has the larger variance, while GP has the smaller variance. This validates the variance comparison across software.

The chart supports the need to discuss the equal-variance assumption before interpreting the pooled-variance t-test row.

R Chart 2: Group Standard Deviation Comparison

The R standard deviation chart confirms that MS is more variable than GP. Since standard deviation is in original G3 score units, this chart is easier to interpret than the variance chart.

This supports the practical explanation that MS scores are more spread out around their mean.

R Chart 3: Boxplot Spread by Group

The R boxplot confirms the spread difference between groups. MS has wider spread, and both groups show lower-end observations. This visual supports the variance comparison.

Boxplot interpretation is useful because it helps readers see spread, center and outliers in one chart.

R Chart 4: Absolute Deviations from Group Median

The R absolute-deviation chart confirms the spread pattern using deviations from group medians. Larger deviations indicate greater variability.

This chart is closely related to robust homogeneity checks and helps explain why median-centered spread measures can be useful when data include outliers.

R Chart 5: F Variance Ratio Distribution

The R F distribution chart validates the observed variance ratio of 2.1322. It shows where the variance ratio falls relative to the F distribution.

The chart is useful because it distinguishes practical spread comparison from formal significance testing. The formal test can become significant with large samples even when a practical rule labels the spread difference as manageable.

R Chart 6: Equal Variances Assumed T Distribution

The R t distribution chart confirms the equal-variances-assumed inference. The observed t statistic is 7.5426, which is far from zero.

This chart visually explains the small p value and the decision to reject the null hypothesis of equal group means.

R Chart 7: Pooled Variance Components for Equal Variances

The R pooled-variance chart confirms how the equal-variances-assumed t test combines group variances. The pooled variance estimate is built from both group variances and their degrees of freedom.

This chart is central to explaining why the equal-variance row differs from Welch’s unequal-variance row.

R Chart 8: Group Statistics Table

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

This table supports the descriptive foundation for both the variance check and the equal-variances-assumed t test.

R Chart 9: Equal Variances Test Table

The R equal-variance table confirms the variance ratio, degrees of freedom, significance value, larger variance group and equal variance status. It supports the same interpretation as the Python table.

The table should be reported with caution because practical and formal interpretations can differ. A complete report should mention the variance ratio and whether Welch’s test was also checked.

R Chart 10: Independent Samples Equal Variances T Test Table

The R equal-variances-assumed t-test table confirms the main mean-comparison decision. The result is significant, and the confidence interval is fully above zero.

This validates the conclusion that GP has a higher average G3 score than MS under the equal-variance t-test row.

Additional Output 1: Group Variance Comparison for Equal Variances

This additional output preserves the complete supplied chart set and repeats the group variance comparison. It confirms that MS has the larger variance.

The repeated visual helps verify that the same spread pattern appears across exported outputs.

Additional Output 2: Group Standard Deviation Comparison

This additional standard deviation chart confirms that MS has greater spread in original G3 units.

The chart helps nontechnical readers understand variance differences without relying only on squared units.

Additional Output 3: Boxplot Spread by Group

This additional boxplot output confirms the visual spread pattern. MS shows wider spread, while GP is relatively tighter.

Boxplots should be kept in the assumption-check workflow because they reveal both spread differences and possible outliers.

Additional Output 4: Absolute Deviations from Group Median

This additional deviation chart confirms the median-centered spread comparison. It helps explain why absolute deviations are useful for robust variance checking.

The chart supports the overall message that equal variance should be evaluated from both numerical and visual evidence.

Additional Output 5: F Variance Ratio Distribution

This additional F distribution chart repeats the formal variance-ratio view. The observed variance ratio is compared with the expected F distribution under equal variances.

The chart is useful for explaining why a variance-ratio test can flag a difference in spread.

Additional Output 6: Equal Variances Assumed T Distribution

This additional t-distribution chart confirms the strong mean-comparison result. The t statistic is far from zero, so the equal-variances-assumed test rejects the null hypothesis.

The chart provides visual support for the p-value result.

Additional Output 7: Pooled Variance Components for Equal Variances

This additional pooled-variance chart repeats the central calculation behind Student’s equal-variance t test.

It reinforces that the equal-variance row uses a shared variance estimate rather than separate Welch variance components.

Additional Output 8: Group Statistics Table

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

It should be retained because it provides the numerical foundation for the entire equal-variance analysis.

Additional Output 9: Equal Variances Test Table

This additional equal-variance table repeats the variance ratio output and equal variance status. It confirms that MS is the larger variance group.

This table is useful because it gives the assumption decision in reportable form.

Additional Output 10: Independent Samples Equal Variances T Test Table

This additional t-test table repeats the final inferential result. It confirms t(647) = 7.5426, mean difference = 1.9264, and 95% CI [1.4249, 2.4279].

The table provides the final evidence for rejecting the null hypothesis of equal means under the equal-variances-assumed row.

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

The same T Test for Equal Variances workflow can be reproduced in SPSS, R, Python and Excel. SPSS provides the equal-variance decision and the equal-variances-assumed independent samples t-test row. R can run var.test() and t.test(..., var.equal = TRUE). Python can calculate the variance ratio, pooled variance and equal-variance t test. Excel can calculate variance ratios manually and run a two-sample equal-variance t test through the Analysis ToolPak.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for T Test for Equal Variances

SPSS Syntax for T Test for Equal Variances

* T Test for Equal Variances in SPSS.
* Dependent variable: G3.
* Grouping variable: school.
* Groups: GP and MS.
* Interpret Equal variances assumed row when assumption is accepted.

TITLE "T Test for Equal Variances: 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
  /STATISTICS DESCRIPTIVES
  /MISSING LISTWISE.

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

Python Code for T Test for Equal 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)

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

# F variance ratio test
if v1 >= v2:
    f_ratio = v1 / v2
    df1 = n1 - 1
    df2 = n2 - 1
else:
    f_ratio = v2 / v1
    df1 = n2 - 1
    df2 = n1 - 1

# Two-tailed p for variance ratio
p_upper = stats.f.sf(f_ratio, df1, df2)
p_value_f = min(1, 2 * p_upper)

# Equal-variance independent samples t test
t_stat, p_value_t = stats.ttest_ind(gp, ms, equal_var=True)

dfree = n1 + n2 - 2
mean_difference = m1 - m2

pooled_variance = (((n1 - 1) * v1) + ((n2 - 1) * v2)) / dfree
pooled_sd = np.sqrt(pooled_variance)
se_difference = pooled_sd * np.sqrt((1 / n1) + (1 / n2))

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

cohens_d = mean_difference / pooled_sd

print("T Test for Equal Variances")
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("F ratio =", f_ratio, "df1 =", df1, "df2 =", df2, "p =", p_value_f)
print("Pooled variance =", pooled_variance)
print("Pooled SD =", pooled_sd)
print("Equal variances assumed t =", t_stat)
print("df =", dfree)
print("p =", p_value_t)
print("Mean difference =", mean_difference)
print("SE difference =", se_difference)
print("95% CI =", (ci_low, ci_high))
print("Cohen's d =", cohens_d)

R Code for T Test for Equal Variances

# T Test for Equal Variances in R

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)

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")

# F variance ratio test
print(var.test(gp, ms))

# Equal-variance Student's t test
result_equal <- t.test(gp, ms, var.equal = TRUE)
print(result_equal)

dfree <- n1 + n2 - 2
pooled_variance <- (((n1 - 1) * v1) + ((n2 - 1) * v2)) / dfree
pooled_sd <- sqrt(pooled_variance)
mean_difference <- m1 - m2
cohens_d <- mean_difference / pooled_sd

cat("Pooled variance =", pooled_variance, "\n")
cat("Pooled SD =", pooled_sd, "\n")
cat("Mean difference =", mean_difference, "\n")
cat("Cohen's d =", cohens_d, "\n")

Excel Formulas for T Test for Equal 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 degrees of freedom:
=GP_n+MS_n-2

Step 8:
Calculate pooled variance:
=(((GP_n-1)*GP_variance)+((MS_n-1)*MS_variance))/(GP_n+MS_n-2)

Step 9:
Calculate pooled standard deviation:
=SQRT(pooled_variance)

Step 10:
Calculate standard error difference:
=pooled_sd*SQRT((1/GP_n)+(1/MS_n))

Step 11:
Calculate mean difference:
=GP_mean-MS_mean

Step 12:
Calculate equal-variances-assumed t statistic:
=mean_difference/standard_error_difference

Step 13:
Calculate two-tailed p value:
=T.DIST.2T(ABS(t_cell),df_cell)

Step 14:
Calculate 95% CI lower:
=mean_difference-T.INV.2T(0.05,df_cell)*standard_error_difference

Step 15:
Calculate 95% CI upper:
=mean_difference+T.INV.2T(0.05,df_cell)*standard_error_difference

Step 16:
Run Excel ToolPak:
Data Analysis > t-Test: Two-Sample Assuming Equal Variances

APA Reporting Wording for T Test for Equal Variances

The T Test for Equal Variances should report the variance check and the equal-variances-assumed t test result. The report should include group means, standard deviations, variance ratio, t statistic, degrees of freedom, p value, mean difference and confidence interval.

Common Mistakes in T Test for Equal Variances

When to Use T Test for Equal Variances

Use a T Test for Equal Variances when comparing two independent group means and the equal variance assumption is acceptable. This is the classic Student’s independent samples t test. It is especially useful when group variances are similar and the groups are independent.

Downloads and Resources

Use the following downloadable resources to reproduce the T Test for Equal 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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