Quick Answer: Two Sample T Test Result

The worked example compares G3 final grade between two independent school groups: GP and MS. The GP group has 423 students with mean G3 approximately 12.5768. The MS group has 226 students with mean G3 approximately 10.6504. The mean difference is approximately 1.9264 points in favor of GP.

Because the MS group has a larger standard deviation and variance than the GP group, the Welch two sample t test is the safer result to report. The Welch result is approximately t(340.49) = 6.75, p < .001, with a 95% confidence interval for the mean difference of about 1.37 to 2.49. This means the GP group scored significantly higher than the MS group on average.

What Is a Two Sample T Test?

A Two Sample T Test is a statistical test used to compare the means of two independent groups. It answers whether the observed difference between two sample means is large enough to conclude that the two population means are different.

The word “independent” is important. The two groups must contain different observations. In this example, GP students and MS students are separate groups, so the correct design is an independent two sample t test. If the same students were measured twice, the correct procedure would be a paired t test instead.

There are two major versions of the two sample t test. The equal variance two sample t test assumes that the two population variances are equal. The Welch two sample t test does not assume equal variances and adjusts the degrees of freedom. In real data analysis, Welch is often safer when group standard deviations or sample sizes differ.

Useful related guides include T Test Assumptions, T Test for Difference Between Means, T Test for Unequal Variances, T Test for Equal Variances, T Test vs ANOVA, P Value, Confidence Interval, Effect Size, and Levene Test.

Two Sample T Test Formula

The general Two Sample T Test equation compares the difference between two means with the standard error of that difference.

Here, x̄₁ is the first group mean, x̄₂ is the second group mean, and SE_difference is the standard error of the difference between the two means.

Equal Variance Two Sample T Test Formula

When equal variances are assumed, the test uses a pooled standard deviation:

The equal variance degrees of freedom are:

Welch Two Sample T Test Formula

When equal variances are not assumed, the Welch two sample t test uses separate group variances:

The Welch degrees of freedom are calculated using the Welch-Satterthwaite equation:

Null and Alternative Hypothesis

A Two Sample T Test begins with the null hypothesis that the two population means are equal. The alternative hypothesis says the two means are different.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade. The independent grouping variable is school, which contains the two groups GP and MS. Because the groups are independent and the outcome is numeric, a two sample t test is appropriate.

Before interpreting the result, inspect descriptive statistics, group distributions, standard deviations, box plots, and the equality of variance pattern. This is why the post includes group means, distribution plots, variance comparison, standard error components, p value comparison and assumption summary tables.

SPSS Output Interpretation for Two Sample T Test

In SPSS, a Two Sample T Test is run through Analyze > Compare Means > Independent-Samples T Test. Put G3 in Test Variable(s), put school in Grouping Variable, define the two groups as GP and MS, then read the Group Statistics and Independent Samples Test tables.

SPSS Output Values to Read

SPSS Chart 1: G3 Distribution by Group

This SPSS distribution chart shows the spread and shape of G3 scores in each school group. The chart helps confirm that the comparison is between two independent groups and that the outcome is numeric.

GP is centered higher than MS, which supports the direction of the two sample t test. The distributions overlap, but the group means differ enough to produce a significant result.

SPSS Chart 2: Group Means with Confidence Intervals

This chart shows that the GP mean is higher than the MS mean. The confidence intervals give a visual estimate of uncertainty around each group mean.

The chart supports the conclusion that the group difference is not only numerical but also statistically meaningful.

SPSS Chart 3: Mean Difference Confidence Interval

This chart focuses on the estimated mean difference. The interval is above zero, which supports rejecting the null hypothesis of equal means.

Because the interval is positive, the practical interpretation is that GP has the higher average G3 score.

SPSS Chart 4: T Statistic Distribution Curve

This chart places the observed t statistic on the t distribution. A statistic far from zero produces a small p value.

The visual confirms why the null hypothesis is rejected: the observed group mean difference is large relative to its standard error.

SPSS Chart 5: G3 Boxplot by Group

The boxplot shows the median, spread and possible outliers for each group. It is useful for checking whether one group has much wider variability than the other.

The spread pattern supports using the Welch interpretation because the MS group shows greater variability.

SPSS Chart 6: Group Standard Deviation Comparison

This chart directly compares group standard deviations. The MS group has a larger standard deviation than the GP group.

This difference matters because the equal variance t test assumes similar population variances. When this assumption is doubtful, the Welch two sample t test is safer.

SPSS Chart 7: Mean G3 by Study Time Within Group

This chart provides context by showing mean G3 across studytime categories within each school group. It helps explain whether the group pattern appears across learning-behavior levels.

The chart is descriptive context. The formal two sample t test still compares the overall GP and MS group means.

Python Chart-by-Chart Interpretation

The Python charts show the complete Two Sample T Test workflow. They include group means, distributions, mean difference confidence interval, Welch t distribution, variance comparison, standard error components, p value comparison, group statistics table, results table and assumption summary table.

Python Chart 1: Two Sample T Group Means

This chart shows the main comparison. GP has a higher mean G3 score than MS.

The group means chart explains the direction of the result before the test statistic is interpreted.

Python Chart 2: Distribution by Group

This chart shows the shape, spread and overlap of G3 scores in GP and MS. It helps verify that the comparison is between two distributions, not just two numbers.

The MS distribution is more spread out, supporting the use of Welch’s unequal variance version.

Python Chart 3: Mean Difference Confidence Interval

This chart focuses on the GP minus MS mean difference. The confidence interval is above zero, so the result is statistically significant.

The chart gives both direction and uncertainty: GP is higher, and the plausible difference remains positive.

Python Chart 4: Welch T Distribution

This chart places the Welch t statistic on the t distribution using adjusted degrees of freedom.

The observed statistic is far from zero, which explains why the p value is below .001.

Python Chart 5: Variance Comparison

This chart compares the group variances. MS has a larger variance than GP.

The variance difference is important because it affects whether the equal variance test or Welch test should be preferred.

Python Chart 6: Welch Standard Error Components

This chart explains the standard error for two sample t test. Welch standard error uses each group variance divided by each group sample size.

The chart is useful because many students memorize the formula but do not understand why unequal variance affects the standard error.

Python Chart 7: P Value Comparison

This chart compares p values from the equal variance test and Welch test. Both are significant, but the Welch test is safer when variances differ.

The chart helps explain that method choice should be based on assumptions, not on choosing the smallest p value.

Python Chart 8: Group Statistics Table

This table gives the descriptive foundation for the test. It shows group sample sizes, means, standard deviations and variances.

Always interpret this table before the hypothesis test because it explains what the t test is comparing.

Python Chart 9: Results Table

This is the main Python results table. It reports the test statistic, degrees of freedom, p value, confidence interval and decision.

The result supports rejecting the null hypothesis and reporting a significant GP versus MS difference.

Python Chart 10: Assumption Summary Table

This table summarizes important assumptions: numeric outcome, two independent groups, independence, variance pattern and appropriate test version.

The table supports the final recommendation to use Welch when the equal variance assumption is doubtful.

R Chart-by-Chart Validation

The R charts validate the same Two Sample T Test conclusion using an independent workflow. The same pattern appears: GP has a higher mean, MS has greater spread, the confidence interval for the mean difference is above zero, and the Welch result is statistically significant.

R Chart 1: G3 Distribution by Group

This R chart validates the distribution pattern seen in Python and SPSS. GP is centered higher, while MS shows greater spread.

The chart supports the same two sample t-test interpretation across software.

R Chart 2: Group Means with Confidence Intervals

This chart confirms that GP has the higher average G3 score. The confidence intervals show uncertainty around each group mean.

The chart is useful because it visually connects group descriptives with the formal t-test result.

R Chart 3: Mean Difference Confidence Interval

The R confidence interval for the mean difference is above zero. This validates the conclusion that GP has a significantly higher mean.

The chart is especially important because confidence intervals show the estimated range of the group difference, not just the p value.

R Chart 4: T Statistic Distribution Curve

This chart shows the observed t statistic on the t distribution. The statistic is far from zero, so the p value is very small.

The curve validates the rejection of the null hypothesis.

R Chart 5: G3 Boxplot by Group

The boxplot validates the spread difference between the two groups. It shows the central tendency, variability and possible outliers.

The plot supports assumption discussion and helps explain why Welch is appropriate.

R Chart 6: Group Standard Deviation Comparison

This chart confirms that the MS standard deviation is larger than the GP standard deviation.

This supports the unequal variance interpretation and helps justify the Welch two sample t test.

R Chart 7: Mean G3 by Study Time Within Group

This chart gives additional context by showing the GP and MS mean pattern across studytime categories.

The chart is descriptive context. It does not replace the formal two sample t-test decision.

R Chart 8: Group Summary Table

This table reports sample size, mean, standard deviation and variance for GP and MS.

The table validates the descriptive values used in the formula and interpretation sections.

R Chart 9: Main Result Table

This table reports the main R t-test result, including the statistic, degrees of freedom, p value and confidence interval.

The result supports the same final conclusion: reject the null hypothesis and report a significant mean difference.

R Chart 10: Welch Result Table

This table gives the preferred Welch result. It uses adjusted degrees of freedom and separate variance components.

The table is the best result to report when group variances are unequal or sample sizes differ noticeably.

SPSS, R, Python and Excel Workflows for Two Sample T Test

The same Two Sample T Test can be reproduced in SPSS, R, Python and Excel. The interpretation remains the same when the same variable, groups, test direction and confidence level are used.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Two Sample T Test

SPSS Syntax for Two Sample T Test

* Two Sample T Test in SPSS.
* Outcome: G3.
* Grouping variable: school.
* Groups: GP and MS.

TITLE "Two Sample T Test: 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 NPPLOT
  /STATISTICS DESCRIPTIVES
  /CINTERVAL 95
  /MISSING LISTWISE
  /NOTOTAL.

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

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Two-Sample-T-Test-SPSS-Output.pdf".

Python Code for Two Sample T Test

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

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

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

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

# Descriptive statistics
summary = pd.DataFrame({
    "group": ["GP", "MS"],
    "n": [gp.count(), ms.count()],
    "mean": [gp.mean(), ms.mean()],
    "sd": [gp.std(ddof=1), ms.std(ddof=1)],
    "variance": [gp.var(ddof=1), ms.var(ddof=1)]
})

# Equal variance two sample t test
student_result = stats.ttest_ind(gp, ms, equal_var=True)

# Welch unequal variance two sample t test
welch_result = stats.ttest_ind(gp, ms, equal_var=False)

mean_diff = gp.mean() - ms.mean()
welch_se = np.sqrt(gp.var(ddof=1) / gp.count() + ms.var(ddof=1) / ms.count())

# Welch-Satterthwaite degrees of freedom
v1 = gp.var(ddof=1) / gp.count()
v2 = ms.var(ddof=1) / ms.count()
welch_df = (v1 + v2) ** 2 / ((v1 ** 2) / (gp.count() - 1) + (v2 ** 2) / (ms.count() - 1))

print(summary)
print("Mean difference:", mean_diff)
print("Equal variance t:", student_result)
print("Welch t:", welch_result)
print("Welch SE:", welch_se)
print("Welch df:", welch_df)

R Code for Two Sample T Test

# Two Sample T Test 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("G3", "school", "studytime")])

# Group summaries
aggregate(G3 ~ school, data = df_model, FUN = function(x) {
  c(n = length(x), mean = mean(x), sd = sd(x), variance = var(x))
})

# Welch two sample t test, default in R
welch_result <- t.test(G3 ~ school, data = df_model)

# Equal variance Student two sample t test
student_result <- t.test(G3 ~ school, data = df_model, var.equal = TRUE)

print(welch_result)
print(student_result)

# Extract key values
welch_result$statistic
welch_result$parameter
welch_result$p.value
welch_result$conf.int
welch_result$estimate

Excel Formulas for Two Sample T Test

Group 1 mean:
=AVERAGE(group1_range)

Group 2 mean:
=AVERAGE(group2_range)

Group 1 standard deviation:
=STDEV.S(group1_range)

Group 2 standard deviation:
=STDEV.S(group2_range)

Mean difference:
=AVERAGE(group1_range)-AVERAGE(group2_range)

Equal variance two sample t test p value:
=T.TEST(group1_range,group2_range,2,2)

Welch unequal variance two sample t test p value:
=T.TEST(group1_range,group2_range,2,3)

Welch standard error:
=SQRT(VAR.S(group1_range)/COUNT(group1_range)+VAR.S(group2_range)/COUNT(group2_range))

Decision:
=IF(p_value_cell<0.05,"Reject H0","Fail to reject H0")

APA Reporting Wording

When reporting a Two Sample T Test, include the test type, group sample sizes, group means, standard deviations, t statistic, degrees of freedom, p value, confidence interval and direction of the result. If the variances differ, report the Welch result.

Common Mistakes

When to Use Two Sample T Test

Use a Two Sample T Test when the dependent variable is numeric and the independent variable has exactly two independent groups.

Downloads and Resources

Use these resources to reproduce the Two Sample T Test workflow. Replace placeholder links with final hosted file URLs after uploading the dataset, scripts, syntax files, workbook and output PDFs to WordPress Media Library.

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