Quick Answer: Independent Samples T Test Result

The Independent Samples T Test showed that students in GP scored significantly higher on G3 final grade than students in MS. The GP group had a mean of 12.577, while the MS group had a mean of 10.650. The mean difference was 1.926 points.

The test statistic was t(647) = 7.54, with p < .001. The 95% confidence interval for the mean difference was [1.425, 2.428], which does not include zero. That means the difference is statistically significant. The estimated effect size was approximately Cohen’s d = 0.62, which is usually interpreted as a medium practical effect.

What Is an Independent Samples T Test?

An Independent Samples T Test compares the means of two unrelated groups on a continuous outcome. It answers a simple but important research question: is the average outcome in one group different from the average outcome in another group?

In this example, the two groups are school categories: GP and MS. The outcome variable is G3, the final exam grade. Because each student belongs to only one school group, the groups are independent. That makes the Independent Samples T Test the correct parametric method.

This test is sometimes called the independent t test, two sample t test, or unpaired t test. It is one of the most widely used procedures in quantitative research because it is easy to apply and easy to interpret.

Before using the test, it is good practice to understand related topics such as the null and alternative hypothesis, p value, confidence interval, standard deviation, and parametric vs nonparametric tests.

Independent Samples T Test Formula

The basic test compares the difference between two sample means relative to the standard error of that difference:

For the equal-variances version of the Independent Samples T Test, the pooled standard deviation is:

The standard error of the mean difference is then:

Null and Alternative Hypotheses for the Independent Samples T Test

The Independent Samples T Test evaluates whether the two population means are equal.

This example uses a two-tailed test because the question is whether the groups differ, not whether one group must be larger in a pre-specified direction. If you want a directional version, see the one tailed t test guide.

Assumptions of the Independent Samples T Test

Like other parametric tests, the Independent Samples T Test relies on a set of assumptions. Good reporting should always discuss them.

The provided chart outputs do not display the numeric Levene statistic directly. However, the standard deviation comparison chart clearly shows that MS has more variability than GP. In a full SPSS table, report the Levene test p-value and choose the equal-variances or unequal-variances row accordingly.

Even so, the group difference is large enough that the practical conclusion is stable. A Welch-style robustness check using the actual group summaries still gives a highly significant result, so the decision would not change.

Dataset and Variables Used

This worked example uses the student performance dataset, where school is the grouping variable and G3 is the dependent variable. The two school groups are:

• GP = Gabriel Pereira
• MS = Mousinho da Silveira

The total number of valid observations in this analysis is 649. The group counts are:

The outcome G3 is the final exam grade. Because it is measured on a numeric scale and the groups are independent, the Independent Samples T Test is a suitable method for comparing the group means.

To understand the raw distribution before testing, you can also review related guides such as descriptive statistics, histogram interpretation, box plot interpretation, and frequency distribution.

Actual Results Summary

The core findings from the Independent Samples T Test are shown below.

SPSS-Style Interpretation of the Independent Samples T Test

If this analysis were reported from SPSS, the output would usually include Group Statistics and an Independent Samples Test table. Based on the actual chart outputs provided, the interpretation would be as follows.

SPSS Group Statistics Interpretation

The GP group had a higher mean G3 score (12.577) than the MS group (10.650). The GP scores were also less dispersed (SD = 2.626) than the MS scores (SD = 3.834). This means GP students not only scored higher on average, but their scores were also somewhat more clustered around the mean.

SPSS Independent Samples Test Interpretation

The main inferential result is t(647) = 7.54, p < .001. Because the p-value is well below .05, the difference between groups is statistically significant. The 95% CI for the mean difference is [1.425, 2.428], so the interval does not cross zero. That confirms the result visually and statistically.

Effect Size Interpretation

The estimated effect size is Cohen’s d ≈ 0.62. In many applied settings, this is considered a medium effect. In plain language, the average difference is meaningful enough to matter, not just statistically detectable.

Levene Test Interpretation

The provided images do not show the numeric Levene statistic, so it should not be invented. However, because the MS group has a noticeably larger standard deviation than GP, the safest reporting practice is:

• Report the Levene test if available from SPSS.
• If Levene is significant, use the unequal-variances row.
• If Levene is not significant, use the equal-variances row.

In this case, the group difference is strong enough that the substantive conclusion would remain the same either way.

Python Chart-by-Chart Interpretation

The Python charts below show how the Independent Samples T Test result looks from multiple visual angles. These charts are very useful because they turn abstract statistics into understandable patterns.

Python Chart 1: G3 Distribution by Group

This histogram shows that the overall GP distribution is shifted to the right of the MS distribution. That means GP students tend to have higher final grades than MS students. The chart legend displays the exact group means: GP mean = 12.577 and MS mean = 10.650.

The plot also shows that the MS group has a wider spread, which matches the larger standard deviation reported later. Even though the two distributions overlap, the center of GP is clearly higher, which visually supports the significant t test result.

Python Chart 2: Group Means with 95% Confidence Intervals

This chart emphasizes the mean difference directly. The GP bar is clearly taller than the MS bar, and the numeric labels repeat the exact means. The confidence intervals around the group means help show sampling uncertainty.

Because the group means are separated by nearly two points, the chart communicates that the group difference is not a tiny statistical artifact. It is visually substantial and consistent with the inferential result.

Python Chart 3: Mean Difference Confidence Interval

This is one of the most important charts in the entire analysis. It shows the estimated mean difference of 1.926 with a 95% CI from 1.425 to 2.428. The dashed horizontal reference line at zero represents “no mean difference.”

Because the entire interval is above zero, the conclusion is clear: GP has a significantly higher mean G3 score than MS. Confidence-interval interpretation is often more informative than a p-value alone because it shows both the direction and plausible magnitude of the difference.

Python Chart 4: t Statistic on Student’s t Distribution

This chart places the observed test statistic t = 7.54 on the t distribution. The critical values for a two-tailed 5% test are shown around ±1.96. The observed t value is far beyond the critical region.

This means the observed group difference is much larger than what we would expect under the null hypothesis of equal means. Graphically, the chart confirms why the p-value is extremely small.

Python Chart 5: G3 Boxplot by Group

The boxplot provides a clean summary of group distributions. The median of GP is higher than the median of MS, which agrees with the mean-based t test result. Both groups show some lower-end outliers, including zero scores, but the central location of GP remains higher.

The MS box also appears more spread out, which fits the larger standard deviation. This chart is particularly helpful for spotting skewness, outliers and group spread at a glance.

Python Chart 6: Group Standard Deviation Comparison

This chart shows that the standard deviation for GP is 2.626, while the standard deviation for MS is 3.834. That means scores in MS are more variable.

This matters for assumption checking because the equal-variances version of the t test works best when group variances are reasonably similar. The difference in SDs suggests that Levene’s test should be reported when available, but it does not overturn the core conclusion of the analysis.

Python Chart 7: Mean G3 by Study Time within Group

This chart adds useful context. Across all study-time categories, GP tends to maintain higher mean G3 scores than MS. In other words, the GP advantage is not limited to only one study-time category.

This does not replace a factorial model, but it is a strong descriptive sign that the group difference is broadly consistent across subgroups. It strengthens the practical interpretation of the mean difference.

R Chart-by-Chart Validation

The R chart set validates the same findings using a different software workflow. This cross-software agreement is important because it shows the result is not dependent on one package.

R Chart 1: G3 Distribution by Group

The R distribution chart confirms the same pattern seen in Python: GP scores are centered at a higher level than MS scores. Although both groups share some overlap, the distribution centers are clearly separated.

This supports the main conclusion that the difference is not random noise. It is visible in the raw score structure.

R Chart 2: Group Means with 95% Confidence Intervals

The R bar chart again shows GP with a higher average G3 than MS. The confidence intervals reinforce that the GP mean is clearly above the MS mean.

When two different programs draw the same picture from the same data, confidence in the interpretation increases.

R Chart 3: Mean Difference Confidence Interval

This chart repeats the crucial inferential message: the mean difference is positive and the 95% interval stays above zero. That is direct visual evidence that GP students scored significantly higher.

Confidence intervals are especially helpful because they provide more than a yes/no decision. They show how large the true difference is likely to be.

R Chart 4: t Statistic on Student’s t Distribution

This R chart explicitly labels Observed t = 7.54 and df = 647. Those values match the Python-based interpretation and confirm that the same inferential conclusion was reached in R.

Because the observed t statistic lies far beyond the critical region, the null hypothesis of equal means is decisively rejected.

R Chart 5: G3 Boxplot by Group

The R boxplot confirms the same structure observed earlier: higher central tendency in GP and greater spread in MS. The lower-end outliers do not erase the consistent upward shift of GP.

That is why robust descriptive visuals and formal hypothesis tests are best interpreted together.

R Chart 6: Group Standard Deviation Comparison

The R standard deviation chart again highlights a larger spread in MS. This repeated pattern is useful for discussing the homogeneity of variances assumption and the relevance of Levene’s test.

Even if the assumption is imperfect, the large sample and sizeable mean difference make the main conclusion robust.

R Chart 7: Mean G3 by Study Time within Group

The R study-time chart tells the same practical story as the Python version. GP tends to score higher across study-time levels, which suggests the school-level difference is fairly stable across these descriptive subgroups.

This cross-check is useful for interpretation because it shows that the GP advantage is not an artifact from only one subgroup.

SPSS, R, Python and Excel Workflows for the Independent Samples T Test

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for the Independent Samples T Test

SPSS Syntax

* Independent Samples T Test for G3 by school.
* Groups: GP and MS.

TITLE "Independent Samples T Test: G3 by School".

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

* Optional descriptive statistics.
MEANS TABLES=G3 BY school
  /CELLS MEAN COUNT STDDEV SEMEAN.

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

Python Code

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

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

# Keep needed columns
data = df[["school", "G3", "studytime"]].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"]

# Descriptive statistics
summary = data.groupby("school")["G3"].agg(["count", "mean", "std"])
print(summary)

# Independent samples t test
t_stat, p_val = stats.ttest_ind(gp, ms, equal_var=True)
print("t =", t_stat)
print("p =", p_val)

# Mean difference and CI
n1, n2 = len(gp), len(ms)
m1, m2 = gp.mean(), ms.mean()
s1, s2 = gp.std(ddof=1), ms.std(ddof=1)

sp = np.sqrt(((n1 - 1) * s1**2 + (n2 - 1) * s2**2) / (n1 + n2 - 2))
se = sp * np.sqrt(1/n1 + 1/n2)
dfree = n1 + n2 - 2
crit = stats.t.ppf(0.975, dfree)

mean_diff = m1 - m2
ci_low = mean_diff - crit * se
ci_high = mean_diff + crit * se

cohens_d = mean_diff / sp

print("Mean difference =", mean_diff)
print("95% CI =", (ci_low, ci_high))
print("Cohen's d =", cohens_d)

R Code

# Independent Samples T Test for G3 by school

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

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

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

# Descriptive statistics
aggregate(G3 ~ school, data = df_model,
          FUN = function(x) c(n = length(x), mean = mean(x), sd = sd(x)))

# T test
result <- t.test(G3 ~ school, data = df_model, var.equal = TRUE)
print(result)

# Effect size
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)

sp <- sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2))
d <- (m1 - m2) / sp

cat("Cohen's d =", d, "\n")

Excel Steps

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

Step 2:
Calculate group means:
=AVERAGE(range)

Step 3:
Calculate group standard deviations:
=STDEV.S(range)

Step 4:
Open Data Analysis ToolPak.
Choose: t-Test: Two-Sample Assuming Equal Variances

Step 5:
Select GP and MS ranges.
Set Hypothesized Mean Difference = 0
Alpha = 0.05

Step 6:
Interpret:
- If p < .05, reject the null hypothesis.
- Check the mean difference and t statistic.
- Report the confidence interval if available.

Step 7:
Create charts:
- Histogram by group
- Mean comparison chart
- Boxplot
- Standard deviation comparison

APA Reporting for the Independent Samples T Test

Here is a ready-to-use APA-style reporting example for this analysis:

If your instructor or journal requires it, you can also add a note about the variance assumption and Levene’s test when the SPSS table is available.

Common Mistakes in the Independent Samples T Test

When to Use the Independent Samples T Test

Use the Independent Samples T Test when:

• You have two unrelated groups.
• Your outcome variable is continuous.
• You want to test whether the group means differ.
• The assumptions are at least reasonably acceptable.

Examples include comparing:

• exam scores between two schools,
• blood pressure between treatment and control groups,
• income between two employment categories,
• customer ratings between two service models.

Downloads and Resources

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