T-Test Examples: Step-by-Step Worked Problems & Real-Life Applications

What Is a T-Test?

William Sealy Gosset developed the t-test in 1908 while working at the Guinness brewery in Dublin, publishing under the pseudonym "Student" — hence the name Student's t-test. He needed a method for comparing small samples of barley yields when σ was unknown. The test Ronald Fisher later formalized is now one of the most widely used procedures in statistics, appearing in clinical trials, psychology studies, quality control, and A/B testing.

The intuition behind any t-test is simple. You expect two things to be equal (or a mean to equal some value). You collect data. The question is whether the gap you observe is genuine or whether sampling variability could plausibly explain it. The t-statistic measures that gap in units of standard error. Large |t| means the gap is large relative to the noise — and a low p-value follows.

The t-distribution has heavier tails than the normal distribution, which accounts for the extra uncertainty that comes from estimating σ from the sample. As sample size grows, the t-distribution approaches the normal distribution — with df > 120, the difference is negligible. Full background on this is in the t-distribution guide at Statistics Fundamentals.

Types of T-Tests

Three variants cover nearly every mean-comparison problem in practice. Choosing the wrong type is one of the most common errors students make. The decision depends on two questions: how many groups are you comparing, and are those groups independent?

One-Sample T-Test

Use this when you have a single group and want to know whether its mean matches a specific value. The "specific value" is a known standard, a regulatory threshold, a historical benchmark, or a manufacturer's claim. You do not need a second group — just a hypothesized value to compare against. For full theory, see the one-sample t-test guide.

Independent Samples T-Test

Use this when you have two separate groups and the subjects in one group have no relationship to the subjects in the other. The two groups must be truly independent — no person or subject appears in both. If the two groups have noticeably different variances (check with Levene's test), switch to Welch's t-test instead. The full procedure is at the two-sample t-test guide.

Paired Samples T-Test

Use this when each observation in one group corresponds to exactly one observation in the other — for example, the same person measured before and after a treatment, or two patients matched on age and health status. The key insight is that you reduce the problem to a one-sample test on the differences. See the complete walkthrough at the paired samples t-test guide.

T-Test Formulas

One-Sample T-Test Formula

Independent Samples T-Test Formula (Equal Variances)

Paired T-Test Formula

Welch's T-Test Formula (Unequal Variances)

Assumptions of a T-Test

Every t-test requires a set of conditions to be met before the results are trustworthy. Violating these assumptions does not always make the test useless — the t-test is robust to mild departures from normality when samples are large — but understanding each assumption helps you decide whether a t-test is appropriate or whether a nonparametric alternative (like the Mann-Whitney U test or Wilcoxon signed-rank test) would be more appropriate.

The TEST Framework — A Structured Approach

The following four-step framework organizes every t-test problem in a consistent, reproducible way. It is designed to prevent the most common errors: choosing the wrong test type, skipping assumption checks, and misinterpreting the conclusion.

One-Sample T-Test Examples

A one-sample t-test compares a sample mean to a specific number you already know — a regulatory standard, a manufacturer's claim, a historical benchmark, or any fixed reference point. The null hypothesis is always H₀: μ = μ₀. The test statistic is t = (x̄ − μ₀) / (s/√n) with df = n − 1.

Example 1 — Light Bulb Lifespan (Manufacturing)

Example 2 — Student Exam Scores (Education)

Example 3 — Daily Calorie Intake (Healthcare)

Independent Samples T-Test Examples

The independent samples t-test (also called the two-sample t-test) compares means from two separate, unrelated groups. The null hypothesis is H₀: μ₁ = μ₂, meaning no difference between the population means. When the two groups have equal variances, you use the pooled formula with df = n₁ + n₂ − 2. The full procedure and assumptions are at the two-sample t-test guide.

Example 4 — Two Teaching Methods (Education)

Example 5 — Drug vs. Placebo (Clinical Research)

Example 6 — Website Conversion Rates (A/B Testing)

Paired Samples T-Test Examples

The paired t-test works by reducing the two-column problem to a one-column problem. You compute the difference for each pair, then run a one-sample t-test on those differences against a hypothesized mean difference of zero. Because you remove between-subject variability, the paired test is more statistically powerful than the independent test when the pairing is justified. Full theory is at the paired samples t-test guide.

Example 7 — Blood Pressure Before and After Treatment

Example 8 — Employee Productivity Before and After Training

Welch's T-Test Examples

Welch's t-test is the appropriate choice when the two groups have unequal population variances. It adjusts the degrees of freedom using the Welch-Satterthwaite equation, resulting in a non-integer df that is always smaller than or equal to the df from the pooled t-test. Many statisticians recommend using Welch's t-test by default for independent comparisons, since it gives the correct Type I error rate whether or not variances are equal. See the detailed comparison at equal vs. unequal variance.

Example 9 — Salary Comparison Between Departments

Example 10 — Crop Yield Comparison (Agriculture)

Which T-Test Should You Use?

Use the statistical test selector for a comprehensive guide covering all tests. For t-tests specifically, this decision tree covers the most common scenarios:

T-Test vs. Z-Test vs. ANOVA

How to Calculate and Interpret Effect Size (Cohen's d)

Statistical significance tells you whether an effect exists. Effect size tells you how large that effect is. These are separate questions. A t-test with 10,000 participants will produce a significant result for a trivial difference; a test with 10 participants may fail to detect a large but real effect. Cohen's d is the most widely reported effect size for t-tests.

The American Statistical Association recommends reporting effect sizes alongside p-values in all published research. A significant p-value with a tiny d (e.g., d = 0.05) means the effect, while real, is too small to matter in practice. Conversely, a non-significant p-value with a large d (e.g., d = 0.90) suggests the study was underpowered — there likely is a real effect, but the sample was too small to detect it reliably.

Real-Life Applications of T-Tests

Running a T-Test in R, Python, and Excel

Every major statistics package handles all three t-test variants with a single function call. The examples below replicate Example 5 (drug vs. placebo, independent t-test). For full software tutorials, see the statistical interpretation guide.

# One-Sample T-Test
x <- c(978,1010,965,982,1024,990,1003,971,1018,985,962,995,1007,973,1012)
t.test(x, mu = 1000, alternative = "two.sided")

# Independent T-Test (Welch's by default)
drug   <- c(18,24,20,25,23,22,19,26,21,24, # 30 values...
placebo <- c(10,14,12,11,13,15, # 30 values...
t.test(drug, placebo, var.equal = FALSE)   # Welch's
t.test(drug, placebo, var.equal = TRUE)    # Pooled

# Paired T-Test
before <- c(145,138,152,149,143,156,148,140,151,147)
after  <- c(133,130,137,143,133,142,139,129,144,134)
t.test(before, after, paired = TRUE, alternative = "greater")

# Effect size (Cohen's d) — install effsize package
library(effsize)
cohen.d(drug, placebo)

from scipy import stats
import numpy as np

# One-Sample T-Test
x = [978,1010,965,982,1024,990,1003,971,1018,985,962,995,1007,973,1012]
t_stat, p_val = stats.ttest_1samp(x, popmean=1000)
print(f"t = {t_stat:.4f}, p = {p_val:.4f}")

# Independent T-Test (Welch's — equal_var=False)
drug    = [22.4, 18.1, 25.3]  # full data...
placebo = [12.3, 10.5, 14.2]  # full data...
t_stat, p_val = stats.ttest_ind(drug, placebo, equal_var=False)

# Paired T-Test
before = [145,138,152,149,143,156,148,140,151,147]
after  = [133,130,137,143,133,142,139,129,144,134]
t_stat, p_val = stats.ttest_rel(before, after)

# Cohen's d (manual)
def cohens_d(g1, g2):
    diff = np.mean(g1) - np.mean(g2)
    n1,n2 = len(g1), len(g2)
    s_pool = np.sqrt(((n1-1)*np.var(g1,ddof=1) + (n2-1)*np.var(g2,ddof=1)) / (n1+n2-2))
    return diff / s_pool

-- One-Sample T-Test in Excel --
Store data in A1:A15. In B1, enter the formula:
=T.TEST(A1:A15, {1000,1000,...}, 2, 1)  ← workaround: compare to constant

Better approach for one-sample:
=T.DIST.2T(ABS((AVERAGE(A1:A15) - 1000) / (STDEV(A1:A15) / SQRT(COUNT(A1:A15)))), COUNT(A1:A15)-1)

-- Independent T-Test (Data Analysis ToolPak) --
Data → Data Analysis → t-Test: Two-Sample Assuming Equal Variances
   OR t-Test: Two-Sample Assuming Unequal Variances (Welch's)

-- Paired T-Test --
Data → Data Analysis → t-Test: Paired Two Sample for Means
   OR formula: =T.TEST(A1:A10, B1:B10, 2, 1)
   (last argument: 1=paired, 2=independent equal var, 3=independent unequal var)

-- Critical Value --
=T.INV.2T(0.05, df)   ← two-tailed critical value
=T.INV(0.95, df)       ← one-tailed upper critical value

Common T-Test Mistakes

T-Test Formula Cheat Sheet

Entity & Terminology Glossary

Interactive T-Test Calculator

Enter your summary statistics below to run a one-sample or two-sample t-test. For a paired t-test, compute the differences first and enter them as a one-sample problem (μ₀ = 0). The full-featured calculator with CSV upload is at statisticsfundamentals.com/calculators/t-test.

Frequently Asked Questions

References & Further Reading

• Gosset, W.S. ("Student") (1908) — "The Probable Error of a Mean." Biometrika, 6(1), 1–25. The original paper deriving the t-distribution and one-sample test.
• Cohen, J. (1988) — Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates. Definitive reference for effect size interpretation.
• NIST Engineering Statistics Handbook — Section 7.3: t-Tests. itl.nist.gov. Authoritative reference from the National Institute of Standards and Technology.
• Welch, B.L. (1947) — "The Generalization of 'Student's' Problem when Several Different Population Variances are Involved." Biometrika, 34(1/2), 28–35. Original paper introducing Welch's t-test.
• OpenStax Statistics — Introductory Statistics, Chapter 10: Hypothesis Testing with Two Samples. openstax.org. Free, peer-reviewed textbook.
• Penn State STAT 415 — Matched Pairs, Two-Sample T-Tests. online.stat.psu.edu. Comprehensive university course material.
• American Statistical Association (2016) — Statement on p-values: context, process, and purpose. The American Statistician, 70(2), 129–133. tandfonline.com
• SciPy Documentation — scipy.stats.ttest_1samp, ttest_ind, ttest_rel. docs.scipy.org. Python implementation reference.