Guides / t test
T-tests: a complete guide, with worked examples
The t-test is the first inferential test most researchers meet and the one most often reported incorrectly. This guide covers the three types and when each applies, the assumptions and what to do when they fail, two worked examples calculated by hand, effect sizes, and exactly how to report the result in APA style.
Written and reviewed by Hafiz Ahmad Tariq, Senior BiostatisticianA t-test is a statistical test that compares means to determine whether an observed difference is larger than would be expected by chance. There are three types: a one-sample t-test compares a sample mean to a known value, an independent samples t-test compares two separate groups, and a paired samples t-test compares two measurements from the same participants.
Definition
What a t-test does
A t-test compares means and asks whether the difference between them is larger than chance alone would comfortably produce. It does this by expressing the difference as a ratio: how big the difference is, divided by how much variability there is in the data.
Because it is a ratio, the same difference between two means can produce a large or a small t depending entirely on how noisy the data are. Two studies reporting an identical four-point difference can reach opposite conclusions, and neither is wrong — the one with tighter data simply has stronger evidence that the difference is real. This is why reporting a difference without its variability tells a reader almost nothing, and why the standard deviation belongs in every results table alongside the mean.
That ratio is the t statistic. A large t means the difference is big relative to the noise; a small t means the difference could easily be noise. The p-value then tells you how often you would see a t that extreme if there were genuinely no difference at all.
A significant t-test does not mean the difference is large, important, or caused by your intervention. It means a difference this size is unlikely to arise by chance. Size is the effect size’s job; importance is yours; causation is the design’s.
Choosing
The three types, and which you need
| Type | Compares | Example | Calculator |
|---|---|---|---|
| One-sample | One sample mean against a known or hypothesised value | Is our cohort’s mean score different from the national average of 50? | — |
| Independent samples | Two separate groups of different people | Do the intervention group and control group differ? | effect size calculator |
| Paired samples | Two measurements from the same people | Did scores change from before to after? | — |
The distinction between independent and paired is where most errors happen, and it is not about the data layout — it is about the design. If each row of your dataset is one person measured twice, you need a paired test. If each row is a different person, you need an independent test.
A paired design analysed with an independent test throws away the pairing, which is usually the most powerful feature of the design. It typically makes a real effect disappear, and an examiner will spot it from the methods section alone.
When a t-test is not the right test
Assumptions
Assumptions, and what to do when they fail
Every parametric test rests on assumptions. Reporting that you checked them is the single most common omission in a results chapter, and it is also the quickest to fix.
| Assumption | How to check | If it fails |
|---|---|---|
| Outcome is continuous | By inspection of the variable | Use a test for ordinal or categorical data |
| Observations are independent | By the design, not a test | Use a paired test, or a multilevel model for clustering |
| Approximate normality | Shapiro-Wilk; Q-Q plot; skew and kurtosis | Mann-Whitney U (independent) or Wilcoxon (paired). Or rely on the central limit theorem if n is large |
| Equal variances (independent only) | Levene’s test | Use Welch’s t-test — and see below |
| No extreme outliers | Boxplot; standardised residuals | Investigate before removing. Report the analysis with and without |
Normality: the assumption people over-worry about
The t-test assumes the sampling distribution of the mean is approximately normal, not that your raw data is. With samples above roughly 30 per group, the central limit theorem makes this reasonable even for fairly skewed data. Shapiro-Wilk is also over-sensitive in large samples, flagging trivial departures as significant.
Practical guidance: inspect a Q-Q plot as well as running a test, and treat a significant Shapiro-Wilk in a large sample as information rather than an instruction.
Equal variances: just use Welch
The classic advice is to run Levene’s test and switch to Welch’s correction if it is significant. The modern recommendation is simpler: use Welch’s t-test by default. It performs as well as Student’s when variances are equal and considerably better when they are not, so the two-step procedure adds a decision without adding accuracy.
SPSS reports both rows in its output. R’s t.test() uses Welch by default, which is why R and SPSS sometimes disagree until you notice which row you read.
Send the dataset and a statistician will check the assumptions, run the right test, and give you the annotated output plus the syntax — so you can rerun it and defend it.
See SPSS data analysisWorked example
Worked example: independent samples
A study compares reading scores between an intervention group and a control group.
| Group | n | Mean | SD |
|---|---|---|---|
| Intervention | 48 | 24.6 | 5.2 |
| Control | 45 | 21.1 | 4.9 |
Step 1 — the pooled standard deviation
Combine the two groups’ variability, weighted by their sample sizes:
sp = √[ ((n₁−1)s₁² + (n₂−1)s₂²) / (n₁+n₂−2) ]= √[ (47 × 27.04 + 44 × 24.01) / 91 ] = √25.62 = 5.06
Step 2 — the standard error of the difference
SE = sp × √(1/n₁ + 1/n₂) = 5.06 × √(1/48 + 1/45) = 5.06 × 0.2075 = 1.050
Step 3 — the t statistic
t = (M₁ − M₂) / SE = (24.6 − 21.1) / 1.050 = 3.5 / 1.050 = 3.33, with df = n₁ + n₂ − 2 = 91.
Step 4 — the p-value and the decision
At df = 91 the two-tailed critical value at α = .05 is about ±1.99. Our t of 3.33 exceeds it comfortably, giving p = .001. We reject the null hypothesis of no difference.
Step 5 — the effect size, which is not optional
d = (M₁ − M₂) / sp = 3.5 / 5.06 = 0.69 — a medium-to-large effect by convention. The 95% confidence interval for d is [0.27, 1.11], which excludes zero and is consistent with the significance test.
Our effect size calculator takes exactly these six numbers and returns Cohen’s d with its confidence interval, Hedges’ g, and the APA sentence.
Worked example
Worked example: paired samples
The paired test works on the differences, not the raw scores — which is why it is more powerful when the pairing is real.
| Participant | Before | After | Difference |
|---|---|---|---|
| P1 | 18 | 23 | +5 |
| P2 | 22 | 24 | +2 |
| P3 | 15 | 21 | +6 |
| P4 | 20 | 20 | 0 |
| P5 | 17 | 22 | +5 |
| P6 | 19 | 25 | +6 |
Mean difference Md = 4.0, standard deviation of the differences sd = 2.28, n = 6.
SE = sd / √n = 2.28 / 2.449 = 0.931t = Md / SE = 4.0 / 0.931 = 4.30, df = n − 1 = 5, giving p = .008.
Effect size for a paired test is dz = Md / sd = 4.0 / 2.28 = 1.75.
Between-person variability is removed entirely. Scores ranged from 15 to 25, but the differences ranged only from 0 to 6 — and it is that smaller variability the test divides by. Analysed as independent groups, this same data would not have come close to significance.
Worked example
Worked example: one-sample
The one-sample t-test is the simplest of the three and the one most often forgotten. It compares your sample mean against a value you already know — a national average, a published benchmark, a clinical threshold, a target set by a funder.
A department reports a mean student satisfaction score of 4.18 (SD = 0.62) from 40 respondents. The sector benchmark is 4.00. Is the department genuinely above benchmark, or is 4.18 within the range you would expect from 40 people by chance?
SE = s / √n = 0.62 / √40 = 0.62 / 6.325 = 0.098t = (M − μ) / SE = (4.18 − 4.00) / 0.098 = 1.84, df = n − 1 = 39.
At df = 39 the two-tailed critical value is about ±2.02. Our t of 1.84 does not reach it, giving p = .073. The difference is in the expected direction but not statistically significant.
Resist it. The honest report is that the department scored above benchmark by 0.18 points, t(39) = 1.84, p = .073, d = 0.29 — a small effect that this sample was too small to distinguish from chance. That is a finding, and it is more useful than a phrase that implies the threshold nearly moved.
Effect size here is d = (M − μ) / s = 0.18 / 0.62 = 0.29. Note how much more the effect size tells you than the p-value: the department is above benchmark by roughly a third of a standard deviation, and a larger sample would settle whether that is real.
Non-parametric options
When the assumptions fail: the alternatives
If normality is genuinely violated in a small sample, the standard alternatives compare ranks rather than means. They answer a slightly different question, and that difference matters when you write up.
| Parametric test | Non-parametric alternative | What it actually compares |
|---|---|---|
Independent samples t | Mann-Whitney U | Whether one group tends to score higher, by ranks |
Paired samples t | Wilcoxon signed-rank | Whether the differences tend to be positive or negative |
One-sample t | One-sample Wilcoxon | Whether the median differs from a value |
The important nuance: Mann-Whitney does not test whether the means differ, and describing it as if it did is a common slip. It tests whether values from one group tend to exceed values from the other. Where the two distributions have a similar shape, that is close to a median comparison; where they do not, it is a statement about stochastic dominance and should be worded carefully.
Reach for non-parametric when
- Small samples with clear, visible skew
- The outcome is genuinely ordinal
- There are extreme outliers you cannot justify removing
- Your discipline expects it for this measure
Stay parametric when
- Samples are reasonably large — roughly 30+ per group
- Shapiro-Wilk is significant but the Q-Q plot looks fine
- You need the effect size and confidence interval a t-test gives
- You will later extend the analysis to regression or ANOVA
It discards information by converting values to ranks, which usually costs power, and effect sizes are less familiar to readers. Switching only because Shapiro-Wilk flagged a large sample is a common over-correction.
Whichever you use, report it consistently: the test statistic, the exact p-value, and an effect size — r = z/√N for Mann-Whitney and Wilcoxon.
Effect size
Effect size — the part people omit
Significance tells you an effect probably exists. Effect size tells you how big it is, and it is the question anyone reading your work asks next. APA has required it for years and reviewers routinely send papers back without it.
| Statistic | Use for | Conventional bands |
|---|---|---|
| Cohen’s d | Independent samples | 0.2 small · 0.5 medium · 0.8 large |
| Hedges’ g | Independent samples, n < ~50 | Same bands; corrects small-sample bias |
| Cohen’s dz | Paired samples | Same bands, but not comparable to independent d |
Cohen said so himself, and warned against exactly the mechanical use they now get. In a field where interventions typically move an outcome by d = 0.15, a d of 0.3 is a substantial result. Judge magnitude against your literature, not a table.
Report the confidence interval around the effect size as well as the point estimate. It carries far more information than the p-value: an interval of [0.27, 1.11] tells the reader the effect is probably real but its size is still uncertain, which a bare p = .001 conceals.
APA 7
Reporting a t-test in APA style
The complete form, with every element APA 7 expects:
t(91) = 3.33, p = .001, d = 0.69, 95% CI [0.27, 1.11]
Written into a results section, with the descriptive statistics that give it meaning:
The intervention group (M = 24.6, SD = 5.2) scored significantly higher than the control group (M = 21.1, SD = 4.9), t(91) = 3.33, p = .001, d = 0.69, 95% CI [0.27, 1.11].
p to three decimals; below .001 report p < .001, never p = .000p = .001, not 0.001Our APA results generator takes your values and returns the correctly formatted line for eight different tests, including both t-tests.
Open the APA generatorCommon problems
Seven mistakes that cost marks
| Mistake | Why it matters | Instead |
|---|---|---|
| Independent test on a paired design | Discards the pairing and usually hides a real effect | Match the test to the design, not the spreadsheet |
| Multiple t-tests instead of ANOVA | Three tests at α = .05 give roughly a 14% false positive rate | Use ANOVA, then post-hoc tests with correction |
p = .000 | No p-value is zero; it is a rounding artefact | p < .001 |
| No effect size | Significance without magnitude answers half the question | Report d or g with its CI |
| Assumptions never mentioned | The reader cannot judge whether the test was valid | State what you checked and what you found |
“Approaching significance” at p = .06 | The threshold is arbitrary but the language is not | Report the exact p and the effect size, and let them speak |
| Causal language from a correlational design | The test cannot support it | “Associated with”, unless you randomised |
Answers
Frequently asked questions
What is the difference between a paired and an independent t-test?
An independent samples t-test compares two separate groups of different people. A paired samples t-test compares two measurements taken from the same people — before and after, or two conditions completed by everyone. The distinction is about your design, not how the data is laid out.
What does the t value actually mean?
It is the size of the difference divided by the standard error of that difference. A t of 3 means the observed difference is three times larger than the typical sampling variability, which makes chance an unconvincing explanation.
How many participants do I need for a t-test?
It depends on the effect you want to detect. To find a medium effect (d = 0.5) with 80% power at α = .05 you need about 64 per group for an independent test. A power analysis before data collection gives the answer for your specific design.
Can I run a t-test on Likert data?
With a single Likert item, usually not — it is ordinal with few categories, so Mann-Whitney U is safer. With a summed multi-item scale, a t-test is generally accepted, though you should say why you treated the total as continuous.
What if my data is not normally distributed?
With reasonably large samples the t-test is robust, because the central limit theorem applies to the sampling distribution of the mean rather than to your raw data. With small or heavily skewed samples, use Mann-Whitney U for independent data or Wilcoxon signed-rank for paired data.
Should I use a one-tailed or two-tailed test?
Two-tailed, almost always. A one-tailed test is only justified when a difference in the opposite direction would be meaningless and you committed to that before seeing the data. Switching to one-tailed after a two-tailed result misses significance is a recognised questionable research practice.
Why do SPSS and R give slightly different results?
R's t.test() uses Welch's correction by default; SPSS reports both rows and people commonly read the Student's row. The Welch result has fractional degrees of freedom, which is the usual giveaway.
Do I have to report an effect size?
Yes, in practice. APA has required it for years, and reviewers and examiners routinely ask for it. Report Cohen's d or Hedges' g with a confidence interval.
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