First identify which t-test was run
The output should tell you whether the result comes from:
- a one-sample t-test, comparing one sample mean with a specified value;
- an independent two-sample t-test, comparing means from separate groups;
- a paired t-test, analyzing within-pair differences;
- a Welch two-sample t-test, which does not impose equal group variances in the same way as the pooled-variance test.
These tests can display similar fields while estimating different quantities. Do not interpret a paired result as if it compared unrelated groups. Do not assume every two-sample result uses pooled variance; R’s standard two-sample function, for example, defaults to Welch’s test unless equal variance is requested.1
The fields and what each one answers
| Field | What it tells you | What to check |
|---|---|---|
| Estimate or means | The observed center or mean difference | Group order and units |
| Standard error | Estimated sampling variability of the difference | Whether the reported model matches the design |
| t statistic | Difference from the null in standard-error units | Sign and null value |
| Degrees of freedom | Reference-distribution parameter | It may be fractional for Welch’s test |
| p-value | Tail probability under the null model and assumptions | One- or two-sided alternative |
| Confidence interval | Range of parameter values compatible with the method and data | Confidence level and null value |
| Effect size | Standardized or otherwise scaled magnitude | Which effect-size definition was used |
Official R output includes the t statistic, degrees of freedom, p-value, confidence interval, estimates, method, and alternative; other software can arrange or name these pieces differently.1 2
A worked reading
Suppose a two-group output reports:
Mean difference = 4.2; t = 2.31; df = 37.6; p = .026; 95% CI [0.52, 7.88].
Read it field by field:
- Difference: the first group’s observed mean is 4.2 units higher under the subtraction order used.
- Interval: compatible differences under the method run from 0.52 to 7.88 units. The range shows substantial uncertainty around 4.2.
- t statistic: the estimate is 2.31 estimated standard errors above the null difference of 0.
- Degrees of freedom: the fractional 37.6 is consistent with a Welch–Satterthwaite calculation, not a data-entry error.1
- p-value: under the null model and stated assumptions, the selected tail probability is .026.
For a matching two-sided procedure, a 95% confidence interval that excludes 0 corresponds to rejecting a zero-difference null at the 5% level.3 How to interpret a p-value explains why this is not the probability that the null is true.
What the sign of t means
The sign follows the direction of the estimated difference. If software calculates mean A minus mean B, a positive t generally points toward A having the higher mean; reversing the subtraction reverses the sign. The two-sided p-value normally stays the same.
This means “t = −2.4” is not a worse result than “t = 2.4.” It indicates direction under the coding and comparison order. Read the named groups or paired-difference definition before describing that direction.
Why degrees of freedom can be fractional
The classic pooled two-sample test uses a simple degrees-of-freedom expression when its variance assumptions apply. Welch’s test estimates degrees of freedom using the sample variances and sizes, so the result need not be a whole number.1 Rounding it to an integer can erase a clue about the method actually used.
If the output gives no method label, the degrees of freedom, equal-variance setting, and software documentation can help identify the test. For output with more than two groups, the ANOVA table guide shows how degrees of freedom participate in a different decomposition.
Check assumptions after identifying the estimand
A t-test does not become valid because its table is complete. The relevant checks depend on the design:
- observations or pairs should match the sampling and dependence structure;
- extreme observations can affect means, standard errors, and the test statistic;
- a paired test concerns the distribution of within-pair differences;
- a pooled two-sample test adds an equal-variance assumption;
- Welch’s test addresses unequal variances but not every design problem.
For categorical-count output rather than mean comparisons, see chi-square output interpretation. If group variance is central, inspect the variance model rather than selecting a test from a single preliminary p-value.
Effect size belongs beside the test
The t statistic is scaled by uncertainty and sample information; it is not a direct measure of practical magnitude. Two studies can have similar mean differences and different t values because their standard errors differ. Report the raw difference and interval, and add a clearly defined effect size when it helps readers compare scales.
P-value interpretation and confidence intervals answer inferential questions. Effect size answers a magnitude question. None replaces an explanation in the original units.
Common reading errors
- Reporting “t was significant” without naming the comparison or direction.
- Treating the p-value as the size of the difference.
- Assuming a fractional df is invalid.
- Ignoring whether the test was paired, pooled, or Welch.
- Reading a two-sided p-value as a directional probability.
- Omitting the estimate and confidence interval.
A defensible reporting sentence
“The first group’s mean was 4.2 units higher than the second group’s (95% CI [0.52, 7.88]); Welch’s t(37.6) = 2.31, p = .026.” Then describe the outcome units, design, assumptions, and practical meaning. The table fields should support that account, not replace it.