Why assumptions matter
Confidence intervals and hypothesis tests don’t work in a vacuum—they depend on assumptions about your data. When you calculate a t-interval for a mean, you’re assuming certain conditions about normality and independence. Understanding these assumptions (and what to do when they’re violated) is key to making valid inferences.

t-Intervals and their assumptions

A classic t-interval for a population mean is built under these assumptions:

• Independence – The data points are independent of each other (usually satisfied by random sampling or random assignment).
• Normality – The population distribution is normal, or the sample size is large enough for the Central Limit Theorem (CLT) to kick in.
• Scale of measurement – Data are quantitative and roughly continuous.

When these assumptions hold, t-intervals give accurate coverage probabilities and can be safely used.

When normality is questionable

Real data often deviate from perfect normality. Here’s how to think about it:

• Small samples (n < 30) – Skewness or heavy tails can cause inaccurate intervals. Normality should be checked with plots or tests.
• Moderate to large samples – By the CLT, the sampling distribution of the mean becomes approximately normal, making t-intervals robust.
• Outliers – Even with large samples, extreme values can distort means and intervals. Outlier diagnostics are essential.

Robust and alternative options

When assumptions fail, robust methods step in:

• Bootstrap confidence intervals – Resample data with replacement and compute the interval empirically. Great for skewed distributions or small samples.
• Nonparametric intervals – Use medians or quantiles instead of means (with binomial or bootstrap-based intervals).
• Transformations – Apply log or square root transforms to stabilize variance and improve normality before applying t-methods.
• Trimmed means – Reduce the impact of outliers by discarding extreme values before analysis.

Practical guidance

• Check assumptions first: visual inspections (histograms, Q-Q plots) are simple but powerful.
• If sample size is large, t-intervals are usually robust to mild departures from normality.
• If data are heavily skewed or contain outliers, bootstrap or nonparametric approaches may be more reliable.
• Always report what method you used and why—it strengthens the credibility of your results.

Helpful resources

For deeper learning, a comprehensive applied statistics textbook can walk through t-interval assumptions and alternatives in detail. For a quick refresher, a statistics quick study guide offers charts and decision trees for checking normality and choosing robust options. If you want hands-on examples, a book on bootstrap methods is a practical companion.

The bottom line

t-intervals are powerful, but only as valid as their assumptions. Normality, independence, and careful handling of outliers are critical. When assumptions break, robust methods like bootstrapping provide reliable alternatives. The best analysts don’t just compute intervals—they check the foundations those intervals rest on.