Every statistical test carries invisible baggage—a set of assumptions about your data that determine whether the results mean anything at all. Run a t-test on data that violates its core requirements, and you might as well flip a coin. The p-value you calculate becomes fiction dressed up as precision.

The problem is that these assumptions rarely get checked. Researchers learn to plug numbers into software, interpret the output, and move on. The software doesn't complain when assumptions fail. It just produces results—results that may be wildly misleading. A statistically significant finding built on violated assumptions is like a building with a cracked foundation: it might stand, or it might collapse spectacularly.

Understanding these hidden requirements transforms how you read scientific literature and evaluate claims. When you know what can go wrong beneath the surface, you start asking better questions—not just what did they find? but should I believe it?

The bell curve haunts statistics education. Students learn that data should be normally distributed for many common tests to work properly. This creates a kind of paranoia—researchers eyeballing histograms, hoping their data looks sufficiently bell-shaped, running normality tests that often tell them nothing useful.

Here's what's rarely explained clearly: the normality assumption usually applies to residuals or sampling distributions, not raw data. The Central Limit Theorem offers a lifeline—with large enough samples, the distribution of sample means becomes approximately normal regardless of the underlying data shape. Sample size can cure non-normality, at least for many parametric tests.

But there are limits. Heavy-tailed distributions with extreme outliers can break things even with generous sample sizes. Skewed data in small samples creates real problems. The question isn't whether your data is perfectly normal—it never is—but whether departures from normality are severe enough to distort your conclusions.

Robust alternatives exist for when normality fails. Rank-based tests like the Wilcoxon signed-rank test or Mann-Whitney U test make weaker assumptions, trading some power for protection against non-normality. Bootstrapping generates confidence intervals without distributional assumptions by resampling your actual data. Transformation—taking logarithms of skewed data, for instance—can sometimes normalize distributions enough that standard methods apply. The key is matching your analytical approach to your data's actual behavior.

Takeaway

Normality matters less for the raw data than for residuals and sampling distributions. Large samples provide protection, but robust alternatives exist when standard assumptions fail—the skill lies in recognizing when to use them.

Statistical independence seems straightforward: each observation should be unrelated to every other. In practice, this assumption gets violated constantly—and the consequences are severe. When observations cluster together, sharing hidden connections, your effective sample size shrinks. You think you have 500 independent data points, but you really have 50 clusters pretending to be 500.

Consider clinical trials where patients are nested within hospitals. Patients at the same hospital share environmental factors, treatment protocols, and unmeasured influences. Treating them as independent artificially inflates precision. Your confidence intervals become too narrow. Your p-values become too small. You find significant effects that aren't really there—a recipe for false discoveries that won't replicate.

Social media analysis faces similar problems on a massive scale. A viral tweet spawns thousands of correlated responses. Users within communities share beliefs and behaviors. Geographic clustering means opinions in Portland aren't independent of other Portland opinions. Standard tests assume each observation adds fresh information, but correlated data keeps telling you the same thing in different voices.

The solutions involve explicitly modeling the dependence structure. Mixed-effects models account for clustering by estimating random effects for groups. Generalized estimating equations handle correlated outcomes in longitudinal studies. Time-series methods address temporal autocorrelation. These approaches require knowing how your data is correlated—nested within groups, repeated over time, connected through networks—and choosing appropriate corrections.

Takeaway

Correlated observations masquerade as larger sample sizes than they really are, inflating false positive rates. The fix requires acknowledging and modeling the dependence structure rather than pretending it doesn't exist.

Assumption checking isn't a formality to skip—it's where statistical analysis actually begins. Before running any test, you need diagnostic tools that reveal whether your data can support the conclusions you're about to draw. The good news: these diagnostics are often visual and intuitive rather than mathematically complex.

For normality, Q-Q plots beat formal normality tests in most situations. Plot your data's quantiles against theoretical normal quantiles; if points fall roughly along a straight line, normality is reasonable. Formal tests like Shapiro-Wilk are overpowered for large samples, flagging trivial departures as significant, and underpowered for small samples where normality matters most. Trust your eyes over arbitrary p-value thresholds for normality assessment.

Independence is harder to visualize but not impossible. Plot residuals over time or observation order—patterns suggest temporal dependence. Check for clustering by group in scatterplots. Durbin-Watson tests detect autocorrelation in regression residuals. For spatial data, variograms reveal distance-based correlation. The key diagnostic question: does knowing one observation's value help predict nearby observations?

Homoscedasticity—equal variance across groups or prediction levels—shows up in residual plots. Fan-shaped patterns, where spread increases with fitted values, signal heteroscedasticity that can bias standard errors. Levene's test formally compares variances, but again, plots often tell you more than test statistics. When assumptions fail, you have options: transform variables, use robust standard errors, switch to non-parametric methods, or explicitly model the violation.

Takeaway

Visual diagnostics like Q-Q plots and residual scatterplots reveal assumption violations more reliably than formal tests. Build the habit of plotting before computing—your eyes catch problems that p-values miss.

Statistical tests aren't magic boxes that produce truth from data. They're tools with specific requirements, and their outputs are only meaningful when those requirements are approximately met. Every result comes with invisible conditions attached.

The practical skill isn't memorizing every assumption for every test—it's developing the habit of asking what could go wrong here? before trusting any analysis. Plot your data. Check residuals. Think about whether observations are truly independent. Know the robust alternatives for when standard methods fail.

Scientists who take assumptions seriously produce more reliable findings. Readers who understand assumptions become better evaluators of evidence. The hidden machinery behind statistical tests, once visible, transforms from a source of hidden errors into a foundation for trustworthy conclusions.