Teaching with the labs

“Once n is above about 30, the sampling distribution is normal.”

The threshold is a rule of thumb for mildly skewed, finite-variance populations. Log-normal data at n = 30 is still visibly skewed, and Cauchy data is never normal at any n because the CLT's finite-variance hypothesis fails.

“If I impute the missing values, I have recovered the missing data.”

Imputation is a device for using the rows you kept, not for recreating information you never observed. Mean imputation fabricates a spike at the centre and shrinks the variance; under MNAR, even a perfectly specified conditional model remains biased because the deletion depends on the value that was deleted.

“Collinearity biases my regression coefficients, so I should drop one predictor.”

Least squares stays unbiased at any ρ < 1; only the variance inflates. Dropping a genuinely relevant predictor introduces real omitted-variable bias, trading a variance problem for a bias problem. If prediction is the goal, collinearity may be harmless; if a single coefficient must be interpreted, the fix is better data or a regularized/identifiable target such as the sum.

“PCA finds the intrinsic structure of my data, whatever units I measure in.”

PCA on the covariance matrix is not invariant to rescaling a single variable. Changing metres to millimetres inflates that variable's variance by a factor of a million and drags PC1 onto it. Only decomposition of the correlation matrix (standardizing first) is invariant, which is why it is the default for variables in different units.

“My classifier is 99% accurate, so it works well.”

With 1% prevalence, predicting “negative” for everyone is 99% accurate and useless. Accuracy weights both error types by their frequency, which hides failures on the rare class that usually matters most. Report sensitivity and precision at a stated threshold, plus AUC or average precision for the ranking, and always give the majority-class baseline for comparison.

“The elbow in the inertia curve tells me the right number of clusters.”

Inertia falls monotonically with k by construction, and on data with no clusters at all it still produces a smooth curve with a visible bend. An elbow is a heuristic about diminishing returns, not evidence about the number of groups. Use it alongside stability across resamples, a validation index appropriate to the cluster shape, and — decisively — subject-matter meaning.

“My model has an AUC of 0.85, so its predicted probabilities are reliable.”

AUC only measures whether positives are ranked above negatives; it is invariant to every monotone rescaling of the scores. A model can achieve any AUC while reporting probabilities that are systematically double the truth. Whenever probabilities feed a decision — treatment thresholds, expected loss, resource allocation — you must report calibration (reliability diagram, ECE, Brier, log loss) alongside discrimination, and recalibrate if needed.

“The variables selected at the cross-validated λ are the true predictors.”

Cross-validation optimizes out-of-sample prediction error, and prediction tolerates extra weakly correlated variables far more happily than it tolerates dropping a real one. So the CV-optimal model habitually includes false discoveries, and it also under-reports the size of true effects because of shrinkage. For inferential claims about which variables matter, use methods designed for that target — knockoffs, stability selection, or debiased inference — rather than reading off the LASSO's support.