The Hidden Danger of Misunderstanding Moivre’s Formula in Educational Data

In a 2007 article Howard Wainer warned that some formulas are dangerous when unknown, citing Einstein’s equation as an example, and highlighted the lesser‑known risk of the Moivre formula, which relates the standard error of the mean to the standard deviation and sample size.

The author collected three years of Brazilian ENEM scores, cleaned the data, and observed that top‑performing schools often have very few students, suggesting that small class sizes appear to yield higher average scores.

Using Moivre’s equation, the article shows that with small samples the average score estimate has high variance, leading to extreme high or low scores, whereas larger schools produce more stable averages.

Statistical concepts such as variance, standard error, confidence intervals, and the central limit theorem are explained, illustrating how a 95% confidence interval is constructed by multiplying the standard error by roughly two and how wider intervals result from smaller samples.

Hypothesis testing is introduced, describing how to compare the means of online versus face‑to‑face courses, compute the difference distribution, and use a z‑statistic to assess significance, noting that non‑overlapping confidence intervals imply a statistically significant difference.

The article clarifies the proper interpretation of p‑values, providing an example where a p‑value of 0.0027 indicates a 0.2% chance of observing such an extreme result if the true difference were zero, and warns against common misstatements about confidence intervals containing the true mean with a certain probability.

Overall, the piece emphasizes the importance of accounting for uncertainty in statistical estimates, especially when sample sizes are small, and demonstrates these ideas with real educational data.

Probability is not just about dice; it is about acknowledging the limits of our knowledge and developing methods to handle our ignorance.

P-value: 0.0027239680835564706