Confidence Intervals for the Mean of Non-Normal Distribution: Transform or Not to Transform
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Bibliographic record
Abstract
In many areas of applied statistics, confidence intervals for the mean of the population are of interest. Confidence intervals are typically constructed as-suming normality although non-normally distributed data are a common occurrence in practice. Given a large enough sample size, confidence intervals for the mean can be constructed by applying the Central Limit Theorem or by the bootstrap method. Another commonly used method in practice is the back-transformation method, which takes on the following three steps. First, apply a transformation to the data such that the transformed data are normally distributed. Second, obtain confidence intervals for the transformed mean in the usual manner, which assumes normality. Third, apply the back- transformation to obtain confidence intervals for the mean of the original, non-transformed distribution. The parametric Wald method and a small sample likelihood-based third order method, which can address non-normality, are also reviewed in this paper. Our simulation results suggest that common approaches such as back-transformation give erroneous and misleading results even when the sample size is large. However, the likelihood-based third order method gives extremely accurate results even when the sample size is small.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it