Estimating Mean and Standard Deviation from the Sample Size, Three Quartiles, Minimum, and Maximum
Why this work is in the frame
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Bibliographic record
Abstract
Background: We sometimes want to include in a meta-analysis data from studies where results are presented as medians and ranges or interquartile ranges rather than as means and standard deviations. In this paper I extend a method of Hozo et al. to estimate mean and standard deviation from median, minimum, and maximum to the case where quartiles are also available. Methods: Inequalities are developed for each observation using upper and lower limits derived from the minimum, the three quartiles, and the maximum. These are summed to give bounds for the sum and hence the mean of the observations, the average of these bounds in the estimate. A similar estimate is found for the sum of the observations squared and hence for the variance and standard deviation. Results: For data from a Normal distribution, the extended method using quartiles gives good estimates of sample means but sample standard deviations are overestimated. For data from a Lognormal distribution, both sample mean and standard deviation are overestimated. Overestimation is worse for larger samples and for highly skewed parent distributions. The extended estimates using quartiles are always superior in both bias and precision to those without. Conclusions: The estimates have the advantage of being extremely simple to carry out. I argue that as, in practice, such methods will be applied to small samples, the overestimation may not be a serious problem.
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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 | 0.007 | 0.137 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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