Estimating Mean and Standard Deviation from the Sample Size, Three Quartiles, Minimum, and Maximum
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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.101 | 0.401 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.008 |
| Bibliometrics | 0.006 | 0.007 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".