Estimating the mean and variance from the median, range, and the size of a sample

Abstract

Background: Usually the researchers performing meta-analysis of continuous outcomes from clinical trials need their mean value and the variance (or standard deviation) in order to pool data. However, sometimes the published reports of clinical trials only report the median, range and the size of the trial.

Methods: In this article we use simple and elementary inequalities and approximations in order to estimate the mean and the variance for such trials. Our estimation is distribution-free, i.e., it makes no assumption on the distribution of the underlying data.

Results: We found two simple formulas that estimate the mean using the values of the median (m), low and high end of the range (a and b, respectively), and n (the sample size). Using simulations, we show that median can be used to estimate mean when the sample size is larger than 25. For smaller samples our new formula, devised in this paper, should be used. We also estimated the variance of an unknown sample using the median, low and high end of the range, and the sample size. Our estimate is performing as the best estimate in our simulations for very small samples (n < or = 15). For moderately sized samples (15 < n < or = 70), our simulations show that the formula range/4 is the best estimator for the standard deviation (variance). For large samples (n > 70), the formula range/6 gives the best estimator for the standard deviation (variance). We also include an illustrative example of the potential value of our method using reports from the Cochrane review on the role of erythropoietin in anemia due to malignancy.

Conclusion: Using these formulas, we hope to help meta-analysts use clinical trials in their analysis even when not all of the information is available and/or reported.

Figure 1

Meta-Analysis of random data. After drawing fifteen samples of random sizes (between 8 and 100) from the Log-Normal [4, 0.3] distribution, we used our estimation formulas to estimate the mean and the variance from the median and the range. Then we performed meta-analysis using STATA, treating the real samples as one subgroup and their estimates as another subgroup to determine the results and heterogeneity.

Figure 2

Actual pooled mean difference and estimated pooled mean difference. The black diamonds represent the actual pooled mean difference using sample means. The red circles represent the pooled mean differences for the same samples using our estimation formulas (we connected the corresponding symbols for clarity). The horizontal axis represents the number of trials in the meta-analysis (from 8 to 100).

Figure 3

An example: Meta Analysis with all eligible trials included. Cochrane investigators [5] were only able to pool two studies to evaluate the effect of Epo on change in hemoglobin in the patients with the baseline level of hemoglobin >12 g/dl who underwent chemotherapy. Their results show that on average Epo increases hemoglobin by 2.05 g/dl. Using our estimation formulas, we were able to include two other studies eligible for this meta-analysis ([9, 10]). The pooled estimate decreased to 1.22 g/dl, i.e., a decrease of approximately 40%.