Simulation with a skewed distribution (Log-Normal, Beta, Exponential and Weibull) To compare the precision of these estimates on average, we collected the results of our simulation in the Additional file 1. However, for larger sample sizes, the formula Range/6 performs the best for this distribution. When the sample size is between 16 and 70, the formula Range/4 is the best estimator of the sample standard deviation, with a relative error between 10–15%. For a very small sample size (up to 15) the formula (16) is performing the best (within 10% of the real sample standard deviation). The variance estimators however show greater distinction. For larger sample sizes, however, the median is more consistent estimator for a normally distributed sample. For sample sizes smaller than 29, formula (5) is actually outperforming the median as a mean estimator. Both estimators for the mean, formulas (4) and (5), are very close to the sample mean (within 4%). Then we graphed the average relative error vs. We drew 200 random samples of sizes ranging from 8 to 100 from a Normal Distribution with a population mean 50 and standard deviation 17. In each case we compared the relative error made by estimating the sample mean with the approximation given by formulas (4) and (5), as well as by the median, and the relative error made by estimating the sample variance by the formulas (12) and (16), as well as the well-known standard deviation estimators Range/4 and Range/6. We also show the results of simulations where the data were selected from a skewed distributions. In the first subsection we present the results of our estimation for a normal distribution, which is what meta-analysts would commonly assume. The size of the sample ranged from 8 to about 100. We drew samples from five different distributions, Normal, Log-normal, Beta, Exponential and Weibull. In order to verify the accuracy of these estimates, we ran several simulations using the computer package Maple where the data were variously distributed, and obtained the tables below. 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. 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. For moderately sized samples (15 70), the formula range/6 gives the best estimator for the standard deviation (variance). Our estimate is performing as the best estimate in our simulations for very small samples ( n ≤ 15). We also estimated the variance of an unknown sample using the median, low and high end of the range, and the sample size. For smaller samples our new formula, devised in this paper, should be used. Using simulations, we show that median can be used to estimate mean when the sample size is larger than 25. 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). Our estimation is distribution-free, i.e., it makes no assumption on the distribution of the underlying data. In this article we use simple and elementary inequalities and approximations in order to estimate the mean and the variance for such trials. However, sometimes the published reports of clinical trials only report the median, range and the size of the trial. 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.
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