According to the central limit theorem, the means of a random sample of size, n, from a population with mean, µ, and variance, σ², distribute normally with mean, µ, and variance, σ²/n. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding.
Mathematical Concepts ; Normal Distribution ; Statistical Distributions

OBJECTIVES: The Study of Small Area Variation(SAV) is most interesting issue in the health care researches. Most studies of SAV have been concluded the existences of variation on the basis of the magnitude of variation without statistical testing. But it is difficult to explain the existence of variation with this way because variation indicies are easily influenced by several parameters and also their distribution are skewed. So, it needs for the study to investigate the distribution of these indices and develop the statistical testing model. METHODS: This study was planned to analyze on the distribution of variation indices such as Extremal Quotient(EQ), Coefficient of Variation(CV), Systematic Component of Variation(SCV) and compare the statistical power among indicies. The simulations was performed on the basis of several assumptions and compared to the empirical data. RESULTS: Main findings can be summarized as follows. 1. If other conditions are constant, the more number of regions, the larger 95 percentile of EQ. But under same situation, 95 percentile of CV and SCV were slightly decreased. 2. If the size of regional population or utilization rate were increased, 95 percentile of all statistics were decreased. Also in the cases of small population size and low utilization rate, 95 percentiles of EQ showed various change contrast to the little change of CV. 3. If the difference at the size of regional population were increased, 95 percentiles of EQ and SCV were increased contrast to the little difference of CV 4. If the utilization rate were increased, 95 percentiles of all indicies were increased. But under the same difference of utilization rate, the power of CV and SCV were increased comparing to no change of the power of EQ. 5. Usually the power of EQ were lower than that of CV or SCV and it is similar between CV and SCV. CONCLUSIONS: Therefore, we suggest that in selecting the variation indicies at the SAV, CV or SCV are superior than EQ in terms of significance level and power.
Health Services Research ; Population Density ; Small-Area Analysis* ; Statistical Distributions*

OBJECTIVETo study the tuberculosis clustering areas and the changing trend, from 2008 to 2010, so as to provider the reference for tuberculosis control.

METHODSGlobal spatial autocorrelation and SaTScan methods were used to detect and analyse the spatial clustering of total tuberculosis notification rate and the new smear-positive pulmonary tuberculosis notification rate, at the provincial level from 2008 to 2010.

RESULTSThe spatial clustering (SC) phenomenon was significant on total notification rate and new smear-positive pulmonary tuberculosis notification rate from 2008 to 2010 (P < 0.01). The coverages of clustering areas on total notification rate showed a reduction from 19 provinces to 14 provinces, distributed in the south, west and north-east areas of China. The coverages of clustering areas on new smear-positive pulmonary tuberculosis notification rate concentrated in 14 provinces which covered the south and north-east of China.

CONCLUSIONThe disease burden and the risk of transmission in the clustering areas of tuberculosis both located in the south and the north-east of China. The disease burden of tuberculosis was high in the west of China, but not the areas with high risk of transmission.

China ; epidemiology ; Cluster Analysis ; Humans ; Statistical Distributions ; Tuberculosis ; epidemiology

OBJECTIVETo detect the spatial distribution characteristics of iodine in drinking water of residents in Shandong province with spatial autocorrelation analysis.

METHODSThe county-based study set Shandong province as a research site. A total of 108 164 water samples from 140 counties were collected. The drinking water iodine data in county-level city between 2008 to 2010 were obtained from Shandong Institute of Prevention and Control for Endemic Disease and was merged with an electronic map to build a spatial database. Global and local Moran's I index were calculated, respectively, and spatial autocorrelation and cluster range of iodine distribution in drinking water in Shandong province were studied by SaTScan software.

RESULTSAll counties were further grouped according to the "criteria of delimitation for IDD endemic areas" and "determination and classification of the areas of high water iodine and the endemic areas of iodine excess goiter", and 90 counties were iodine deficiency (< 10 µg/L), 31 were iodine suitable (10 - 150 µg/L), and 19 (> 150 µg/L) were high iodine. For the overall study area, the iodine distribution in drinking water in Shandong province existed spatial autocorrelation (Moran's I = 0.52, Z = 7.4, P < 0.01). For the local scale, the drinking water iodine in 18 counties of Dezhou, Liaocheng and Heze city in western Shandong province was clustered, the local Moran's I were between 0.22 - 1.00 (P < 0.01), which were all high-high clusters, indicating the positive spatial correlation. Spatial analysis using SaTScan software detected two cluster areas including 20 counties, which the centers located in Xiajin and Dingtao county, the cluster radiuses were 57.47 km and 65.58 km respectively. The analysis results were consistent with the results of local spatial autocorrelation.

CONCLUSIONThere are apparent spatial autocorrelation and strong spatial heterogeneity existed in the iodine distribution in drink water in Shandong province.

China ; Cluster Analysis ; Drinking Water ; analysis ; Iodine ; analysis ; Spatial Analysis ; Statistical Distributions

The configurational properties of tail-like polymer chains with one end attached to a flat surface are studied by using dynamic Monte Carlo technique. We find that the probability distribution of the free end in z direction P(R(z)) and the density profile rho(z) can be scaled approximately by a factor beta to be a length independent function for both random walking (RW) and self-avoiding walking (SAW) tail-like chains, where the factor beta is related to the mean square end-to-end distance . The scaled P(R(z)) of the SAW chain roughly overlaps that of the RW chain, but the scaled rho(z) of the SAW chain locates at smaller betaz than that of the RW chain.
Computer Simulation ; Models, Chemical ; Models, Molecular ; Models, Statistical ; Molecular Conformation ; Molecular Structure ; Monte Carlo Method ; Polymers ; chemistry ; Statistical Distributions

OBJECTIVETo establish the methods of calculating and analyzing the multi-coefficient of variation significance test for the toxicology study.

METHODSThe paper aimed to confirm the significance level with the method of Bonferroni and then compared the methods of calculating and analyzing of the experiment groups with the control group respectively.

RESULTSThe significance level of multi-coefficient of variation significance test was confirmed as alpha1=0.0167. Compared with the control groups, the activity of ALT in serum both in 30 mg/kg and 60 mg/kg groups did not change in the average significance test, which was not statistically significant (P>0.05), while it changed in the variation significance test, which was of statistical significance (P<0.0167). The activity of AST in serum in 60 mg/kg group did not change in the average significance test (P>0.05), while it changed in the variation significance test (P<0.0167).

CONCLUSIONThe complete changes of the indexes can only be shown by use of both the average significance test and the variation significance test together.

Alanine Transaminase ; blood ; Animals ; Aspartate Aminotransferases ; blood ; Disease Models, Animal ; Female ; Lead Poisoning ; enzymology ; Rats ; Rats, Wistar ; Statistical Distributions

OBJECTIVETo determine the best statistical distribution of concentration data of major air pollutants in Shanghai.

METHODSFour types of theoretic distributions (lognormal, gamma, Pearson V and extreme value) were chosen to fit daily average concentration data of PM10, SO2 and NO2 from June 1, 2000 to May 31, 2003 in Shanghai by using the maximum likelihood method. The fit results were evaluated by Chi-square test.

RESULTSThe best-fit distributions for PM10, SO2 and NO2 concentrations in Shanghai were lognormal, Pearson V, and extreme value distributions, respectively.

CONCLUSIONThe results can be further applied to local air pollution prediction and control, e.g., the probabilities exceeding the air quality standard and emission source reduction of air pollutant concentration to meet the standard.

Air Pollutants ; analysis ; Air Pollution ; statistics & numerical data ; China ; Environmental Monitoring ; Nitric Oxide ; analysis ; Particle Size ; Statistical Distributions ; Sulfur Dioxide ; analysis

OBJECTIVE: To determine whether pre-operative perfusion skewness and kurtosis derived from normalized cerebral blood volume (nCBV) histograms are associated with progression-free survival (PFS) of patients after partial resection of newly diagnosed glioblastoma. MATERIALS AND METHODS: A total of 135 glioblastoma patients who had undergone partial resection of tumor (resection of < 50% of pre-operative tumor volume or surgical biopsy) confirmed with immediate postsurgical MRI and examined with both conventional MRI and dynamic susceptibility contrast (DSC) perfusion MRI before the surgery were retrospectively reviewed in this study. They had been followed up post-surgical chemoradiotherapy for tumor progression. Using histogram analyses of nCBV derived from pre-operative DSC perfusion MRI, patients were sub-classified into the following four groups: positive skewness and leptokurtosis (group 1); positive skewness and platykurtosis (group 2); negative skewness and leptokurtosis (group 3); negative skewness and platykurtosis (group 4). Kaplan-Meier analysis and multivariable Cox proportional hazards regression analysis were performed to determine whether clinical and imaging covariates were associated with PFS or overall survival (OS) of these patients. RESULTS: According to the Kaplan-Meier method, median PFS of group 1, 2, 3, and 4 was 62, 51, 39, and 41 weeks, respectively, with median OS of 82, 77, 77, and 72 weeks, respectively. In multivariable analyses with Cox proportional hazards regression, pre-operative skewness/kurtosis pattern (hazard ratio: 2.98 to 4.64; p < 0.001), Karnofsky performance scale score (hazard ratio: 1.04; p = 0.003), and post-operative tumor volume (hazard ratio: 1.04; p = 0.02) were independently associated with PFS but not with OS. CONCLUSION: Higher skewness and kurtosis of nCBV histogram before surgery were associated with longer PFS in patients with newly diagnosed glioblastoma after partial tumor resection.
Adult ; Aged ; Aged, 80 and over ; Brain Neoplasms/*mortality/pathology/*surgery ; Chemoradiotherapy ; Disease-Free Survival ; Female ; Glioblastoma/*mortality/pathology/*surgery ; Humans ; Image Interpretation, Computer-Assisted ; Image Processing, Computer-Assisted ; Kaplan-Meier Estimate ; Magnetic Resonance Imaging/methods ; Male ; Middle Aged ; Regression Analysis ; Retrospective Studies ; Statistical Distributions ; Tumor Burden ; Young Adult