Bounds for the Ratio of Range to Standard Deviation

The roots of unity have many uses. Let us find the roots of Jacobi's identity (Equation [5n] of [5, p.106]Setting a j = 2j + 1, AcknowledgementsThe authors would like to thank the editor and the referee for their careful reading and for comments which improved the paper. References[1] Chen H. On the summation of subseries in closed form. Int J Math Ed Sci Tech. 2010;41:538-547. [2] Chen Z, Wei S, Xiao X. Finding sums for an infinite class of alternating series. Int J Math Ed Sci Tech. 2012;43:694-702.The present study ensembles the variation of standard deviation in terms of range for small sample. Smaller ranges having minimum upper limit of the sample standard deviation and sample mean deviation have been found. A comparative study is conducted between their dispersions and evaluated by a couple of examples.

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“…Mcleod and Henderso, [7] Thompson, [10] and Joarder and Latiff [6] derived the lower bound for standard deviation such as…”

Section: Preliminariesmentioning

confidence: 99%

Shorter variation of standard deviation for small sample

Adnan

Joarder

2013

International Journal of Mathematical Education in Science and

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“…Also, it is necessary to specify the initial value of t, say t 0 . In particular, the midpoint of the Thomson (1955) bounds is used for t 0 . 2.…”

Section: Mcmc Algorithmmentioning

confidence: 99%

“…The I_RL values indicate no convergence problems and all the Nprec values are below 5000 indicating that the results are at least so precise as the precision required Bayesian Influence for the Mean and Standard Deviation 95 (r ¼ 0.0075 and s ¼ 0.95). The 95% Bayesian confidence interval constructed from the quantiles 0.025 and 0.975 and the 90% Bayesian confidence interval constructed from the quantiles 0.05 and 0.95 both include the true values of the parameters m and s. In particular, for the standard deviation the MCMC point estimates are all inside the interval defined by the bounds determined by Thomson (1955).…”

Section: Simulated Datamentioning

confidence: 99%

“…In order to make inference about the mean of a normal population, Lord (1947) replaces the use of the sample standard deviation by a function of the sample range in the ttest. David et al (1954) showed the existence of bounds for the ratio of range to standard deviation for all populations with non-zero variance and Thomson (1955) determined upper and lower bounds for this ratio.…”

Section: Introductionmentioning

confidence: 98%

“…Let R denote the range and S the standard deviation, Thomson (1955) found the following bounds for R S for any distribution with non-zero variance: R S (2(n À 1)) From these results it is possible to determine bounds for the standard deviation in terms of the range. Bland et al (1966) present results on the distributions of the range and mean range for samples from a normal population.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bayesian inference for the mean and standard deviation of a normal population when only the sample size, mean and range are observed

Alba

Fernández-Durán

Gregorio-Domínguez

2006

Journal of Applied Statistics

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Consider a random sample X1, X2,…, Xn, from a normal population with unknown mean and standard deviation. Only the sample size, mean and range are recorded and it is necessary to estimate the unknown population mean and standard deviation. In this paper the estimation of the mean and standard deviation is made from a Bayesian perspective by using a Markov Chain Monte Carlo (MCMC) algorithm to simulate samples from the intractable joint posterior distribution of the mean and standard deviation. The proposed methodology is applied to simulated and real data. The real data refers to the sugar content (oBRIX level) of orange juice produced in different countries.Bayesian estimation, range, order statistics, MCMC,

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