The Probable Error of a Mean

1992 ◽  
pp. 33-57 ◽  
Author(s):  
Keyword(s):  
Probable Error

QMC integration errors and quasi-asymptotics

2020 ◽  
Vol 26 (3) ◽  
pp. 171-176
Author(s):  
Ilya M. Sobol ◽  
Boris V. Shukhman
Keyword(s):  
Monte Carlo ◽  
Error Estimate ◽  
Numerical Experiments ◽  
Probable Error ◽  
Quasi Monte Carlo ◽  
Integration Error ◽  
Asymptotic Error ◽  
Dimensional Cube ◽  
Integration Errors ◽  
Mc Method

AbstractA crude Monte Carlo (MC) method allows to calculate integrals over a d-dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as {O(1/\sqrt{N})}. The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d-dimensional functions contains a multiplier {1/N}. However, the multiplier {(\ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor {1/N}. However, our numerical experiments show that using quasi-random points of Sobol sequences with {N=2^{m}} with natural m makes the integration error approximately proportional to {1/N}. In our numerical experiments, {d\leq 15}, and we used {N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, {d\leq 2^{14}} and {N\leq 2^{63}}.

EXPERIMENTAL INVESTIGATION OF LANGMUIR PROBES

1967 ◽  
Vol 45 (10) ◽  
pp. 3199-3209 ◽  
Author(s):  
R. M. Clements ◽  
H. M. Skarsgard
Keyword(s):  
Magnetic Field ◽  
Microwave Radiometer ◽  
Resonant Cavity ◽  
Boltzmann Distribution ◽  
Probable Error ◽  
Langmuir Probes ◽  
Noticeable Effect ◽  
Probe Field ◽  
Probe Radius ◽  
Weakly Ionized

Electron temperatures and densities measured in a weakly ionized helium afterglow with cylindrical double probes are compared with measurements obtained using a gated microwave radiometer and a microwave resonant cavity. The pressure was varied from 0.1 to 8.5 Torr. At low pressure, magnetic fields up to 0.11 T were applied. Independent of the values of the electron Larmor radii or particle mean free paths relative to the probe radius, the probes correctly measured the electron temperatures within an estimated random probable error of ±4% and a systematic error not exceeding ±4%. This demonstrates the validity, for the range of conditions studied, of a fundamental assumption of probe theory—that electrons in a retarding probe field are in a Maxwell–Boltzmann distribution at a temperature unaffected by the presence of the probe. Towards higher pressure the measurements show an increasing depression of the plasma density near the probe, associated with the diffusion to it. The applied magnetic field had no noticeable effect on the densities measured with the probes as compared with the cavity measurements.

The composition of milk. Part II: Variation in the percentage of mineral constituents in cow’s milk, and their relationship with the solids not fat and protein content

1927 ◽  
Vol 17 (1) ◽  
pp. 72-93 ◽  
Author(s):  
H. T. Cranfield ◽  
D. G. Griffiths ◽  
E. R. Ling
Keyword(s):  
Standard Deviation ◽  
Phosphoric Acid ◽  
Protein Content ◽  
Ash Content ◽  
Probable Error ◽  
Cow’S Milk ◽  
Frequency Distributions ◽  
Cow's Milk ◽  
Mineral Constituents

1. 670 samples of the mixed milk from 15 herds were analysed, and the average percentages of total ash, soluble ash, insoluble ash, lime and phosphoric acid are given.2. Tables showing frequency distributions are also given, with the standard deviation, mean and probable error of mean for each constituent determined.3. Various correlations of these constituents with solids not fat and protein have been prepared, and these correlations are illustrated by graphs.It is observed that the total ash falls with the solids not fat until low values of solids not fat are reached, when the ash content appears to rise. This variation is confirmed by a curve illustrating the variation in ash content of samples of individual cow’s milk. Soluble ash rises as the solids not fat falls, but the insoluble ash shows a reverse variation. Lime and phosphoric acid both fall with the solids not fat.

Probable Error in Pig-Feeding Trials

1915 ◽  
Vol 7 (2) ◽  
pp. 137-141
Author(s):  
Charles Crowther
Keyword(s):  
Statistical Analysis ◽  
Live Weight ◽  
Probable Error ◽  
Weight Increase ◽  
Feeding Experiment ◽  
Feeding Experiments ◽  
Feeding Trials

In an earlier issue of this Journal, Robinson and Halnan have communicated the results of a statistical analysis of three sets of pig-feeding experiments from which they conclude that “the probable error of one animal in a pig-feeding experiment is in the region of 10 per cent, of the average live-weight increase.”

MEASUREMENT OF THE DIFFUSION LENGTH OF THERMAL NEUTRONS IN GRAPHITE

1947 ◽  
Vol 25a (1) ◽  
pp. 15-25 ◽  
Author(s):  
H. G. Hereward ◽  
H. R. Paneth ◽  
G. C. Laurence ◽  
B. W. Sargent
Keyword(s):  
Density Distribution ◽  
Diffusion Length ◽  
Probable Error ◽  
Rectangular Parallelepiped ◽  
Thermal Neutrons ◽  
Dysprosium Oxide

The theory and method of measuring the diffusion length of thermal neutrons in graphite are discussed in detail. The graphite pile was a rectangular parallelepiped, 185.8 cm. square and 153.6 cm. high. The source was 1.2 gm. of radium mixed with beryllium. Part of the pile was used to slow down the neutrons to thermal velocities, and the density distribution of thermal neutrons was explored in the remainder of the pile with detectors of dysprosium oxide. The diffusion length of thermal neutrons in this graphite of average pile density 1.55 gm. per cc. was found to be 51 cm., with a probable error of 3 cm.

Comparison of Estimates of Circular Probable Error

1959 ◽  
Vol 54 (288) ◽  
pp. 794 ◽  
Author(s):  
P. B. Moranda
Keyword(s):  
Probable Error

SPECIAL ARTICLE

PEDIATRICS ◽  
1954 ◽  
Vol 14 (5) ◽  
pp. 563-564
Author(s):  
Keyword(s):  
Standard Deviation ◽  
Visual Assessment ◽  
Probable Error ◽  
Oral Examination ◽  
Frequency Curve ◽  
Written Examination ◽  
Lower Figure ◽  
Normal Frequency ◽  
The Mean ◽  
Selection Of

THE WRITTEN examination of January 15, 1954, was taken by 515 candidates, a larger number than in any previous year except 1953 when there were 607 candidates Grades ranged from a lowest mark of 32.0 to a highest mark of 89.5 Inspection of the range resulted in the decision to place the passing mark at 51. On this basis there were 32, or 6.2%, who failed and were therefore ineligible for oral examination. The distribution of the grades earned by the 515 candidates is presented in the form of a histogram. As an aid to visual assessment of the nature of the distribution, a normal frequency curve computed from the mean and standard deviation of the data has been superimposed on the diagram. The distribution of the grades is clearly and impressively skewed to the left, that is, the scores tend to be massed at the high end of the scale and spread out at the low end. An examination of this type is relatively sensitive in the zone of poorer scholarship where the selection of failures is to be made and relatively insensitive in the range of higher scholarship. The intrinsic reliability of the examination has again been assessed by comparing the grade made by each candidate on his odd-numbered questions with that earned on his even-numbered questions. The comparison reveals a "probable error of estimate," P.E.m, of 2.39, a lower figure and therefore a higher degree of reliability than yielded by any previous examination. The Committee is pleased that the increase in reliability was accomplished in spite of the fact that the length of the examination was decreased from 250 grading points in 1953 to 200 grading points.

scholarly journals On Mathematical Statistics and its application to Political Economy and Insurance. (Continued from page 189.)

1873 ◽  
Vol 17 (5) ◽  
pp. 355-369
Author(s):  
Theodor Wittstein
Keyword(s):  
Political Economy ◽  
Mathematical Expectation ◽  
Probable Error ◽  
Mathematical Statistics ◽  
Most Probable Number ◽  
Probable Number ◽  
To Receive

Besides the probable error above considered, there is another quantity of some importance in practice, which may be called the Risk of a departure from the most probable number of survivers. It is obtained by adding together the products of all the possible positive deviations in the number of the survivers from its most probable value, each multiplied into the probability of its occurrence; or by adding together the products of all the possible negative deviations in the number of the survivers from its most probable value, each multiplied into the probability of its occurrence; in this case, however, taken without sign. For if we consider the deviations from the most probable number of survivers, taken without regard to sign, as represented by proportional sums of money, then the quantity just defined evidently expresses either the mathematical expectation of gain of a person who is to receive those sums, or the mathematical expectation of loss of a person who is to pay them. The latter of these two ideas more generally occurs in practice.

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