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# Spherical Probable Error

## Abstract

THE problem of determining an estimate which may reflect light on the accuracy of a weapon system directed against an attacking aircraft has long been recognized. Although efficient methods are well known to mathematicians, the personnel assigned to the execution may not know, and therefore may not utilize, the raw data to form an efficient estimate. Particularly, its importance is to be realized when the data are not complete. This leads me to suggest a new conception, namely, ‘spherical probable error’. This is the three-dimensional analogue of the probable error of a single variate. Just as the probable error measures the half-width of the mean-centred interval which includes 50 per cent of the normal probability mass, the spherical probable error measures the radius of the mean-centred sphere which includes 50 per cent of a trivariate normal probability mass. The spherical probable error is denned as : where c = 1.20645 and σ is the common unknown standard deviation of the three orthogonal rectangular normal variates with means equal to zero. First two moments of S are given for two cases, namely, when the number of the largest unmeasured observations is : (1) not known, (2) known. Since the spherical probable error is related to σ, the problem of estimating S reduces to that of estimating σ only. The unbiased estimate of σ for the above two cases is given by: where σi is the maximum likelihood estimate of the standard deviation of the i-th variate as given by Singh1. The estimate given in section (2.2) of ref. 1 corresponds to the case (1) and that in section (2.1) to case (2). The asymptotic variance of S under the assumption that σi's are uncorrelated is given by: where:  will be given by equations (15) and (9) of ref. 1 for case (1) and (2) respectively. It is important and interesting to note that when points of truncation are equidistantly distributed about the mean point of impact, the σi will simply be given by si, the sample standard deviation, and by for both the cases, n being the size of sample measured.

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## References

1. 1

Singh, Naunihal, J. Roy. Stat. Soc., 22, B, No. 2, 307 (1960).

## Author information

### Affiliations

1. #### Defence Science Laboratory, Metcalfe House, Delhi, 6

• NAUNIHAL SINGH

### DOI

https://doi.org/10.1038/193605a0 