DIY History | Transcribe | Scholarship at Iowa | Theory of least squares applied to the problems arising in our observatory by Arthur George Smith, 1895 | Theory of Least Squares Applied to the Problems Arising in our Observatory by Arthur George Smith, 1895, Page 12

Transcribe
Translate

Theory of least squares applied to the problems arising in our observatory by Arthur George Smith, 1895

Theory of Least Squares Applied to the Problems Arising in our Observatory by Arthur George Smith, 1895, Page 12

[[page#]]8[[/page#]]
of the arithmetic means should be affected in each case, with the weight, the product of the reciprocal of the square of its mean error, and the number of intervals entering the mean difference. 

The mean that is found by affecting a series of observations each with its respective weight and dividing the sum of the products thus obtained by the sum of the weights, is known as the [[underline]]most probable, general or weighted mean[[/underline]] of the observations. 

The formula for the mean error of this most probable or weighted mean of the observations is 
E[[subscript]]0[[/subscript]] = E[[subscript]]1[[/subscript]] / (square root of [P])) where E[[subscript]]v[[/subscript]] is the desired error.
The mean error corresponding to a unit weight, and [P] the sum of the respective weights.

Saving...

[[page#]]8[[/page#]] of the arithmetic means should be affected in each case, with the weight, the product of the reciprocal of the square of its mean error, and the number of intervals entering the mean difference. The mean that is found by affecting a series of observations each with its respective weight and dividing the sum of the products thus obtained by the sum of the weights, is known as the [[underline]]most probable, general or weighted mean[[/underline]] of the observations. The formula for the mean error of this most probable or weighted mean of the observations is E[[subscript]]0[[/subscript]] = E[[subscript]]1[[/subscript]] / (square root of [P])) where E[[subscript]]v[[/subscript]] is the desired error. The mean error corresponding to a unit weight, and [P] the sum of the respective weights.