Control of Standard Deviation of Unit Weight

This is one of several global tests that treat the entire system as a whole. It provides an indication of how well the adjustment has performed, as all observations along with stochastic and functional models are included in the calculation.

Warning

The weakness is that a passed or failed test does not directly indicate the quality of the data. The test also does not reveal the exact problem when it fails.

T-test for Standard Deviation

The T-test compares the assumed (a priori) standard deviation of unit weight with the calculated standard deviation (\(S_0\)).

Calculation of Standard Deviation

The standard deviation of unit weight is calculated as follows:

\(S_0 = \sqrt \frac {\Sigma_{i=1}^n vi_i^2P_i} {n-m} = \sqrt \frac {\Sigma_{i=1}^n vi_i^2P_i} {f}\)

Where: * \(n\) = number of measurements * \(m\) = number of unknowns * \(f\) = number of redundancies (degrees of freedom)

Info

The standard deviation of unit weight is an estimate of the standard deviation for a measurement with weight 1.

\(S_0\) should neither be too large (\(S_0 >> 1\)) nor too small (\(S_0 << 1\)).

Statistical Significance

Statistically significant deviations are determined by the tolerances from a chi-square test, typically with 95% confidence level and n degrees of freedom:

\(x_{max}=\frac{x_{95\%,(n-m)}^2}{n-m}\), \(x_{min}=\frac{1}{x_{max}}\)

\(x_{max} ≥ s_0 ≥ x_{min}\)

Info

Gemini Terrain calculates the values for the confidence interval (control value range) and displays these in the summary.

Interpretation of Results

Warning

Significant upward deviation typically indicates either: * Gross errors in the measurements, or * Too optimistic weighting (measurement uncertainty is larger than assumed)

In such cases, further analyses should be performed as described in the section on gross error detection.

Reporting

The test results are displayed in the summary section of the various analysis and calculation reports.