# Combined and Simultaneous Measurements

Chapter

## Abstract

Combined and simultaneous measurements, as pointed out in Chap. 1General Concepts in the Theory of Measurementschapter.1.6, are measurements performed to find values of several quantities related by a known equation. In either case, a measurement experiment involves multiple measurements, with each individual measurement producing one equation instance. Typically, the number of measurements is such that there are more equations than the unknowns (the parameters and measurands). Because of measurement errors, it is impossible to find values of the unknowns such that all equations would be satisfied simultaneously. Under these conditions, the estimated values of the unknowns usually are found with the help of the method of least squares.

The method of least squares is a widely employed computational technique that makes it possible to handle the inconsistency of experimental data. This method is easily implemented with the help of computers, and good least-squares software is available.

There is extensive literature on the method of least squares, and it has been well studied. It is known that the estimates obtained with this method satisfy the requirements for estimates from Sect. 3.2Requirements for Statistical Estimatessection.3.2.85 only if all the errors in the measurements are random and normally distributed. Nevertheless, the method of least squares is widely employed, because it is simple and in general, the biasness of the estimates obtained is usually not significant even when the above condition does not hold. Moreover, in measurement practice, the least-squares method is also used to reduce the systematic errors if the measurement experiment can be organized in such a way that different measurements of the same quantities have different systematic errors.

## References

1. 37.
M.F. Malikov, Foundations of Metrology (Committee on Measures and Measuring Devices at the Council of Ministers of the USSR, Moscow, 1949).Google Scholar