Measurement and Propagation of Error (Part 2)

Wed, 14 Nov 2018 03:08:14 +0100

#### Linear Combinations of Dependent Measurements :

In the case of dependent measurement, to quantify the uncertainty, we need to know the value of covariance for all the possible pairs of measurements. This is practically not feasible. In this case, an upper bound can be placed on the uncertainty. If \(X_1, X_2, ..., X_n\) are \(n\) dependent measurements and \(c_1, c_2, ..., c_n\) are constants, then the uncertainty of \(c_1X_1 + c_2X_2 + ... + c_nX_n\) can be bounded as:

Measurement and Propagation of Error (Part 1)

Wed, 14 Nov 2018 02:18:29 +0100

Any measurement in a scientific or an engineering process, consists of two error parts: systematic error or bias and random error. The bias is the part of the error that is same for every measurement. Random error varies from measurement to measurement and averages out to 0 in the long run. Hence, the measured value can be written as: \[Measured \ Value = True \ Value + Bias + Random \ Error\]

The mean \(\mu\) of the population represents the part of the measurement that is the same for every measurement. Hence, \(\mu\) is the sum of the true value and the bias. The smaller the bias, the more accurate the measuring process is. If the mean is equal to the true value, the measuring process is said to be unbiased. The standard deviation \(\sigma\) of the population is the the standard deviation of the random error. The precision of the measurement is determined by the standard deviation of the measurement process. The smaller the value of \(\sigma\), the more precise the process. \(\sigma\) is often referred to as the uncertainty in the measuring process.

Propagation of Error on Amit Rajan

Measurement and Propagation of Error (Part 2)

Measurement and Propagation of Error (Part 1)