Before 1978, there were various ways of defining the uncertainty. Uncertainty in act was some function of different sources of errors. The errors were classified into random errors and systematic errors. Those used to be defined as follows.

## Random Error

An error varies in an unpredictable manner both in magnitude and in sign, when a large number of measurements of the same quantity are made under essentially the same conditions. These errors follow the Gaussian (normal) distribution with zero mean. However, for small sample (smaller number of observations), statistical results which are based on normal distribution are corrected by means of Student’s t0 factor. These errors may be due to uncontrollable environmental conditions, personal judgement of the observer and inherent instability of the measuring instrument or any other cause of random nature.

## Systematic Error

An error is due to the system (including the standards used for the measurement) and cannot be reduced by taking larger number of observations if the equipment and conditions of measurement remain unchanged. These errors may be due to the inability in detection of the measuring system, constant bias, error in the value of the standard, a physical constant and property of the medium or conversion factor used. The value and the sign of this error do not change with the given measuring system. Systematic errors can be broadly classified into (1) constant and (2) variable. Constant systematic errors are those which do not change with respect to time but sometimes, may vary with the magnitude of the measured quantity. Zero setting error in an instrument is a constant systematic error while inaccuracy in the calibration scale may depend upon the magnitude of the quantity measured. Variable systematic errors do depend upon the time; say value of a resistor, which may vary with time because of ageing effect. These may also occur due to insufficient control of environmental conditions.

## Calculation of Random Uncertainty ur

The best estimate of the expected value of a random variable of n independent observations x1; x2; x3; : : :; xn obtained under same conditions of measurement is the arithmetic mean of n observations.

The mean is given as

The measure of dispersion is variance. The best estimate of the population variance

from the sample of size n is given

Standard deviation – the positive square root of variance is given by

Standard deviation of the mean is given by

From the standard deviation of the mean of the sample of size n, population standard deviation was calculated by multiplying it by the student t factor. The value of student t for chosen level of confidence is taken from the student t Table A.5 by taking n 1 as the degree of freedom. The random standard uncertainty ur due to single input quantity is given as

The above calculations are based upon the assumption that measured value of the input variable follows the Gaussian (Normal) distribution and f x is represented as

Source Linear calibration function : measurement uncertainty