Applying Mathcad to the Calculation of Probable Propagated Error

An important consideration in any experiment is the estimation of the uncertainty in the result due to random errors in measurement.

Consider a calculated result, u, which is a function of the experimentally measured variables x, y, and z. Then

u= f(x, y, z)

The uncertainty in each of the measured variables is taken to be the standard deviation of a series of measurements of that variable, and is symbolized by , s_i. Therefore the uncertainties in x, y, and z are s_x, s_y, and s_z, respectively. The probable propagated error in the calculated value of u due to the uncertainties in x, y, and z is also called the standard deviation of u, and is given by

(1)

The evaluation of s_u greatly facilitated by the use of Mathcad. Consider the following examples.

Example 1. The length and radius of a solid metal cylinder are 12.0 cm and 2.3 cm, respectively. The uncertainties in the length and radius are each 0.1 cm. What is the probable propagated error in the calculated volume of the cylinder?

The Mathcad page is shown below:

Example 2. A sample of oxygen gas is confined to a volume of 2.0 L at a temperature of 25^oC and pressure of 0.78 atm. The uncertainties in the temperature and volume measurements are 1^oC and 0.10 L, respectively. The uncertainty in the pressure measurement is 1%. What are the absolute and the relative uncertainties in the mass of oxygen calculated from the ideal gas law?

The Mathcad page is shown below:

Example 3. The relationship between the resistance, R, of a thermistor, and the absolute temperature , T, is . For a certain thermistor, A = 0.0183, and B = 4000 K. If the uncertainty in the temperature is ± 0.05^oC, what is the absolute uncertainty in the resistance? What is the relative uncertainty at 298 K?