Precision Error

DET:091205

When we run an experiment, we often need to know the precision error of the calculated results for the measurand. If we have an analytical model for the measurement, we can estimate the error as explained in the following example.

Example 1: Measuring Volume Flowrate

Consider the experimental setup shown in Figure 1where we are measuring the volume flowrate using a graduated cylinder and a stopwatch.

Figure 1. Experimental apparatus for measuring volume flowrate.

The formula for calculating flowrate is Q = V/t, and there are errors of measurement associated with both the measurement of volume, e_V, and time, e_t. Note that the errors can not be calculated as a simple ration, or mathematically, e_Q ≠ e_V/e_t. Instead, we use the Root-Sum-Square (RSS) method:

Consider the following numerical data:

V = 1000ml ± 5ml

t = 50 sec ± 1 sec

Nominally, Q = V/t = 20ml/sec. But what is the error of the measurement? Using the above formula, we can compute this as

Note that the errors are very sensitive to the time measurement!

Now that we have seen a concrete example, we shall generalize the concept of the RSS method. Consider a calculated measurement R±e_R. If the measurement R is a function of measured quantities R(x_i) = f{x₁,x₂, …, x_n}, and each have uncertainties of measurement (or standard deviations from calibration) e_i= {e_1, e₂, …, e_n}, then we may calculate the standard deviation of the measurement of R as

Example 2: Measuring a Voltage

Consider the experimental setup shown in Figure 2.

Figure 2. Experimental setup.

We have a Digital Multimeter (DMM) for measuring voltage and current, and can estimate the resistance value and its accuracy based on its color code as defined in Figure 3. Resistance values are only 2-digits and tolerances are generally either gold (5%) or silver (10%).

Figure 3. Color Code for Resistors, showing a 1kΩ, 10% resistor.

The measured values are

E_in = E_in ± e_E = 5.27 ± 0.010 volts (0.010v was from ±1 digit on DMM)

i = i ± e_i = 0.516ma ± 0.001ma (again from 1 digit on DMM)

R = R ± eR = 10k ± 1k

If we compute a value for the measurement of the voltage rom the resistance and current measurements, we get

Ein = iR = 5.16volts

or Ein = 5.16volts ± 0.516 volts

Thus, the calculated precision of the voltage input is almost completely dominated by the precision of the resistor. It is also noteworthy that a direct measurement of the voltage is within the calculated error of measurement.