Uncertainty in Measured Quantities

1. Uncertainty in Measured Quantities • Measured values are not exact • Uncertainty must be estimated • simple method is based upon the size of the gauge’s gradations and your estimate of how much more you can reliably interpolate • statistical method uses several repeated measurements • calculate the average and the variance • choose a confidence level (95% recommended) • use t-table to find uncertainty limits • Propagation of uncertainty • Uncertainty in calculated values • when you use a measured value in a calculation, how does the uncertainty propagate through the calculation • Uncertainty in values from graphs and tables

2. Comparing a Measured Value (x) to aTheoretical or “Known” Value (Y) • Compute , the uncertainty in x, as already described • If (x - ) < Y < (x + ) • there is no significant difference between x and Y at the confidence level used to find . • Otherwise • x and Y are not equal at the confidence level used.

3. Comparing Two Measured Values • Suppose x was measured using two different instruments as an example • Approach #1 • find 1 and 2 as previously described • suppose x1 > x2 • then if (x2 + 2) > (x1 -1) there is no significant difference between the two at the confidence level used to find the uncertainties

4. Comparing Two Measured ValuesA Second Approach • Calculate tcalc • Look up ttable for (N1 + N2 - 2) degrees of freedom at the desired confidence level • There is no significant difference between the two values if

5. Uncertainty in Values Read from Graphs • Suppose x was measured or calculated and now y is being determined • Note that low and high are not equal • My personal preference is to take whichever is larger and use it as both the low and high uncertainty

6. Uncertainty from Charts with Parameters • Suppose x and p were measured or calculated and y is now desired • Method shown is a worst case uncertainty • assumes maximum of both errors • errors often offset each other

7. Uncertainty in Values from a Table • As before,low and high are not equal

8. Error Propagation in Calculations • Suppose x, y, and z were measured (or calculated from other measured values) • this means the uncertainty for each is known • Now want to calculate A which is a function of these measured values • also want to know the uncertainty in the calculated value of A • A = f(x,y,z)

9. Error Propagation • For infinitesimal errors (dx, dy, and dz) • Assuming the errors are small enough that the partials of f are not affected, the actual errors (A, x, y and z) can be substituted for the infinitesimal ones (dA, dx, dy, and dz)

10. Errors aren’t known; Uncertainties are known • The uncertainties are the maximum values of the errors, not the actual errors • It is likely that some errors will cancel each other out • and that most errors will be smaller than the maximum • Square both sides of the previous equation, then average over all possible errors (assuming a normal distribution)

11. Formula for Uncertainty in a Calculated Value • Resulting equation for Uncertainty: • Example: Suppose a cylinder has a radius of 3.3 ± 0.1 cm and a length of 10.8 ± 0.2 cm. What is its volume and what is the uncertainty in that volume?

12. Solution