CE 428 LAB IV 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

CE 428 LAB IV Comparing a Measured Value (x) to a Theoretical 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.

CE 428 LAB IV 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 x 1 > x 2 –then if (x ) > (x 1 -  1 ) there is no significant difference between the two at the confidence level used to find the uncertainties

CE 428 LAB IV Comparing Two Measured Values A Second Approach Calculate t calc Look up t table for (N 1 + N 2 - 2) degrees of freedom at the desired confidence level There is no significant difference between the two values if

CE 428 LAB IV 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

CE 428 LAB IV 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

CE 428 LAB IV Uncertainty in Values from a Table As before, low and high are not equal

CE 428 LAB IV 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)

CE 428 LAB IV 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)

CE 428 LAB IV 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)

CE 428 LAB IV 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?

CE 428 LAB IV Solution