x x tv P P Sx n Degrees of freedom used for the standard deviation calculation the confidence In general the degrees of freedom for an estimate is equal to the number of values minus the number of parameters estimated en route to the estimate in question values can be changed independently of the others 1 N 2 Sx x x i s N 1 i 1 1 2 Prediction Intervals concerned with values that may be sampled from a population in the future Large sample x x zS x P Small sample x x tv P S x 1 1 n P Example A sample of 10 concrete blocks manufactured by a certain process had a mean compressive strength of 1312 MPa and a standard deviation of 25 MPa Find a 95 prediction interval for the next block to be measured 95 x 1312 2 262 25 1 10 95 x 1312 59 95 x 1312 t9 95 25 1 10 Uncertainties always exist Have reliable unified approaches standards defined for evaluating and expressing uncertainty in measurement calibrating standards and instruments maintaining quality control and quality assurance in measurement complying with and enforcing laws and regulations conducting basic and applied research x x u x P Contains uncertainties associated with all known error sources x x u x P Often several measurements will be made to obtain a final result How do uncertainties propagate through to the answer For most applications and unless otherwise specified you will use 95 confidence intervals 95 confidence also used in public opinion polls ie 70 of people think that the Bills will in the Super Bowl with a 3 margin of error x 70 3 95 Pollsters are 95 confident that the value lies between 67 and 73 Ok we need to make some assumptions We assume a normal distribution of errors and reporting of uncertainty Unless otherwise stated we assume errors are independent or uncorrelated We also need to report results correctly Significant Figures in Uncertainty Analysis x x u x P If ux is a statement of uncertainty should not report to too many significant figures m g 9 82 0 02385 not correct s g 9 82 0 02 m s a 4321 353 70 m No s a 4320 70 m Yes s Review How do we combine uncertainties Suppose we make a measurement of a the room with a yardstick x1 36 1 in We take 3 measurements and get x 36 1 in 2 x3 10 1 in 95 95 95 What is the expected value for the distance across the room What is the uncertainty Case A flaw in yardstick and all measurements may be off by the same amount correlated L 82 3 95 Case B random and normally distributed errors ux u1 2 u2 2 u3 2 ui2 i L 82 3 95 Design stage uncertainty ud combines instrument uncertainty with zero order uncertainty Zero order uncertainty instrument resolution 1 u0 resolution 95 2 Be sure to temper this with some common sense it depends on the type of readout that you have Instrument uncertainty linearity hysteresis uc u1 2 u2 2 u3 2 Assumes perfect control of test conditions and measurement procedures Before we get to more complicated uncertainty analysis we have to look at how error propagates General idea I measure the length of the side of a cube 10 times and get Sx 95 L x tv P N I m actually interested in finding the volume V y u y 95 Error propagation how does the uncertainty in one variable translate into another We can say that y u y f x u x Now consider the Taylor Series Expansion 1 2 f x0 x f x0 x f x0 x f x0 2 Applying this we get 2 dy 1 2 d y y u y f x ux u x 2 dx 2 dx y f x We have dy u y ux dx Derivative evaluated at mean value of x Here we keep only the first term Say y f x1 x2 x3 Then y f x1 x2 x3 y 2 y 2 y 2 And u y x u x1 x u x 2 x u x 3 1 2 3 or n 2 u y i ui i 1 1 2 1 2 Back to the cube example suppose we get L 10 1 0 1 in 95 How do we express the volume and uncertainty What is the expected volume V 10 1 1030 3 in 3 3 3 V L What is the uncertainty dV 3L2 dL So uv 3L2 uL 3 10 1 2 0 1 30 603 We can express our answer as V 1030 30 in 3