

Almost no scientific quantities are known exactly
– there is almost always some degree of uncertainty in the value
Value ± Uncertainty
– Values that are measured experimentally
– Values that are calculated
» from an equation
» using other values which have their own uncertainty
– A value might be determined both ways: calculate it and measure it

 Two components of Uncertainty
– Measured value ± systematic errors ± random errors
 Precision of a measurement
– variations due to random fluctuations
– power supply, angle of view of a meter, etc.
 Accuracy of a measurement
– includes uncertainty in precision
– also includes systematic errors
» incorrect experimental procedure, uncalibrated instrument, use a ruler with only 9 mm per cm
Add four bullseyes here next time

 Assume that systematic errors have been eliminated
 Simple Estimate
– Analog Gauge or Scale
» How finely divided is the readout, and how much more finely do you estimate that you can interpolate between those divisions?
– Digital Readout
» What is the smallest stable digit?

 Suppose you repeated the exact same measurement at the exact same conditions an infinite number of times
– Not every measurement will be the same due to random errors
– Instead there will be a distribution of measured values
 Could use the results to construct a frequency distribution or probability function

60
50
40
30
20
10
0
23
.5
25
.5
27
.5
29
.5
31
.5
33
.5
35
.5
37
Measured Value (+/- 0.5)
.5
39
.5
0.12
0.1
0.08
0.06
0.04
0.02
0
23
.5
25
.5
27
.5
29
.5
31
.5
33
.5
35
.5
Measured Value
37
.5
39
.5
41
.5


With a finite number of measurements you get a frequency distribution
– Probability of a measurement falling within a given box is number in that box divided by total number
With an infinite number you get a probability function
– Plot of P(x) versus x
– P(x) is the probability of a measurement being between x and x + dx



Certain kinds of experiments may naturally lead to a certain kind of probability function
– For example, counting radioactive decay processes leads to a
Poisson Distribution
Often, however, it is assumed that the errors are by a
Normal Distribution Function
1 M x


2
 exp
2

2 N
–  is the mean (average) of the infinite number of measurements
–  is the standard deviation of the infinite number of measurements

 z

F
H
  I
K

1 

P(xµ) is normalized:
– That is, the total area under the P curve equals 1.0
If you knew
 and

(and so you knew P) you could find the limits between which 95% of all measurements lie.
Insert plot with shaded area at left
– Noting that P is symmetric about µ you could say with 95% confidence that the measured value lies between

-
 and

+

– That is, the value is  ±  at the
95% confidence level

 You can only make a finite number of measurements
– Therefore you do not know  or

 You can calculate the average and variance for your set of measurements x

1
N i
N 

1 x i
S
2 
N
1

1 i
N 

1





Use Student’s t-Table to relate the two:
– Pick a confidence level, 95%
– Define degrees of freedom as N-1
– Read value of t
» Be careful, t-Tables can be presented in two ways
» One is such that 95% will be less than t
 In this case if you want 95% between -t and t you need 97.5% less that t
(the curves are symmetric)
» Another is such that 95% will be between -t and t
Uncertainty limits are then found from the variance
– value = average of the data set ± 
   tS
N

Add shaded bell curve here


The value of t from this form of the table corresponds to 95% of all measurements being less than

+
 and therefore 5% being greater than

+

Note that if you want 95% of all measured values to fall between

-
 and

+

– then 97.5% of all measured values must be less than

+

(or 2.5% will be greater than

+

)
– and then due to the symmetry of P
97.5% will also be greater than

-

(or another 2.5% will be less than

-

)
– so 95% will be between 
-
 and

+

Add abbreviated t-table here

Add shaded bell curve here
 The value of t from this form of the table corresponds to 95% of all measurements being between

-
 and

+

 Therefore 5% are either
– greater than 
+

– or less than 
-

Add abbreviated t-table here

 Add Problem Statement here
– preferably use data from one of the experiments they are doing

 Add solution here




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
Next lecture
– 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