Transcript Uncertainty

Measurement Uncertainties
Physics 161
University Physics Lab I
Fall 2010
Measurements
• What do we do in this lab?
• Perfect Measurement
• Measurement Techniques
• Measuring Devices
• Measurement Range
Uncertainties
• Types of Errors
• Random Uncertainties: result from the randomness of
measuring instruments. They can be dealt with by
making repeated measurements and averaging. One
can calculate the standard deviation of the data to
estimate the uncertainty.
• Systematic Uncertainties: result from a flaw or
limitation in the instrument or measurement technique.
Systematic uncertainties will always have the same sign.
For example, if a meter stick is too short, it will always
produce results that are too long.
Uncertainty
• Difference between uncertainty and
error.
• Blunder is not an experimental error!
• Calibration
Mean and STD
• Multiple
Measurements
• Average
• Standard Deviation
N
1
x   xi
N i 1
N
1
2
   ( xi  x )
N i 1
Reporting a Value
• To report a single value for your
measurement use Mean and Standard
Deviation
• Measured value= mean of multiple
measurements ± Standard Deviation
Expressing Results in terms of
the number of σ
•In this course we will use σ to represent the uncertainty in a
measurement no matter how that uncertainty is determined
•You are expected to express agreement or disagreement
between experiment and the accepted value in terms of a
multiple of σ.
•For example if a laboratory measurement the acceleration
due to gravity resulted in g = 9.2 ± 0.2 m / s2 you would say
that the results differed by 3σ from the accepted value and
this is a major disagreement
•To calculate Nσ
N 
accepted  exp erimental


9.8  9.2
3
0.2
Example
Trial
1
2
3
4
5
6
7
8
9
10
Period(s)
1.45
1.67
1.34
1.44
1.55
1.41
1.52
1.61
1.42
1.57
sum
average
x-xbar
-0.05
0.17
-0.16
-0.06
0.05
-0.09
0.02
0.11
-0.08
0.07
(x-xbar)^2
0.00230
0.02958
0.02496
0.00336
0.00270
0.00774
0.00048
0.01254
0.00608
0.00518
x^2
2.1025
2.7889
1.7956
2.0736
2.4025
1.9881
2.3104
2.5921
2.0164
2.4649
14.98
1.498
0.09496
0.00950
22.53500
2.25350
1.50
0.09745
0.10
std
0.097447 sqrt(x^2bar-xbar^2)
0.10
Gaussian Distribution
Standard Deviation
Accuracy vs. Precision
• Accurate: means correct. An accurate
measurement correctly reflects the size of
the thing being measured.
• Precise: repeatable, reliable, getting the
same measurement each time. A
measurement can be precise but not
accurate.
Accuracy vs. Precision
•
Precision depends on Equipment
• A more precise measurement has a smaller σ.
9.4±0.7 m/s/s
9.5±0.1 m/s/s
•
Accuracy depends on how close the
measurement is to the predicted value.
9.4±0.7 m/s/s
•
9.5±0.1 m/s/s
Which one is a better measurement?
9.4±0.7 m/s/s
9.5±0.1 m/s/s
Comparison of Measurements
Absolute and Percent
Uncertainties (Errors)
If x = 99 m ± 5 m then the 5 m is referred to as an absolute
uncertainty and the symbol σx (sigma) is used to refer to it. You
may also need to calculate a percent uncertainty ( %σx):
% x
 5m 

 99 m 
  100%  5%


Please do not write a percent uncertainty as a decimal ( 0.05)
because the reader will not be able to distinguish it from an
absolute uncertainty.
Error Propagation
Addition z  x  y
 z   x2   y2
(uncertainty or error)
Multiplyz  x * y
% z  % x2   % 2y
% z 
z
z
100%
(%uncertaintyor %error)
Propagation of Uncertainties with
addition or Subtraction
If z = x + y or z = x – y then the absolute uncertainty in z is
given by
 z   x2   y2
Example:
Propagation of Uncertainties with
Multiplication or Division
If z = x y or z = x / y then the percent uncertainty in z is given
by
% z  % x2  % y2
Propagation of Uncertainties
in mixed calculations
Error Propagation
P ower rule : z  x
% z  2% x
2
Average Rule z  (x  y  ...)/N
z 
x
N
if and only if all the measurements have the same
st andard deviation.
Error Propagation
3
2
1
2
let z  x 3 y (or z  x y ),
Given theuncertaites
i for xand y (i.e. x and  y )
Find  z
Error Propagation
We use multiplication and power rules :
let z  u , so u  x 3 y. If we also let w  x 3 , then
u  wy ,
so % u  % w2  % y2 , but since
% w  3% x , then% u  %(3 x ) 2  % y2 , on
theotherhand % z  (1 / 2)% u , therefore
% z  (1 / 2) (3% x ) 2  % y2
Example
• Let x=4±.4 and y=9±.9 then
so  x  0.4,  y  0.9, % x  10% and % y  10%
% z  (1 / 2) (3% x ) 2  % y2
% z  0.5 (3 *10%)2  (10%)2
% z  0.5 *10% * 10  15.81%
since z  ( x 3 y)0.5  (439) 0.5  24
 z  z * % z / 100  24(15.81) / 100  3.79
Special Functions
•
•
•
•
•
z=sin(x) for x=0.90±0.03
Sin(0.93)=0.80
Sin(0.90)=0.78
Sin(.87)=0.76
Z=0.78±0.02
Least Square Fitting
• In many instances we would like to fit a number
of data points in to a line.
• Ideally speaking the data points should be on a
line; however, due to random and systematic
errors they are shifted either up or down from
their ideal points.
• Least squares is a statistical model that
determines the slope and intercept for a line by
minimizing the sum of the residuals.
Least Squares
35
30
Y
25
20
15
10
5
0
0
2
4
6
X
8
10
Least Squares
y = 2.9167x + 4.6389
35
2
R = 0.9786
30
Y
25
20
15
10
5
0
0
2
4
6
X
8
10
Least Squares
Slope
m
N  ( xi yi )   y j  xk
i
j
k
N  xk2   x j  xk
k
j
k
Intercept
b
2
x
 i  y j   xi  ( x j y j )
i
j
i
j
i
j
k
N  xi2   x j  xk
Least Squares
2
i
N  ( xi yi )   y j  xk
i
j
b
k
N  x   x j  xk
2
k
k
X
1
2
3
4
5
6
7
8
9
j
Y
8
9
14
15
20
24
26
27
30
i
XY
j
8
18
42
60
100
144
182
216
270
1
4
9
16
25
36
49
64
81
j
i
j
k
45
m
m
173
1040
285
((9*1040)-(173*45))/((9*285)-(45*45))
2.916667
j
y = 2.9167x + 4.6389
R2 = 0.9786
30
25
20
15
10
5
0
0
Sum
j
i
35
X*X
i
j
N  xi2   x j  xk
k
Y
m
 x  y   x  (x y )
2
4
6
8
10
X
b
((285*173)-(45*1040))/((9*285)-(45*45))
4.638889
Percent Difference
Calculating the percent difference is a useful way to compare
experimental results with the accepted value, but it is not a
substitute for a real uncertainty estimate.
 acceptedvalue- experimental value
 100%
% diff  
accepted
value


Example: Calculate the percent difference if a measurement
of g resulted in 9.4 m / s2 .
 9.8m 2  9.4 m 2 

s
s  100%  4%
% diff  

9.8 m 2


s


Significant Figures
•
Addition Rule:
– Use the least number of sig. figs after the
decimal point.
– 12.3456 + 8.99= 21.3356  21.34
•
Multiplication Rule:
– Use the same number of sig. figs as the
number with least sig. figs.
– 12.447*2.31 =28.75257  28.8