Error Analysis Dr. Forrest
PHYS 3110 University of Houston
http://wps.prenhall.com/wps/media/objects/165/169061/SW-comart/fig1_2_5.gif http://chemwiki.ucdavis.edu/@api/deki/files/430/Measuring_pencil.jpg?size=bestfit&width=423&height=241&revision=1
Types of Error
• Instrumental
• Observational
• Environmental
• Theoretical
Types of Error
• Instrumental – Accuracy limits of instrument – Poorly calibrated instrument – Fluctuating signal – Broken instrument
• Observational • Environmental • Theoretical
http://www.designworldonline.com/wp-content/uploads/2010/02/digital-meter-reading.jpg
Types of Error
• Instrumental • Observational
– Parallax – Misused instrument
• Environmental • Theoretical
Types of Error
• Instrumental • Observational • Environmental
– Electrical power brown-out, causing low current
– Local magnetic field not accounted for – wind
• Theoretical
Types of Error
• Instrumental • Observational • Environmental • Theoretical
– Effects not accounted for or incorrectly ignored
– Friction – Error in equations
Types of Error
• Random – Can be quantified by statistical
analysis
• Systematic – Try to identify and get rid of – Hopefully found during
analysis; may need to repeat experiment!
True Value
True Value
Poor precision
Poor accuracy
Error for Different Types of Quantities
• Measured Quantities – N Independent measurements of the same
physical quantity – Measurement of a series of N quantities that are
dependent on an independent value, i.e. x(t) or I(V).
• Calculated quantities – Propogation of error
Statistical Analysis of Random Error
• For n independent measurements, they should group around the true value. For large n, the average should tend to the true value
∑=
=
→n
iixn
x
xx
1
1
• If the measurements are independent, can find the standard deviation, σ. σ proportional to width of the distribution.
∑=
−−
=σn
ii )xx(
n 1
2
11
precision
Standard deviation of the mean
• σm = σ/n1/2
• For n > 1 measured values, report : – If σm = 0 or < roughly 1/10 precision, use
accuracy of measurement device for reported error.
• If there is no systematic error, there is a ~2/3 probability that the true value is within +σm
∑=
−−
=σn
iim )xx(
)n(n 1
2
11
mxx σ±=
Reporting Error
• Significant figures: – σm: one (sometimes two) sig. figs. – xave: same accuracy as σm
mx σ±
e = 1.602 176 6208 x 10-19 + 0.000 000 0098 x 10-19 C NIST, Fundamental Constants, Physics.nist.gov/cgi-bin/cuu/Value?e, accessed 8/29/16. Length = 1.53 + 0.05 m
σ and σm
• σ represents the error in one measurement • σm represents the error in the mean of n
measurements
Gaussian Distribution • Plot of measured
value, versus number (N) of times that value was measured.
• IF error is random, for large n, this distribution tends to a Gaussian distribution.
N(x) = n2π σ
e−(x−x )2
2σ 2
1 2 3 4 5 6 7 8 9 100
2
4
6
8
Nx
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
N
x
Fwhm = 2σ
Gaussian Distribution
• Probability of one measurement being x, P(x) = N(x)/n
• The probability of a measurement being within +σ of xave:
• Probability of being within nσ:
+1σ = 68.3% +2σ = 95.5% +3σ = 99.7%
P(within σ ) = P(x)dxx−σ
x+σ
∫
P(x) = 12π σ
e−(x−x )2
2σ 2
Poisson Distribution
• Applies to processes described by an exponential, such as radioactive decay
• standard deviation σ = √x • For large xave, i.e. for long
counting times, the Poisson distribution tends to a Gaussian distribution with the same ave. and σ.
€
P(x) =x ( )x e−x
x!
00.010.020.030.040.050.060.070.080.09
0.1
0 5 10 15 20 25 30 35
x
P(x
)
Poisson
Gaussian 1
Gaussian 2
00.010.020.030.040.050.060.070.080.090.1
0 5 10 15 20 25 30 35
x
P(x)
Reporting Error from a Poisson Process
• When measuring a physical process that you expect to follow a Poisson distribution, the error in one measurement is x = x + σ = x + √x.
• Example: – Measuring the intensity of radiation emitted by
an α source and scattered by gold nuclei, at an angle θ, over 30 seconds.
Propagation of Errors • Determining the error in a quantity
calculated from measured data. • Let x, y, z be measured values • Let δx, δy, δz be the corresponding
estimated errors in the measurements. x + δx, etc.
• If one measurement, δx = precision of the instrument. If n measurements of x, then use x +σx, where σx is the standard deviation of the mean of x.
Propagation of Errors
• Let w(x,y,z) be a function of measured values. We want to find δw, the error in w.
• If the errors are uncorrelated, we assume dx ≈ δx,
• If the errors are correlated, there are cross terms like
€
differential :
dw =∂w∂x
dx +∂w∂y
dy +∂w∂z
dz
€
δw = ∂w∂x δx( )2 + ∂w
∂y δy( )2
+ ∂w∂z δz( )2
δw2 = ∂w∂xiδxi( )2
i∑
),cov(yx yxww δδ∂∂
∂∂
Propagation of Errors
• w = ax + by + cz
€
δw = aδx( )2 + bδy( )2 + cδz( )2
€
δw = awδxx( )2 + bwδy
y( )2
+ cwδzz( )2
• During lab, for a quick error estimate, just use the biggest term.
€
δw2 = ∂w∂xiδxi( )2
i∑
• w = k xa yb zc
w/ x = a k xa-1 yb zc w/ x = aw/x
€
δww
= aδxx( )2 + bδy
y( )2
+ cδzz( )2
€
∂
€
∂
€
∂
€
∂
Some Examples:
% Error
• precision = % error = (δx/x) *100% • % error compared to accepted
= |xcalculated – xaccepted|/xaccepted * 100% • % difference = |x1 – x2|/(|x1 + x2| /2) * 100%
Example: Density of a Cylinder
ρ = m/V ρ = m/(πr2h)
€
δρ = ∂ρ∂m δm( )2 + ∂ρ
∂d δd( )2 + ∂ρ∂h δh( )2
h
d hd
m2
4π
=ρ
Check your units!
€
δρ = ρm δm( )2 + 2ρ
d δd( )2 + ρh δh( )2
Density of a Cylinder
Assume, after measuring d, h, and m three times each, you get m = 492.0 + 0.5 g
h = 11.00 + 0.01 cm d = 4.00 + 0.02 cm, ρ = 3.559 g/cm3
But precession of tools are 0.5 g and 0.15 cm.
hdm2
4π
=ρ
€
δρ = ρm δm( )2 + 2ρ
d δd( )2 + ρh δh( )2
δρ = 3.559 gcm3
0.5 g492.0g( )
2+ 2×0.02 cm
4.00cm( )2+ 0.15 cm
11.00cm( )2
δρ = 3.559 gcm3 0.0012 + 0.0102 + 0.0142 = 0.0613 g
cm3
ρ = 3.56 + 0.06 g/cm3, δρ/ρ = 0.02, 2% error
Density of a Cylinder
Which term contributes most to error? How can you reduce the error?
hdm2
4π
=ρ
€
δρ = ρm δm( )2 + 2ρ
d δd( )2 + ρh δh( )2
Rule of thumb: If more than 10 measurements of a single variable are needed, use a better instrument or method to reduce σm.
δρ = 3.559 gcm3
0.5 g492.0g( )
2+ 2×0.02 cm
4.00cm( )2+ 0.15 cm
11.00cm( )2
δρ = 3.559 gcm3 0.0012 + 0.0102 + 0.0142 = 0.0613 g
cm3
Weighted Average
• n values, each with their own error,xi + σi
• The error in this weighted mean can be found from propagation of error δx =
( 1σ i)4σ i
2
i=1
n
∑
( 1σ i)2
i=1
n
∑
δx =( 1σ i)2
i=1
n
∑
( 1σ i)2
i=1
n
∑= ( 1σ i
)2i=1
n
∑"
#$
%
&'
−1/2
∑
∑
=σ
σ== n
i
n
ii
)(
)(xx
i
i
1
21
21
1
€
∂x ∂xi
=( 1σ i)2
( 1σ i)2
i=1
n
∑
Weighted Mean Examples
• You measure 5 sets of (E, T) data to try to find σ in the Stephan-Boltzmann Law, E=σT4
• You can’t average E & T and find one σ from the average.
• Find 5 σ’s, find the weighted mean using the 5 δσ’s calculated from propagation of error, then find the uncertainty of the weighted mean.
Homework
• Find the density of your Cougar One Card (UH ID card), and find the estimated error in the density. Take at least 3 measurements of all values.
Density of a cup
( )hdHD
m22
4−
π=ρ
( ))xH(dHD
m
−−π
=ρ22
4
H h
d
D
€
δρ = ∂ρ∂m δm( )2 + ∂ρ
∂D δD( )2 + ∂ρ∂H δH( )2 + ∂ρ
∂d δd( )2 + ∂ρ∂x δx( )2
What if your Goal is 0.2% error? What do you need to do?
Got δρ/ρ = 0.011. Want δρ/ρ = 0.002. Let’s consider the biggest term in δρ/ρ only, what δd would you
need? Set term equal to the desired δρ/ρ:
00202 .dd
=δ
( ) cm ./cm ../d.d 00402004002020020 ===δ
0040.n
d d =σ
=σ=δ
00400350 .n.d ==δ 77
00400350 2
=⎟⎠
⎞⎜⎝
⎛=..n
Rule of thumb: If more than 10 measurements of a single variable are needed, use a better instrument or method to reduce σm.
03500203
. ,.before, =σ∴=σ
!!!