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

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∂

€

∂

€

∂

€

∂

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!

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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

!!!