Error Analysis in Physics: Types, Statistical Analysis, and Propagation - Prof. Rebecca Fo, Study notes of Physics

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Error Analysis

PHYS 3110

Types of Error

• Instrumental • Observational • Environmental • Theoretical

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

•^

Random^ – Can be quantified by statistical

analysis

•^

Systematic^ – Try to identify and get rid of^ – Hopefully found during

analysis; may need to repeatexperiment!

True Value True Value

Statistical Analysis of Random

Error

•^

For n measurements,they should grouparound the true value.For large n, theaverage should tendto the true value

=

n i

i

x

n

x

x

x

1

•^

If the measurementsare independent, canfind the standarddeviation,

σ

.^

σ

is the

width of thedistribution.

n i

i^

x

x(

n

1

2

Reporting Error

m

x

σ ±

•^

Significant figures:^ –

σ

m

: one (sometimes two) sig. figs.

• x

ave

: same accuracy as

σ

m

e = 1.602 176 5 x 10

0.000 000 1 x 10

C

and

m

-^

represents the error in one measurement

-^

m

represents the error in the mean of n

measurements

Gaussian Distribution

•^

The probability of a measurement being within+

σ

of x

ave

Probability of being within:

1

σ

= 68.3% 2

σ

= 95.5% 3

σ

= 99.7%

σ+ ∫ σ−

x x

dx) x( P

) σ in

with( P

Poisson Distribution

! x

e

x

) x( P

x

x^

−

=

0 0.10.090.080.070.060.050.040.030.020.

0

5

10

15

20

25

30

35

x

P(x)

Poisson Gaussian 1 Gaussian 2

0 0.10.090.080.070.060.050.040.030.020.

0

5

10

15

20

25

30

35

x

P(x)

•^

Applies to processesdescribed by anexponential, such asradioactive decay

-^

σ

x

•^

For large x

ave

, i.e. for long

counting times, thePoisson distribution tendsto the Gaussiandistribution.

xave

= 20

Gaussian 1:

σ

=

√

20 = 4.

Gaussian 2:

σ

= 6

Propagation of Errors

• Determining the error in a quantity

calculated from measured data.

• Let x, y, z be measured values • Let

x,

y,

z be the corresponding

estimated errors in the measurements. x +

x, etc.

• If one measurement,

x = precision of the

instrument. If n measurements of x, thenuse x +

, wherex

x^

is the standard

deviation of the mean of x.

Propagation of Errors

•^

Let w(x,y,z) be afunction of measuredvalues. We want tofind

δ

w, the error in

w.

-^

If the errors areuncorrelated, andwith dx

δ

x,

•^

If the errors arecorrelated, there arecross terms like

dz w z

dy w y

dx w x

dw

al

differenti

∂^ ∂

∂ ∂

∂ ∂

:

(^

)^

(^

)^

(^

)

(^

)

∑

∂ ∂

∂ ∂

∂ ∂

∂ ∂

=

=

i

i

w x

w z

w y

w x

x

w

z

y

x

w

i

2

2

2

2

cov( y x^

y x

w w

∂ ∂ ∂ ∂

Density of a Cylinder

ρ

= m/V ρ

= m/(

π

(^2) r h)

(^

)^

(^

)^

(^

(^2) )

2

2

h

d

m

h

d

m

δ

δ

δ

δρ

ρ

ρ

ρ

∂ ∂

∂ ∂

∂^ ∂

=

h

d

h m d

2 π^4 = ρ

Check yourunits!

(^

)^

(^

)^

(^

(^2) )

2

2

2

h

d

m

h

d

m

δ + δ + δ =

δρ

ρ

ρ

ρ

Density of a Cylinder

h m^2 πd 4 = ρ^

(^

)^

(^

)^

(^

(^2) )

2

2

2

h

d

m

h

d

m

δ + δ + δ =

δρ

ρ

ρ

ρ

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/cm

3

(^

)^

(^

)^

(^

)

3

3 3

(^0356). 0

(^0009). 0

(^01). 0

(^001). 0

(^559). 3

(^559). 3

2

2

2

2

(^00). 11

(^01). 0

2

(^02). 0 00. 4 2

2 (^0). 492

g 5

. 0

g cm

g cm

cmcm

cmcm

g

g cm

=

=

=

×

ρ^

= 3.56 +

0.04 g/cm

3 ,

δρ

/ρ

= 0.011, 1% error