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Handling & Propagation of Errors : A simple approach

Level 1 Laboratories

1

University of Surrey, Physics Dept, Level 1 Labs, Oct 2007

Always include Units ! !

(but if the quantity is

dimensionless, say so)

Alwaysquote main value to the same number

of decimal places as the uncertainty

Never quote uncertainty to more than 1 or 2

significant figures (this would make no sense)

General Remarks

- Every physical quantity has :
- A value or size
- Uncertainty (or ‘Error’)
- Units
- Without these three things, no physical quantity is complete.
- When quoting your measured result, follow the simple rules :
- e.g. A = 1.71 0.01m

2

A reminder of terminology: ‘Uncertainty’ and ‘Error’A reminder of terminology: ‘Uncertainty’ and ‘Error’

- The terms Uncertaintyand Error are used interchangeably to describe a measured range of possible true values.
- The meaning of the termError is :
- NOT the DIFFERENCE between your experimental result & that predicted by theory, or an accepted standard result !
- NOT a MISTAKE in the experimental procedure or analysis !
- Hence, the term Uncertainty is less ambiguous.

Nevertheless, we still use terms like ‘propagation of errors’, ‘error bars’, ‘standard error’, etc.

- The term “human error” is imprecise - avoid using this as an explanation of the source of error.

3

e.g., if z = xn, then :

Hence, for this particular function,

the percent (or fractional) error in z is :

D

D

æ

z

ö

æ

x

ö

=

n

ç

÷

ç

÷

è

ø

è

ø

z

x

Error Propagation using Calculus

Functions of one variable

If uncertainty in measured x is Δx, what is uncertainty in a derived quantityz(x)?

Error propagation is just calculus – you do this formally in the “Data Handling” course

Basic principle is that, if (Δx)/xis small, then to first order:

or...... just n times the percent error in x

4

But, combining errors ALWAYS INCREASES total error - so make sure terms add with the same sign :

or ....

Later we will show that it turns out to be better to add in quadrature

i.e. “the root of the sum of the squares” :

2

2

æ

ö

¶

¶

æ

ö

z

z

(

)

(

)

2

2

ç

÷

=

D

+

D

ç

÷

x

y

ç

÷

¶

¶

x

y

è

ø

è

ø

Error Propagationusing Calculus

Functions of more than one variable

Suppose uncertainties in two measured quantities x and yare : Δx andΔy , what is the uncertainty in some derived quantity z(x,y)?

For such functions of 2 variables we use partial differentiation

We can usually always handle error propagation in this way by calculus,

(see Lab Handbook, Section 6, for details)

but next we give some short cuts for common cases...

5

Simplified Error Propagation

A short-cut avoiding calculus!

Instead of differentiating z/x, z/y etc, a simpler approach is also acceptable :

1. In the derived quantity z, replace x by x +Δx, say

2. Evaluate Δz in the approximation that Δx is small

Ex. 1 : z = x + a , where a = constant

Ex. 2 : z = bx , where b = constant

Ex. 3 : z = bx2 , where b = constant

(same as obtained earlier

for z = xn , with n=2)

6

Ex. 4 : z = x + y

Prob.

Prob.

Dy

Dy

Dz

Dx

Dx

Dy

Dx

Dz

Dz

Dy

Dx

Simplified Error Propagation

Functions of more than one variable

i.e., this would mean :

=

+

But in many cases the actual probability distribution is not rectangular but Normal ( or Gaussian)

Then, the above method would overestimateDz

What is Dz for such

prob. distributions?

=

+

Answer : the errors add

in quadrature :

Mnemonic :

7

D

D

D

z

x

y

»

+

z

x

y

Exercise : Show that for z = x/ythen(try using the “min-max” method for this example)

Simplified Error Propagation

Functions of more than one variable

Ex. 5 : z = x y

i.e. percent error in z

sum of percent errors in x & y

For a more accurate estimate, use the quadrature formula :

- Aside : For complicated expressions for z, as a last resort, you can always try the “Min-Max” method to get Dz : given any errors in x, y, ....etc
- 1. Calculate max. possible valuez can take: e.g. zmax = z(x+Dx, y-Dy,....)
- 2. Similarly, calc. min. poss. valuez can take, zmin = z(x-Dx, y+Dy,....)
- 3. Then error can be estimated as : Dz ½ (zmax – zmin)

Remember : Even a crude estimate of an error is always better than no estimate at all !

8

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