Physics 114: Lecture 12 Error Analysis, Part II

Dale E. Gary

NJIT Physics Department

Mar 8, 2010

Method of Propagation of Errors

Start with original relation, in this case: Use the chain rule:

Here, dI represents the deviations of individual measurements :

We now square both sides to give the square deviations:

Finally, average over many measurements:

12

1.

dV dV dV VdI Vd VdR dR

R R R R R

i idI I I x x

2 2 2

2 22 2 3

( ) 2 .i

dV V dV V VdI I I dR dR dVdR

R R R R R

.V

IR

2 2

2 22 3

22 2 2

2 4 3

( ) 2

12

i

I V R

dV V VdI I I dR dVdR

R R R

V VdVdR

R R R

Mar 8, 2010

A Great Simplifier—Relative Error Notice that we can take:

And divide through by I2 = V2/R2:

What about this term ? This is the product of random fluctuations in voltage and resistance. When we multiply these random fluctuations and then average them—they should average to zero!

Thus, we have the final result:

The general formula, developed in the text, is

22 2 2

2 4 3

12I V R

V VdVdR

R R R

22 2

2 2

12VI R dVdR

I V R VR

Square of relative error in I

dVdR

2 2

2 2VI R

I V R

2 22 2 2 22x u v uv

x x x x

u v u v

Mar 8, 2010

Other Examples For homework, you should have calculated some uncertainties for

quantities made up of calculations of other quantities with their own measurement uncertainties.

Let me point out a few patterns:

You see the quadratic nature of the resultant error—the errors behave like the pythagorean theorem. This is a good way to think about errors, and can be a big help in estimating errors. Note the effect of powers, which expand the influence of errors of a quantity.

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2 22

2 2 2

4

x u v

x u v

x u v

x u v

x u v

x uvx u v

ux

v x u v

x uvx u v

error in verror in uerro

r in x

All covariancesassumed zero

(i.e. u and v notcorrelated)

Mar 8, 2010

MatLAB Examples Make random arrays u = randn(1,1000); and v = randn(1,1000); then

plot v vs. u (i.e. plot(u,v,’.’)). Set ‘axis equal’ to see them on the same scale. The cloud of points form a circular pattern, concentrated in the center.

Now multiply v by 2, and plot again with axis equal. The clouds form an oval cloud of points.

Add 10 to u and 15 to v and replot with axis equal. Adding constants does not change the errors, only the mean.

Now calculate the relative errors for u, v, and x=u.*v. Do they obey the expected relationship?

Now calculate the relative errors for x=u./v. They should obey the same relationship. Do they?

Look at hist(u,20), hist(v,20), hist(u.*v,20), hist(u./v,20). Now subtract 13 from v and calculate relative errors for x=u./v. What

is wrong?

2 2 2

2 2 2x u vx uv

x u v

Mar 8, 2010

Estimating Error As noted, the way to think about errors is to consider relative

uncertainty (or percent error). If we have a 10% error in both u and v, what is the percent error in x for this expression?

What if u has a 1% relative error and v has a 10% relative error? What is percent error in x then?

So thinking in terms of the Pythagorean Theorem, errors are dominated by the biggest term, unless both have similar errors.

What if both u and v have 10% errors, and x = uv4?

2 2 2

2 2 2x u vx uv

x u v

2 22 2

2 214%, . . 0.1 0.1 0.1 2 0.14x u vi e

x u v

2 22 2

2 210%, . . 0.01 0.1 0.1 1.01 0.1x u vi e

x u v

2 2 24

2 2 2

16x u vx uvx u v

40%, in this case

Mar 8, 2010

Reducing Error Knowing how errors propagate is very important in reducing sources

of error. Let’s say you do an experiment that results in a measurement whose errors are a factor of 2 too high. You have a budget that allows you do reduce the errors in one quantity by a factor of 2. Where do you put your resources?

If the relationship is

then if the relative errors are equal you are doomed to failure, because you can only reduce the total by 1.4 (square root of 2). But if the relative error in u is much smaller than that in v, you put your resources into reducing the error in v. Likewise, if the relationship is

then you put your resources into reducing the error in v, unless the relative error in v is already 4 times smaller than in u.

2 2 2

2 2 2x u vx uv

x u v

2 2 24

2 2 2

16x u vx uvx u v