ERROR ANALYSIS:

1) How errors add:

Independent and correlated errors affect the resultant error in a calculation differently.  For example, you made one measurement of one side of a square metal and found it to be 1.001 in.  Furthermore, you find that the error in this measurement is 0.001 in.  To find the area we multiply the width (W) and length (L).  The area then is

L x W = (1.001 in) x (1.001 in) = 1.002001 in2 which rounds to 1.002 in2.  This gives an error of 0.002 if we were given that the square was exactly super-accurate 1 inch a side.

This is an example of correlated error (or non-independent error) since the error in L and W are the same.  The error in L is correlated with that of in W.  Now, suppose that we made independent determination of the width and length separately with an error of 0.001 in each.  In this case where two independent measurements are performed the errors are independent or uncorrelated. Therefore the error in the result (area) is calculated differently as follows (rule 1 below).  First, find the relative error (error/quantity) in each of the quantities that enter to the calculation, relative error in width is 0.001/1.001 = 0.00099900.  The resultant relative error is

Relative Error in area =

Therefore the absolute error is (relative error) x (quantity) = 0.0014128 x 1.002001=0.001415627. which rounds to 0.001.

Therefore the area is 1.002 in2± 0.001in.2.

This shows that random relative errors do not simply add arithmetically, rather, they combine by root-mean-square sum rule (Pythagorean theorem).  Let’s summarize some of the rules that applies to combining error when adding (or subtracting), multiplying (or dividing) various quantities.  This topic is also known as error propagation.

2. Error propagation for special cases:

Let σx denote error in a quantity x.  Further assume that two quantities x and y and their errors σx and σy are measured independently.  In this case relative and percent errors are defined as

Relative error = σx / x, Percent error = 100 (σx / x)

1. Multiplying or dividing with a constant.

The resultant absolute error also is multiplied or divided.

1. Multiplication or division, relative error.

1. Addition or subtraction: In this case, the absolute errors obey Pythagorean theorem.  If a and b are constants,

If there are more than two measured quantities, you can extend expressions provided above by adding more terms under the square root sign.

1. Square or cube of a measurement :

The relative error can be calculated from

where a is a constant.

Example 1:

Determine the error in area of a rectangle if the length l=1.5 ±0.1 cm and the width is 0.42±0.03 cm.  Using the rule for multiplication,

Example 2:

The area of a circle is proportional to the square of the radius.  If the radius is determined as r = 10.0 ±0.3 cm, what is the uncertainty in the area?

CORRECTION NEEDED HERE(see lect. notes)!!

Therefore,