Calculating with Powers

Suppose you make a measurement, `x`, whose absolute
uncertainty is .
What will be the uncertainty in `p`, if `p =
x`^{2}? Since p = (x)(x), the product
rule applies, and the relative
uncertainty in `p` will be twice the relative
uncertainty in `x`.

This idea can be easily extended for any power, n:

The relative uncertainty in x^{n}is n times the relative uncertainty in x.

For example, suppose that the side, s, of a cube
measures.
(This is a 0.5 % measurement.) What is the volume of the cube?
Certainly the volume V = s^{3}, so the best estimate for V =
(20.0 cm)^{3} = 8.00 x 10^{3} cm^{3}. The
relative uncertainty in V is 3 times the relative uncertainty in s =
3(0.5%) =1.5%. The volume of the cube is, therefore:

Calculating with Roots:

Suppose that , and that x has a value of 100 with a relative uncertainty of 2% (and some units). What will be the uncertainty in p? Well, the power rule applies here - the square root is equivalent to the one-half power. The value of p will be

The relative error in is (relative error in x) divided by n

Calculating with Exact Numbers (Counts):

I hope that you noticed that whenever you do arithmetic with measurements, the precision of the result always seems to be less - often alarmingly less - than the precision of the original measurements. Is there ever a situation in which the precision increases? Well, yes and no.

Remember that counts and pure numbers are assumed to be absolutely
precise - that is, if you count 10 periods of a pendulum, 10 is
considered to be **exact**, not 10 plus or minus
anything. Suppose that you measure the time for 10 periods of a
pendulum as ,
what is the time for 1 period? It is!
Why is the absolute uncertainty also divided by 10? Since 10 is an
exact value, both the original time measurement and the resulting
period measurement will have the **same relative
uncertainty** (1.6%).

If k is an exact value, and x is a measurement, the relative error of kx or x/k is the same as the relative error of x.

A Practical Note on Experimental Design

When you calculate with measurements, you either add absolutte or relative uncertainties to obtain the precision of the result. What this means for the planning of experiments is that:

- One low precision measurement in an experiment (calculation) will often negate all of the hard work and time it took to get all of the other high precision measurements. For example, suppose you have three 1% measurements and one 20% measurement in a calculation. The result (assuming multiplication and/or division) will be a 23% value.
- In the same vein, it doesn't pay to go to a lot of trouble to get one really precise measurement if the other measurements are low precision. In other words, suppose you have two 10% measurements in an experiment and one 0.2% measurement. A result calculated with these values would be a 20% measurement.

copyright © 1997, 1998 by Jerry L. Stanbrough. All rights reserved. last update January 7, 1998 by JL Stanbrough (jstanbro@venus.net)