Calculating With Measurements

Addition and Subtraction

We have discussed what measurements are, what sorts of things affect the precision of measurements, and how to estimate the precision of a measurement. However, physicists often use these measurements to calculate quantities that are not directly measured - for instance, we might measure a distance and the time that it takes a car to travel the distance, then use the distance and the time to calculate the car's average velocity. We know how to handle significant digits in the calculation, and we can make estimates of the precision for the distance and time measurements, but what is the precision of the average velocity that was calculated from them?

Adding & Subtracting Measurements

Suppose we need to add the measurements and . The measurement means that we believe that the value is close to 10 cm, and is probably in the range from 9 cm to 11 cm. Similarly, the measurement means that we believe that the "true value" of the measurement must lie between 18 cm and 22 cm.

The best estimate for the sum of these two measurements is 10 cm + 20 cm = 30 cm

The smallest sum that you could get in adding these two measurements is 9 cm + 18 cm = 27 cm.

The largest sum that you could get in adding these two measurements is 11 cm + 22 cm = 33 cm.

Therefore, we believe that the sum of the two measurements is about 30 cm, and is probably in the range from 27 cm to 33 cm. Therefore, the sum can be expressed as .

You can verify for yourself that subtracting measurements works the same way.

The shortcut is:

When adding or subtracting measurements, add their absolute uncertainties.

Symbolically, if the most probable value of the sum of 2 measurements x and y is s (s = x + y), and if and are the absolute uncertainties of x and y respectively, then the uncertainty in s, , is:

Going Deeper...

First of all, using the rule about addding & subtracting measurements given above is perfectly acceptable for your work in this course. In many cases, though, this rule tends to over-estimate the probable uncertainty in the result. Here's why:

In order for the uncertainty in the 30 cm measurement to be 3 cm, the uncertainty in BOTH the 10 cm measurement AND the 20 cm measurement must BOTH equal their maximum values at the same time, and BOTH must be positive or negative AT THE SAME TIME. If these two measurements are independent (that is, the value of one measurement does not affect the value of the other measurement) and random, then this is VERY unlikely.

A more mathematically sophisticated approach is this: If s is the most probabable value for the sum (or difference) of two measurements x and y, and if and are the probable uncertainties in x and y respectively, then the most probable error in s, , is:

This even has a fancy name - it is called "addition in quadrature". If you are interested, you can find a derivation and justification for this formula in the references.

Notice that will always be less than , since, , and make up the legs and hypotenuse of a right triangle.

In the example above, then, addition by quadrature gives:

so the sum could be expressed as .

Which Rule Do I Use?

You can use either one - just be sure that you report which method you are using. Certainly the "simple addition" rule is easier to use, and it will give perfectly acceptable results for this course.

last update January 8, 1998 by JL Stanbrough (jstanbro@venus.net)