Question

When using differentials, what is meant by the terms propagated error and relative error?

Short answer

Propagated error refers to the concept of an error in an input quantity affecting the overall output, especially in mathematical calculations involving values used to calculate another value. Relative error, on the other hand, is the ratio of the absolute error to the exact or true value, allowing for a standardized comparison of errors.

Step-by-step solution

Definition of Propagated Error

In mathematical calculations, particularly those involving measurement data, propagated error refers to the concept of an error in an input quantity affecting the resulting output. Whenever values are used to calculate another value, there is a potential for error to propagate through the calculation from the original values. Suppose there's a slight mistake in one measurement; this inaccurate information then spreads and potentially affects the whole result.

Example of Propagated Error

Suppose you're trying to find the volume of a cylindrical tank with radius r and height h. If there are minor measurement inaccuracies in either r or h, the volume computed would also contain these errors, and this is the essence of error propagation. If \( r = 10 \pm 0.1 \) m and \( h = 20 \pm 0.1 \) m, the calculated volume, \( \pi r^2h \), would contain a propagated error.

Definition of Relative Error

The relative error is another way to describe errors in measurement. It is the ratio of absolute error (the magnitude of the difference between the exact value and the approximation) to the exact or true value. It provides a measure of the absolute error in relation to the size of the thing being measured, allowing for a more standardized comparison.

Example of Relative Error

If someone measures a room to be 200 square feet, but the actual measurement is 205 square feet, the absolute error is 5 square feet. The relative error is then this absolute error divided by the actual measurement, in this case 5/205, or 0.024. You could also express this as a percentage by multiplying by 100, indicating a relative error of 2.4%.