PHYS101: Introduction to Mechanics

Example 1.2 Calculating Percent Uncertainty: A Bag of Apples

A grocery store sells 5-lb bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements:

Week 1 weight: 4.8 lb

Week 2 weight: 5.3 lb

Week 3 weight: 4.9 lb

Week 4 weight: 5.4 lb

You determine that the weight of the 5-lb bag has an uncertainty of $\pm$ 0.4 lb. What is the percent uncertainty of the bag's weight?

Strategy

First, observe that the expected value of the bag's weight, $A$, is 5 lb. The uncertainty in this value, $\delta A$, is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight:

$\% \mathrm{unc}=\frac{\delta A}{A} \times 100 \%.$

Solution

Plug the known values into the equation:

$\% \text { unc }=\frac{0.4 \mathrm{lb}}{5 \mathrm{lb}} \times 100 \%=8 \%.$

Discussion

We can conclude that the weight of the apple bag is 5 lb $\pm 8 \%$. Consider how this percent uncertainty would change if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100%. If you do not do this, you will have a decimal quantity, not a percent value.