## Errors and Uncertainties

 Course Support Lab Contents

## Comparison of two values

a) Absolute Difference

Comparisons can be made by taking the absolute difference (eg. my value is $$10~joules$$ less than the accepted value). In this case either your value or the value compared needs to be stated for the difference to be put into context (An expected value of $$100,000~joules$$ would mean the my value was quite good. Similarly an expected value of $$20~joules$$ would say the opposite)

b) Percentage Difference

The percentage difference expresses the difference between two values as a percentage (eg. the difference was $$10\%)$$. For this to be an accurate value one must also state from which value the percent was taken.

For example; to compare the number of students in a morning class of $$30~students$$ with an evening class of $$25~students$$  there is two possible ways of using percent difference (both correct). Which way the calculation is made becomes evident from how a statement is written with percent difference.

 $\text{percent difference}=\left ( \frac{30 - 25}{30\text{ in the morning}}\right )\times100\%=17\%$ The evening class has $$17\%$$ less students than the morning class. $\text{percent difference}=\left ( \frac{30 - 25}{25\text{ in the evening}}\right )\times100\%=20\%$ The morning class has $$20\%$$ more students than the evening class.

A simple approach to remembering how to correctly write a statement using percent difference is to realize the value quoted at the end of the statement is the target value (the one divided by).

Often (but not always) the "accepted value" is the one targeted (divided by). However one can divide by either value. The context in which percent difference is used defines which value was targeted. Be aware of this subtlety as it is sometimes used to promote a position by hiding the context of how percent difference was calculated.
For example, in comparing 100 returning fish with those originally in a stream, a statement "there is a 30% difference" is ambiguous as it can have two meanings. This ambiguity is removed by defining the target value.
 30 fish difference 43 fish difference The original number of fish is $$30\%$$ more than the 100 returning fish. The returning 100 fish is $$30\%$$ less than the original number of fish. ($$(130-100)/100)\times100\% = 30\%$$ ($$(143-100)/143)\times100\% = 30\%$$

Note that percent difference is calculated without uncertainties. The difference, as either an absolute or a percent, is often used to compared with the uncertainties in the values in considering the agreement of the values.