Propagating Uncertainty: Addition and Subtraction - IB Physics

Summary

Highlights

• Understanding the rule that the uncertainty of a sum or difference is the sum of the absolute uncertainties! 📚
• Example rich lecture illustrating converting uncertainties from percent to absolute before operations. 🔄
• Clarifying common mistakes like subtracting uncertainties incorrectly during subtraction. 🚫

Key Takeaways

• Always convert uncertainties to absolute before adding or subtracting! 📏
• Remember: Add the uncertainties even when subtracting the numbers. 🔄
• Convert percent or fractional uncertainties before calculation for accurate results. 🎯

Overview

Chapters

• 00:00 - 00:30: Introduction to Propagating Uncertainty This chapter provides an introduction to propagating uncertainty in mathematical operations, particularly focusing on addition and subtraction. The lecturer proposes a teaching method by presenting the rule for propagating uncertainty followed by numerous examples to clarify the concept. The key takeaway is that the uncertainty of the sum or difference of numbers equals the sum of the absolute uncertainties of each individual number involved in the operation.
• 00:30 - 03:30: Addition Examples The chapter titled 'Addition Examples' discusses the concept of absolute uncertainty in addition. It emphasizes that the uncertainty of a result can be found by summing the absolute uncertainties of the involved numbers. The chapter suggests that if there's any confusion regarding 'absolute uncertainty,' viewers should refer to an earlier lecture on absolute, fractional, and percent uncertainty. The speaker plans to solve six addition and six subtraction problems. For the first problem, the uncertainties are already given as absolute, thus simplifying the process to just adding the numbers and their uncertainties.
• 03:30 - 05:30: Conversion of Uncertainties and Mixed Examples This chapter focuses on the conversion of uncertainties in measurements and provides mixed examples to explain the concepts. It illustrates how to handle absolute uncertainties in calculations, specifically in addition or subtraction tasks. For example, when adding two numbers with absolute uncertainties, the uncertainties are combined by direct addition. An example is provided where the computation of 5 ± 2 and 8 ± 4 results in 13 ± 6, keeping the unit consistent. Furthermore, the chapter stresses that when dealing with absolute uncertainties, one simply adds the uncertainties to reach the final uncertainty in combined measurements. The example of computing 9.5 with combined uncertainties is also explored.
• 05:30 - 09:30: Subtraction Examples and Common Mistakes In the chapter titled "Subtraction Examples and Common Mistakes," the primary focus is on dealing with uncertainties in mathematical problems, particularly those involving subtraction. A significant error highlighted is the confusion between absolute and percent uncertainties. The transcript emphasizes the importance of converting percent uncertainties to absolute uncertainties before performing any additions or subtractions. The conversion is explained as multiplying the percentage (expressed as a decimal) by the original number to obtain the absolute uncertainty, ensuring accurate and rule-compliant results.

Propagating Uncertainty: Addition and Subtraction - IB Physics Transcription

• 00:00 - 00:30 in this lecture i'll show you how to propagate uncertainty when you're adding and subtracting numbers with uncertainty the best way i know of teaching propagating uncertainty is to just give you the rule and then give you a ton of examples that i work through so that's what i'm going to do in this video when you're adding or subtracting numbers with uncertainty the uncertainty of their sum or difference equals the sum of the absolute uncertainties of each individual number so if the sum or difference y is equal to a plus or minus b that means that the uncertainty of y
• 00:30 - 01:00 the uncertainty of that result is equal to the sum of the absolute uncertainties of the two numbers and just as a heads up if you don't know what i mean by absolute uncertainty specifically you should go back and watch my lecture on absolute fractional and percent uncertainty that i've linked in the description i'll go through the six addition problems and then the six subtraction problems in problem number one on the left i can see that the uncertainties are already written as absolute uncertainties so all i need to do is add the two numbers together and then add the two
• 01:00 - 01:30 absolute uncertainties together and this is my result so 5 plus or minus 2 plus 8 plus or minus 4 is equal to 13 plus or minus 6. and i'm keeping that unit the same and in the correct spot so again if you're adding or subtracting any two numbers with uncertainty you just add their absolute uncertainties so i just added two plus four to get the new absolute uncertainty of six in number two we do the same thing because those are both absolute uncertainties so i add the two numbers and then add their uncertainties together and this is the result that i get 9.5
• 01:30 - 02:00 plus or minus 0.8 meters in problem number three we have an issue because those uncertainties are not absolute uncertainties and i need them to be absolute uncertainties to follow the rule so before i can add these i need to convert these back to absolute uncertainties instead of percent uncertainties so to do that i just multiply the percentage as a decimal by the original number that's how you get from percent uncertainty to absolute uncertainty and when i do that these are the absolute uncertainties that i get for
• 02:00 - 02:30 each number so those two numbers with uncertainty are the same as the original two numbers they're just expressed with absolute uncertainty instead of percent uncertainty but they're the same values and now that they're expressed in absolute uncertainty i can add them together using the rule and when i do that this is what i get in number four i can see that those uncertainties are fractional uncertainties i can tell because the unit is to the right of the number not to the right of the uncertainty and again i explained how to tell that an uncertainty is fractional in the lecture linked in the description
• 02:30 - 03:00 so i have to start by converting fractional uncertainties to absolute uncertainties so i multiply the fraction by the original number and when i do that this is what i get for the absolute uncertainties and now that this is an absolute uncertainty i can just add the uncertainties together and solve the problem in number five i have a mixed problem where one of my numbers is an absolute uncertainty and another is in percent uncertainty so i'm going to keep the absolute uncertainty the same and convert the percent uncertainty to absolute uncertainty and when i do
• 03:00 - 03:30 that this is what i get and then when i add the absolute uncertainties together this is my answer number six is similar i have a number with absolute uncertainty and a number with fractional uncertainty so i only convert the fractional uncertainty number to be absolute and when i do that this is what i get and when i add the two absolute uncertainties together this is my result so that's how you add two numbers with uncertainty and if you wanted to express any of those results in fractional or percent uncertainty you can just follow the normal rules for
• 03:30 - 04:00 converting from absolute to fractional to percent going into the subtraction examples when you're subtracting two numbers you still add their absolute uncertainties to each other a common mistake students make is to subtract them but you still add the uncertainties so in number one 10 plus or minus 2 minus 5 plus or minus 4 is going to be equal to 10 minus 5 plus or minus the sum of the absolute uncertainties so that's going to be 5 plus or minus 6. in number 2 i follow the same rule where i subtract the numbers
• 04:00 - 04:30 but add their absolute uncertainties so the result there will be 8 plus or minus 3. in number 3 i have the numbers expressed with percent uncertainty so i need to convert those to absolute uncertainties by multiplying the percent as a fraction times the original number and when i do that these are the absolute uncertainties that i get so subtracting one number from the other and adding their absolute uncertainties gets me this answer in number four i have two numbers with fractional uncertainties because of the position of the unit so to convert from fractional to
• 04:30 - 05:00 absolute i just multiply the fraction by the original number and then i subtract the original numbers and add their absolute uncertainties and this is the number that i get in number five i have one number with an absolute uncertainty and one number with a percent uncertainty so i only convert the percent uncertainty to be absolute this is what i get when i do that and then i add the absolute uncertainties together subtract the numbers and this is my answer and just notice that i rounded that uncertainty to the correct number of decimal points finally number six i have an absolute
• 05:00 - 05:30 and a fractional uncertainty so i convert the fractional to absolute and this is what i get i add the uncertainties together and this is my result so again that's how you propagate uncertainty when you're adding or subtracting numbers you add or subtract the numbers themselves but you always add the absolute uncertainties together and if your numbers aren't expressed with absolute uncertainty you need to convert to absolute uncertainty before solving the problem