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Percent uncertainty

One method of expressing uncertainty is as a percent of the measured value. If a measurement A size 12{A} {} is expressed with uncertainty, δA size 12{δA} {} , the percent uncertainty    (%unc) is defined to be

% unc = δA A × 100% . size 12{%" unc = " { {δA} over {A} } times "100"%} {}

Calculating percent uncertainty: a bag of apples

A grocery store sells 5-lb size 12{ "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 size 12{ "4.8 lb" } {}
  • Week 2 weight: 5.3 lb size 12{ "5.3 lb" } {}
  • Week 3 weight: 4.9 lb size 12{ "4.9 lb" } {}
  • Week 4 weight: 5.4 lb size 12{ "5.4 lb" } {}

You determine that the weight of the 5-lb size 12{ "5-lb" } {} bag has an uncertainty of ± 0 . 4 lb size 12{ +- 0 "." 4`"lb"} {} . What is the percent uncertainty of the bag’s weight?


First, observe that the expected value of the bag’s weight, A size 12{A} {} , is 5 lb. The uncertainty in this value, δA size 12{δA} {} , is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight:

% unc = δA A × 100% . size 12{%" unc = " { {δA} over {A} } times "100"%} {}


Plug the known values into the equation:

% unc = 0 . 4  lb 5  lb × 100% = 8 % . size 12{%" unc = " { {0 "." 4" lb"} over {5" lb"} } times "100"%=8%} {}


We can conclude that the weight of the apple bag is 5 lb ± 8 % size 12{5`"lb" +- 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.

Uncertainties in calculations

There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the method of adding percents    can be used for multiplication or division. This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation . For example, if a floor has a length of 4 . 00 m size 12{4 "." "00"" m"} {} and a width of 3 . 00 m size 12{3 "." "00"" m"} {} , with uncertainties of 2% size 12{2%} {} and 1% size 12{1%} {} , respectively, then the area of the floor is 12 . 0 m 2 size 12{"12" "." 0" m" rSup { size 8{2} } } {} and has an uncertainty of 3% size 12{3%} {} . (Expressed as an area this is 0 . 36 m 2 size 12{0 "." "36"" m" rSup { size 8{2} } } {} , which we round to 0 . 4 m 2 size 12{0 "." 4" m" rSup { size 8{2} } } {} since the area of the floor is given to a tenth of a square meter.)

A high school track coach has just purchased a new stopwatch. The stopwatch manual states that the stopwatch has an uncertainty of ± 0 . 05 s size 12{ +- 0 "." "05"`s} {} . Runners on the track coach’s team regularly clock 100-m sprints of 11.49 s size 12{"11.49 s"} {} to 15.01 s size 12{"15.01 s"} {} . At the school’s last track meet, the first-place sprinter came in at 12 . 04 s size 12{"12" "." "04"" s"} {} and the second-place sprinter came in at 12 . 07 s size 12{"12" "." "07"" s"} {} . Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not?

No, the uncertainty in the stopwatch is too great to effectively differentiate between the sprint times.

Precision of measuring tools and significant figures

An important factor in the accuracy and precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter, while a caliper can measure length to the nearest 0.01 millimeter. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise and accurate the measurements can be.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Selected chapters of college physics for secondary 5. OpenStax CNX. Jun 19, 2013 Download for free at http://legacy.cnx.org/content/col11535/1.1
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