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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.

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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.

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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

How do you convert 0.0045kgcm³ to the si unit?
how many state of matter do we really have like I mean... is there any newly discovered state of matter?
Falana Reply
I only know 5: •Solids •Liquids •Gases •Plasma •Bose-Einstein condensate
Alright Thank you
Which one is the Bose-Einstein
can you explain what plasma and the I her one you mentioned
u can say sun or stars are just the state of plasma
but the are more than seven
what the meaning of continuum
Akhigbe Reply
What state of matter is fire
Thapelo Reply
fire is not in any state of matter...fire is rather a form of energy produced from an oxidising reaction.
Isn`t fire the plasma state of matter?
all this while I taught it was plasma
How can you define time?
Thapelo Reply
Time can be defined as a continuous , dynamic , irreversible , unpredictable quantity .
unpredictable? but I can say after one o'clock its going to be two o'clock predictably!
what is the relativity of physics
Paul Reply
How do you convert 0.0045kgcm³ to the si unit?
What is the formula for motion
Anthony Reply
V=u+at V²=u²-2as
they are eqns of linear motion
v=u+at s=ut+at^\2 v^=u^+2as where ^=2
Explain dopplers effect
Jennifer Reply
Not yet learnt
Explain motion with types
Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
Alabi Reply
Scalar quantity Because acceleration has only magnitude
acleration is vectr quatity it is found in a spefied direction and it is product of displcemnt
its a scalar quantity
velocity is speed and direction. since velocity is a part of acceleration that makes acceleration a vector quantity. an example of this is centripetal acceleration. when you're moving in a circular patter at a constant speed, you are still accelerating because your direction is constantly changing.
acceleration is a vector quantity. As explained by Josh Thompson, even in circular motion, bodies undergoing circular motion only accelerate because on the constantly changing direction of their constant speed. also retardation and acceleration are differentiated by virtue of their direction in
respect to prevailing force
What is the difference between impulse and momentum?
Momentum is the product of the mass of a body and the change in velocity of its motion. ie P=m(v-u)/t (SI unit is kgm/s). it is literally the impact of collision from a moving body. While Impulse is the product of momentum and time. I = Pt (SI unit is kgm) or it is literally the change in momentum
Or I = m(v-u)
Calculation of kinetic and potential energy
dion Reply
K.e=mv² P.e=mgh
K is actually 1/2 mv^2
what impulse is given to an a-particle of mass 6.7*10^-27 kg if it is ejected from a stationary nucleus at a speed of 3.2*10^-6ms²? what average force is needed if it is ejected in approximately 10^-8 s?
speed=velocity÷time velocity=speed×time=3.2×10^-6×10^-8=32×10^-14m/s impulse [I]=∆momentum[P]=mass×velocity=6.7×10^-27×32×10^-14=214.4×10^-41kg/ms force=impulse÷time=214.4×10^-41÷10^-8=214.4×10^-33N. dats how I solved it.if wrong pls correct me.
what is sound wave
Nworu Reply
sound wave is a mechanical longitudinal wave that transfers energy from one point to another
its a longitudnal wave which is associted wth compresion nad rearfractions
what is power
it's also a capability to do something or act in a particular way.
Newton laws of motion
power also known as the rate of ability to do work
power means capabilty to do work p=w/t its unit is watt or j/s it also represents how much work is done fr evry second
what does fluorine do?
Cheyanne Reply
strengthen and whiten teeth.
a simple pendulum make 50 oscillation in 1minute, what is it period of oscillation?
Nansing Reply
length of pendulm?
what is the difference between temperature and heat transfer?
Bonga Reply
temperature is the measurement of hotness or coldness of a body... heat transfer is the movement of heat from one body to another
U get it right
heat is an energy possesed by any substance due to random kinetic energy possesed by molecules while temperature is driving force which gives direction of flow heat
Practice Key Terms 6

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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