Accuracy and precision in measurements - · Scientists aim towards designing experiments that can give a “true value” from their measurements, but due to the limited precision

Accuracy and precision in measurements

© www.cgrahamphysics.com 2015

�  Scientists aim towards designing experiments that can give a “true value” from their measurements, but due to the limited precision in measuring devices, they often quote their results with some form of uncertainty.


� Limit of reading is the smallest graduation of the scale of an instrument

� Uncertainty is at least 1/2  of the limit of reading

� In most cases the uncertainty is greater than the limit of reading


Uncertainties: “All scientific knowledge is uncertain. When the scientist tells you he does not know the answer, he is an ignorant man. When he tells you he has a hunch about how it is going to work, he is uncertain about it. When he is pretty sure of how it is going to work, he still is in some doubt. And it is of paramount importance, in order to make progress, that we recognize this ignorance and this doubt. Because we have the doubt, we then propose looking in new directions for new ideas.” – Feynman, Richard P. 1998. The Meaning of It All: Thoughts of a Citizen-Scientist. Reading, Massachusetts, USA. Perseus. P 13.



� Accuracy: An indication how close a measurement is to the accepted value

� Precision: An indication of the agreement among a number of measurements



Random and systematic errors The following game where a catapult launches darts with the goal of hitting the bull’s eye illustrates the difference between precision and accuracy.

Trial 1 Trial 2 Trial 3 Trial 4

Low Precision •Hits not grouped Low Accuracy •Average well below bulls eye

Low Precision •Hits not grouped High Accuracy •Average right at bulls eye

High Precision •Hits grouped Low Accuracy •Average well below bulls eye

High Precision •Hits grouped High Accuracy •Average right at bulls eye © www.cgrahamphysics.com 2015

� Random and Systematic errors

� Mistakes on the part of the individual such as - misreading scales - poor arithmetic and computational skills - wrongly transferring raw data to the final report - using wrong theory and equations are not considered experimental errors


� Causes random set of measurements to be spread out about a value rather than the accepted value

� It is a system or instrumental error � Reproducible


� Badly made instruments � Poorly calibrated instruments � An instrument having a zero error � Poorly timed actions � Instrument parallax error


Correcting systematic error improves accuracy � Most systematic errors should be eliminated

before carrying out the investigation � Repeated measurements will not eliminate

systematic error


� This is due to the variations in performance of the instrument or the operator


� Vibrations and air convection currents in mass readings

� Temperature variations � Misreadings � Variation in thickness of

surface being measured (thickness of a wire)

� Not collecting enough data � Using a less sensitive

instrument when a more sensitive is available

� Human parallax error

2 data ≠𝑒𝑛𝑜𝑢𝑔ℎ


� …create a series of measurements which are scattered around a mean value Increasing the number of observations reduces uncertainty in the mean value and improves the precision


ACCURACY PRECISION

�  Low systematic error �  Expressed as relative or %

error

�  Low random error �  Expressed as absolute

error

Repeating experiments reduces random error, but not systematic error


Random and systematic errors PRACTICE: SOLUTION: • This is like the rounded-end ruler. It will produce a systematic error. • Thus its error will be in accuracy, not precision.


� Absolute error in measurement is the size of an error and its unit In most cases this is not the same as the maximum degree of uncertainty b/c it can be larger than half the limit of reading

0.600 cm ±0.013 𝑐𝑚 0.6 cm ±0.1 𝑐𝑚


� Take experimental variations into account such as reaction time or the object traveling a different path for each measurement

� What is your reaction time?

http://www.humanbenchmark.com/tests/reactiontime

Reaction time is usually recorded as 0.2seconds


One is your first recording, your observed value absolute error The second is the processed, final value after you do error calculation


Guidance: • Analysis of uncertainties will not be expected for

trigonometric or logarithmic functions in examinations

Data booklet reference: • If y = a ± b then Δy = Δa + Δb • If y = a · b / c then Δy / y = Δa / a + Δb / b + Δc / c • If y = a n then Δy / y = | n · Δa / a |


�  Fractional or relative error = 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑒𝑟𝑟𝑜𝑟/𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡  For example a length measurement is recorded as

9.8 cm ± 0.2 cm or (9.8 ± 0.2 )cm

Then ∆𝑙= 0.2/9.8  = 0.02 cm �  Percentage error = relative error x 100%

= 0.02 x 100% = 2%

�  Percentage discrepancy = (𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 −𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙)𝑣𝑎𝑙𝑢𝑒/𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 x100%

�  This is used in conclusion to show how much a result deviates from the accepted value

Don’t mix the two up

Errors are stated to one or two significant digits only © www.cgrahamphysics.com 2015

� Limit of reading: 0.1 cm � Uncertainty of limit of reading: 0.05cm � Absolute uncertainty: 0.2cm � Relative uncertainty: 0.02cm � Percentage error: 2%

There is no hard or fast rule for estimating uncertainties as long as it is sensible


� Measuring the angle of refraction for a constant incidence angle, the following results were obtained using a protractor with precision of ± 1↑0 : 45↑0 , 47↑0 , 46↑0 , 45↑0 , 44↑0 

How should we express the angle of refraction?

Mean: 45+47+46+45+44/5  = 45.4 Range: 47−44/2 =1.5

Since precision is ±1, 𝑒𝑟𝑟𝑜𝑟 𝑛𝑒𝑒𝑑𝑠 𝑡𝑜 𝑏𝑒 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 ~ 45. Error also needs to be rounded up to whole number as not to minimize uncertainty

𝐹𝑖𝑛𝑎𝑙 𝑎𝑛𝑠𝑤𝑒𝑟:(45±2)↑0 


� …can be used instead of range:

Measurement 45 47 46 45 44

Mean 45 45 45 45 45

Deviation from mean

0 2 1 0 1

Largest deviation from mean is error


Absolute, fractional and percentage uncertainties

PRACTICE: A student measures the voltage shown. What are the absolute, fractional and percentage uncertainties of his measurement? Find the precision and the raw uncertainty. SOLUTION: • Absolute uncertainty = 0.001 V. • Fractional uncertainty = 0.001/0.385 = 0.0026. • Percentage uncertainty = 0.0026(100%) = 0.26%. • Precision is 0.001 V. • Raw uncertainty is 0.001 V.


Absolute, fractional and percentage uncertainties PRACTICE:

SOLUTION: • Find the average of the two measurements: (49.8 + 50.2) / 2 = 50.0. • Find the range / 2 of the two measurements: (50.2 – 49.8) / 2 = 0.2. • The measurement is 50.0 ± 0.2 cm. © www.cgrahamphysics.com 2015

� Calculate the absolute, fractional and percentage uncertainties for the following measurements of a force F: 2.5N, 2.8N, 2.6N

� Solution: Mean= 2.63N à result can’t be more accurate than collected data = 2.6N Range = 0.15N à rounded up to 0.2N Absolute error: 0.2N Relative error: 0.077N % uncertainty: 7.7%


• “One aim of the physical sciences has been to give an exact picture of the material world. One achievement of physics in the twentieth century has been to prove that this aim is unattainable.” – Jacob Bronowski.

• Can scientists ever be truly certain of their discoveries?

Jacob Bronowski was a mathematician, biologist, historian of science, theatre author, poet and inventor. He is probably best remembered as the writer of The Ascent of Man.


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Accuracy and precision in measurements - · Scientists aim towards designing experiments that can give a “true value” from their measurements, but due to the limited precision