Uncertainty & ErrorUncertainty & Error“Science is what we have

learned about how to keep from fooling ourselves.”

― Richard P. Feynman

Types of Errors Types of Errors http://ibguides.com/physics/notes/measurement-and-uncertainties

ErrorUncertainty is NOT error! Difference between measured result and true value.Illegitimate errors

Blunders resulting from mistakes in procedure. You must be careful. Computational or calculational errors after the experiment. Not paying attention!

Bias or Systematic errors An offset error; one that remains with repeated measurements (i.e. a

change of indicated pressure with the difference in temperature from calibration to use).

Systematic errors can be reduced through calibration Faulty equipment--such as an instrument which always reads 3% high Consistent or recurring - observer bias

This type of error cannot be evaluated directly from the data but can be determined by comparison to theory or other experiments.

““Human Error”Human Error” Giraffes don’t do science!! Of course you’re a human. You probably mean “systematic error” OR . . . You mean uncertainty, which

isn’t an error at all. It is you being honest.

Explain what you mean! (Example: Parallax on Meter Stick)

Letting it fall isn’t human error, it is not following the procedure.

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Types of UncertaintiesTypes of Uncertainties

AKA- “Plus/Minuses” , +/- , Tolerance , Standard Deviations

Random Uncertainties: result from the randomness of measuring instruments. They can be dealt with by making repeated measurements and averaging. One can calculate the standard deviation of the data to estimate the uncertainty.

Systematic Uncertainties: result from a flaw or limitation in the instrument or measurement technique. Systematic uncertainties will always have the same sign. For example, if a meter stick is too short, it will always produce results that are too long.

6

Accuracy vs. PrecisionAccuracy vs. PrecisionAccurate:

•How close a measurement is to an accepted / “true” value.

•An accurate measurement correctly reflects the size of the thing being measured.

•Must know the correct answer beforehand!

Precise:

•How close a measurement is to another.

•repeatable, reliable, getting the same measurement each time. A measurement can be precise but not accurate.

Bias, Precision, and Total Bias, Precision, and Total ErrorError

Bias Error

Total Error

Precision Error

X True X measured

Uncertainty 9

Percent Difference: It’s Percent Difference: It’s Accuracy!Accuracy!

%100 valueaccepted

valuealexperiment- valueaccepteddiff%

Calculating the percent difference is a useful way to compare experimental results with the accepted value, but it is not a substitute for a real uncertainty estimate.

%4%100

sm8.9

sm4.9

sm9.8

diff%2

22

Example: Calculate the percent difference if a measurement of g resulted in 9.4 m / s2 .

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Absolute and Percent Absolute and Percent UncertaintiesUncertainties

If x = 99 m ± 5 m then the 5 m is referred to as an absolute uncertainty and the symbol σx (sigma) is used to refer to it. You may also need to calculate a percent uncertainty/fractional uncertainty ( %σx):

%5%100m99

m5%

x

NO UNITS!

Uncertainty AnalysisUncertainty Analysis The estimate of the error is called the uncertainty.

It includes both bias and precision errors. We need to identify all the potential significant errors for the

instrument(s). All measurements should be given in three parts

Mean value Uncertainty Confidence interval on which that uncertainty is based

(typically 95% C.I.) Uncertainty can be expressed in either

absolute terms (i.e., 5 Volts ±0.5 Volts) or in percentage terms (i.e., 5 Volts ±10%) (relative uncertainty = V / V x 100) We will use a 95 % confidence interval throughout this course

(20:1 odds).

Use Statistics to Estimate Random Use Statistics to Estimate Random Uncert.Uncert.

Mean: the sum of measurement values divided by the number of measurements.

Deviation: the difference between a single result and the mean of many results.

Standard Deviation: the smaller the standard deviation the more precise the data Large sample size

Small sample size (n<30) Slightly larger value

x 1

Nxi

i 1

N

xxd ii

1

nxi x 2

1

2

s 1

n 1x i x 2

1

2

The PopulationThe Population Population: The collection of all items

(measurements) of the group. Represented by a large number of measurements. Gaussian distribution*

Sample: A portion of (or limited number of items in) a population.

*Data do not always abide by the Gaussian distribution. If not, you must use another method!!

x i x 1 n 68.3% of the time

x i x 2 n 95.4% of the time

x i x 3 n 99.7% of the time

3 - 2 - x 2 3

Uncertainty 14

Standard DeviationStandard Deviation

Uncertainty 15

Standard DeviationStandard Deviation

N

iix

Nx

1

1

N

ii xx

N 1

2)(1

1

The average or mean of a set of data is

The formula for the standard deviation given below is the one used by Microsoft Excel. It is best when there is a small set of measurements. The version in the book divides by N instead of

N-1.

Unless you are told to use the above function, you may use the Excel function ‘=stdev(B2:B10)’

Uncertainty 16

Expressing Results in Expressing Results in terms of the number of terms of the number of σσ

•In this course we will use σ to represent the uncertainty in a measurement no matter how that uncertainty is determined

•You are expected to express agreement or disagreement between experiment and the accepted value in terms of a multiple of σ.

•For example if a laboratory measurement the acceleration due to gravity resulted in g = 9.2 ± 0.2 m / s2 you would say that the results differed by 3σ from the accepted value and this is a major disagreement

•To calculate Nσ 3

2.0

2.98.9exp

erimentalacceptedN

Uncertainty 17

Uncertainty resulting Uncertainty resulting from averaging N from averaging N measurementsmeasurements

Nx

avg

If the uncertainty in a single measurement of x is statistical, then you can reduce this uncertainty by making N measurements and averaging.

Example: A single measurement of x yields

x = 12.0 ± 1.0, so you decide to make 10 measurements and average. In this case N = 10 and σx = 1.0, so the uncertainty in the average is

3.010

0.1

Nx

avg

This is not true for systematic uncertainties- if your meter stick is too short, you don’t gain anything by repeated measurements.

Propagation of Error Propagation of Error Used to determine uncertainty of a

quantity that requires measurement of several independent variables. Volume of a cylinder = f(D,L) Volume of a block = f(L,W,H) Density of an ideal gas = f(P,T)

IB Does this on a worst case scenario!

Uncertainty 19

Uncertainty when a number is Uncertainty when a number is raised to a powerraised to a power

%251728%)3.8(31728%)3.812( 33 z

400170043017283 z

%1.446.3%)3.8(2

146.3%)3.812( 2

12

1z

Example: If z = 12 ± 1.0 = 12.0 ± 8.3 % then

If z = xn then %σz = n ( % σx )

14.46.321

z

Uncertainty 20

Uncertainty when calculation Uncertainty when calculation involves a special functioninvolves a special function

Example: If θ = 120 ± 2.00

sin(140) = 0.242

sin(120) = 0.208

sin(100) = 0.174

For a special function, you add and subtract the uncertainties from the value and calculate the function for each case. Then plug these numbers into the function.

And thus sin(120 ± 20 ) = 0.208 ± 0.034

0.034

0.034