Physics 310

Errors in Physical Errors in Physical MeasurementsMeasurements

Error definitionsError definitions

Measurement distributionsMeasurement distributions

Central measuresCentral measures

Physics 310

Errors are uncertainties...Errors are uncertainties...

EveryEvery physical measurement has an physical measurement has an uncertaintyuncertainty, i.e., it has an , i.e., it has an errorerror..Random uncertainties (errors)Random uncertainties (errors)

Can be reduced but Can be reduced but notnot eliminated. eliminated. No measurement is infinitely precise.No measurement is infinitely precise. Determines theDetermines the precision precision of the data. of the data.

Systematic uncertainties (errors)Systematic uncertainties (errors) Can be reduced Can be reduced and and eliminatedeliminated Produces systematic shifts in the dataProduces systematic shifts in the data Determines the Determines the accuracyaccuracy of the data. of the data.

The problem: The problem: What is the best way to find the values for What is the best way to find the values for ak which which

most likely represent the parent population from which our sample is most likely represent the parent population from which our sample is obtained?obtained?

Physics 310

Errors are uncertainties...Errors are uncertainties...

Errors areErrors are not not blunders or mistakes.blunders or mistakes.Blunders or mistakes -Blunders or mistakes -

Must be found and corrected.Must be found and corrected. Are Are not not quoted in error estimates on measurements.quoted in error estimates on measurements. “ “Human Error” isHuman Error” is not not a valid error. a valid error.

Errors are quoted as Errors are quoted as x ± xx needs to be estimated - needs to be estimated -

From the data.From the data. From the measurement.From the measurement. May have asymmetric values about May have asymmetric values about x

Physics 310

The parent population...The parent population...

If you are measuring a physical quantity If you are measuring a physical quantity (x) e.g., the distance a neutron penetrates into a e.g., the distance a neutron penetrates into a given material, repeating the experiment given material, repeating the experiment N times produces slightly different values for times produces slightly different values for (x), i.e., , i.e., (xi) :where :where i goes from goes from 1 to to N..

If If N goes to goes to ∞, then this is the total of , then this is the total of allall possible measurements of this quantity - possible measurements of this quantity - and this set is called the and this set is called the Parent PopulationParent Population..

Physics 310

The sample population...The sample population...

If If N is is finitefinite, this will be , this will be a samplea sample of the of the total number of total number of allall possible measurements possible measurements of this quantity of this quantity (x) - and this smaller set of - and this smaller set of measurements is called the measurements is called the

Sample PopulationSample Population.. We will make notational distinctions We will make notational distinctions

between these populations.between these populations.

Physics 310

The sample population...The sample population...

If If N is is largelarge, the results obtained from the , the results obtained from the sample population of measurements will sample population of measurements will approach those of the parent population, but-- approach those of the parent population, but--

We will We will never knownever know the actual values from the the actual values from the parent population, even though we seek them. parent population, even though we seek them.

Our goal is to find the best estimates of the Our goal is to find the best estimates of the parent population values - parent population values -

Physics 310

The sample population...The sample population...

……and to estimate the and to estimate the precisionprecision and and accuracyaccuracy of of our estimate of our estimate of (x) ..

This latter exercise is calledThis latter exercise is called error analysiserror analysis.. Results are often reported as Results are often reported as

x ± x ± x

where where x is the best representative value of is the best representative value of (x), , x

is the estimate of the random error, and is the estimate of the random error, and x is the is the

estimate of the systematic error.estimate of the systematic error.

Physics 310

Deviation...Deviation...

Because we Because we cannot knowcannot know (x) from the parent from the parent population (the “population (the “truetrue” value) we cannot formally ” value) we cannot formally compare our value with a “true” measurement of compare our value with a “true” measurement of (x). .

There areThere are fewfew quantities whose value is predicted quantities whose value is predicted exactly from theory. For example:exactly from theory. For example: The charge on 1 electron is 1.6 x 10The charge on 1 electron is 1.6 x 10-19-19 C. C. The speed of light is 2.998 x 10The speed of light is 2.998 x 1088 m/s. m/s. The gravitational constant is 6 x 10The gravitational constant is 6 x 10-11-11 N m N m22/kg/kg22

All measured!All measured!

Physics 310

Deviation...Deviation...

Therefore, our measurement of Therefore, our measurement of (x) can only be can only be compared with the another measurement of compared with the another measurement of (x), , each of which has an uncertaintyeach of which has an uncertainty (experimental (experimental error), and neither of which is the “true” value. error), and neither of which is the “true” value.

Such a comparison results in a Such a comparison results in a deviationdeviation between the two measured quantities - or between the two measured quantities - or between a measured quantity and a theoretically between a measured quantity and a theoretically expected quantity.expected quantity.

Physics 310

Deviation...Deviation...

However, on the basis of measurement theory, However, on the basis of measurement theory, we may postulate what the we may postulate what the expected form of the expected form of the distributiondistribution of measurements of measurements (xi) should be should be

expected to be.expected to be. A plot of A plot of v vs vs t for a freely falling object. for a freely falling object. A plot of the distribution (A plot of the distribution (histogramhistogram) of ) of

measurements of neutron depth in indium.measurements of neutron depth in indium. A plot of the angular distribution of photons from A plot of the angular distribution of photons from e+

e- annihilation. annihilation.

Physics 310

Deviation...Deviation...

It is therefore useful to compare not only the It is therefore useful to compare not only the best estimate of best estimate of (x), but also the, but also the distribution distribution of measured values of of measured values of (x) ..

If the distributions do not appear to agree, If the distributions do not appear to agree, what does this mean?what does this mean? A problem with the experiment?A problem with the experiment? A problem with the theory?A problem with the theory? Both?Both?

Physics 310

Quantitative representations...Quantitative representations...

Given a set ofGiven a set of N measurements, what are measurements, what are quantitative ways of expressing results?quantitative ways of expressing results? The mean, The mean, = <x> The deviation, The deviation, d = (x - <x>) The variance, The variance, 2 = <(x - <x>)2> The standard deviation The standard deviation = = √<(x - <x>)2>

Each quantity has Each quantity has physical unitsphysical units! Don’t forget to ! Don’t forget to include them!include them!

Know how to compute each.Know how to compute each.

Physics 310

Computatons: the Computatons: the meanmean

_ The The samplesample mean is defined as: mean is defined as:

_ The The parent populationparent population mean is then: mean is then:

x =

1N

xii

N

∑

=

LimN → ∞

1N

xii

N

∑⎡⎣⎢

⎤⎦⎥

Physics 310

Computatons: the Computatons: the deviationdeviation

_ The The samplesample deviation is defined as: deviation is defined as:

_ The The parent populationparent population deviation is then: deviation is then:

d =

LimN → ∞

1N

dii

N

∑⎡⎣⎢

⎤⎦⎥=0

di =xi − x

Physics 310

Computatons: the Computatons: the variancevariance

_ The The samplesample variance is defined as: variance is defined as:

_ The The parent populationparent population variance is then: variance is then:

2 =

LimN → ∞

1N

(xi −)2i

N

∑⎡⎣⎢

⎤⎦⎥

s2 =

1N −1

(xi − x)2i

N

∑⎡⎣⎢

⎤⎦⎥

Physics 310

...the ...the standard deviationstandard deviation

_ The The samplesample standard deviation is defined standard deviation is defined as:as:

_ The The parent populationparent population standard deviation is standard deviation is then: then:

s=±

1N −1

(xi − x)2

i

N

∑⎡⎣⎢

⎤⎦⎥

=±

LimN → ∞

1N

(xi −)2

i

N

∑⎡⎣⎢

⎤⎦⎥

Physics 310

Distributions...Distributions...

Take a set of Take a set of N measurements. measurements. Form a Form a histogramhistogram of the measurements. (This of the measurements. (This

gives the gives the distribution distribution of the measurements.)of the measurements.) This gives the number of measurements of between This gives the number of measurements of between

x and and x + x as as nj, , j = 1,k where where k is the total number is the total number

of bins. (of bins. (x is the fixed bin width.) is the fixed bin width.) Now, with this you can estimate Now, with this you can estimate and and because because nj

represents a distribution function for the represents a distribution function for the measurements measurements xi . How do you do it? . How do you do it?

Physics 310

Distributions...Distributions...

The The meanmean and and standard deviationstandard deviation from a from a distribution are:distribution are:

x =nj

j

k

∑ xjmidpoint

njj

k

∑

s2 =nj(

j

k

∑ xjmidpoint− x)2

njj

k

∑

Physics 310

Normalized Distributions...Normalized Distributions...

From a From a histogramhistogram of the of the N measurements, you can measurements, you can form a form a normalized distribution normalized distribution of the measurements.of the measurements. Take each value Take each value nj, , j = 1,k and divide it by and divide it by N. This will . This will

give the fractional number give the fractional number fj of all measurements in the of all measurements in the

bin bin j. The sum of all . The sum of all fj will be will be 1, and hence the , and hence the

distribution function distribution function fj is a is a normalized (discrete) normalized (discrete)

distributiondistribution.. IfIf N is very large, his concept can be extended to a is very large, his concept can be extended to a

continuous probability functioncontinuous probability function P(x).

Physics 310

Normalized distributions...Normalized distributions...

The The meanmean and and standard deviationstandard deviation from a from a normalizednormalized discretediscrete distribution are: distribution are:

x = fj

j

k

∑ xjmidpoint

s2 = fj(

j

k

∑ xjmidpoint− x)2

fj

j

k

∑ =1

Physics 310

Normalized distributions...Normalized distributions...

The The meanmean and and standard deviationstandard deviation from a from a continuouscontinuous distribution are: distribution are:

dN(x)=N • P(x)dx

x=x

−∞

+∞

∫ dN(x)

dN(x)−∞

+∞

∫

Physics 310

Normalized distributions...Normalized distributions...

The The denominator denominator is just:is just:

dN(x)

−∞

+∞

∫ = N • P(x)d(x)−∞

+∞

∫

=N P(x)d(x)

−∞

+∞

∫ =N

Physics 310

Normalized distributions...Normalized distributions...

The The standard deviation standard deviation is then:is then:

2 =x− x( )2

−∞

+∞

∫ dN(x)

dN(x)−∞

+∞

∫

Physics 310

Normalized distributions...Normalized distributions...

……or in terms of a probability function or in terms of a probability function P(x):

2 = x− x( )2

−∞

+∞

∫ P(x)dx

x = x

−∞

+∞

∫ P(x)dx

Physics 310

Normalized distributions...Normalized distributions...

Or forOr for any any function: function:

2 = g(x)− g(x)[ ]

2

−∞

+∞

∫ P(x)dx

g x( ) = g(x)

−∞

+∞

∫ P(x)dx