INTRODUCTORY LECTURE 3Lecture 3: Analysis of Lab Work

Electricity and Measurement (E&M) BPM – 15PHF110

In This Lecture…1. Objectives

2. Multiple Measurement Uncertainty

3. Standard Deviation

4. Random and Systematic Uncertainty

5. Combining Uncertainties in Calculations

Analysing Data - Objectives Explain how to produce the uncertainty in a set of multiple readings using

a Normal (Gaussian) Distribution and the Standard Deviation, calculate the mean, standard deviation and standard error in the mean,

Understand the difference between, the standard deviation of a sample from a population and the standard deviation of the entire population.

Be able to contrast how uncertainty effects the accuracy and precision of the data collected, linking accuracy to systematic uncertainty, linking precision to random uncertainty.

Combine uncertainties during calculations to provide the overall uncertainty in the calculated value, when adding or subtracting data, when multiplying and dividing data, when power terms, etc. are involved.

More simply…• How do we calculate the mean

• How do we calculate the Standard Deviation

• How do we calculate the Standard Error

• How do we combine errors where we have separate errors in different readings

• We will do the basics

• Practice on this

• Using your calculator

Reading d /cm1 3.42 3.93 3.24 2.75 3.56 3.57 3.58 3.99 3.0

10 3.211 3.712 3.413 3.914 3.415 3.5

Reducing Uncertainty With Multiple MeasurementsConsider a set of measurements of a rings diameter. • Check for inconsistent results – anomalies• Remove them from the calculation to get a better estimate of

the mean• Number of measurements taken into consideration, • Calculate the mean to get an estimate of the diameter.

𝑑= 1𝑛∑

𝑖=1

𝑛

𝑑 ­𝑖=3.5 cm

𝑑=3.4+3.9+3.2+3.5+3.5+3.5+3.9+3.0+3.2+3.7+3.4+3.9+3.4+3.5

14=3.5

Usually written as

Reading d /cm1 3.42 3.93 3.24 2.74 3.55 3.56 3.57 3.98 3.09 3.2

10 3.711 3.412 3.913 3.414 3.5

Reducing Uncertainty With Multiple MeasurementsConsider a set of measurements of a rings diameter. • Check for inconsistent results – anomalies• Remove them from the calculation to get a better estimate of

the mean• Number of measurements taken into consideration, • Calculate the mean to get an estimate of the diameter.

Usually written as 𝑑= 1𝑛∑

𝑖=1

𝑛

𝑑 ­𝑖=3.5 cm

∑ ¿Is the symbol for “sum the following”, i.e. add them all together

Indicates that each of the values of from to are added together

1𝑛∑𝑖=1

𝑛

𝑑 ­𝑖 divide the sum of the previous step by the number of values,

𝑑=3.4+3.9+3.2+3.5+3.5+3.5+3.9+3.0+3.2+3.7+3.4+3.9+3.4+3.5

14=3.5

0

2

4

6

8

diameter, d /cm

freq

uen

cy

3.0 3.3 3.6 3.9

Spread of DataTo find the uncertainty in the mean value, the spread of the data is analysed:• Minimum value of d = 3.0 cm• Maximum value of d = 3.9 cm

Splitting this spread into intervals gives:• measurements ()• 7 measurements (*)• measurements (#)

Reading d /cm1 3.42 3.93 3.24 3.55 3.56 3.57 3.98 3.09 3.2

10 3.711 3.412 3.913 3.414 3.5

3.5

*

***

*

**

#

#

#

#

Gaussian or Normal Distribution

Standard Deviation and Normal Distributions

𝜎 𝑛−1

𝜎 𝑛−1

𝜎 𝑛−1

𝜎 𝑛−1

If the spread of data is closer to the mean, is smaller and the peak is higher.

Standard Deviation shown on the normal distribution.

If the spread of data is further from the mean, is bigger and the peak is lower.

More Precise

Less Precise

68% of measured values (green dots) are within the blue zone

We use the standard deviation for a sample, , as an indicator that shows how far 68% (about 2/3) of readings are from the true value.

𝜎 𝑛−1

Standard Deviation and UncertaintyIt is considered a reasonable assumption that the uncertainty quoted for a mean value will lie within the region where 68% (about 2/3) of measured values are found, i.e. within one standard deviation of the mean.

To calculate the standard deviation,

• Subtract the mean from each reading to give the deviation,

• Square this and add it to the square of all the other deviations,

• Divide this sum by one less than the total number,

• Find the square root of the result. 𝜎 𝑛−1=√ 1𝑛−1

∑𝑖=1

𝑛

(𝑥 𝑖−𝑥 )2

∑𝑖=1

𝑛

(𝑥𝑖−𝑥 )2

𝑥−𝑥

1𝑛−1

∑𝑖=1

𝑛

(𝑥 𝑖−𝑥 )2

Two Types of Standard Deviation

𝜎 𝑛=√ 1𝑛∑𝑖=1

𝑛

(𝑥 𝑖−𝑥 )2

If n is the total population under test we use this version.

Excel formula:

=STDEV.P(specify range­here)

(Population standard deviation)

𝜎 𝑛−1=√ 1𝑛−1

∑𝑖=1

𝑛

(𝑥 𝑖−𝑥 )2𝜎 𝑛−1=√ 1𝑛−1

∑𝑖=1

𝑛

(𝑥 𝑖−𝑥 )2

If n is a sample from a much larger population we use this version

– for physics experiments this one is usually best.

Excel formula:

=STDEV.S(specify­range­here)

(Sample standard deviation)

Use this one

Uncertainty and Normal DistributionsTo quote the level of uncertainty in the mean we use:• Standard Error in Mean (SEM)

o Applying this to our ring diameter data gives,

o Giving a value for

• Relative Error in the Mean (REM)

SEM=𝜎𝑛− 1

√𝑛

Reading d /cm1 3.42 3.93 3.24 3.55 3.56 3.57 3.98 3.09 3.2

10 3.711 3.412 3.913 3.414 3.5

3.50.3

SEM=0.3

√14=0.08 so ∆𝑑=0.1cm

REM=∆ 𝑑𝑑

=0.13.5

=0.03

fractionalpercentage

Example,Find the standard error in the mean for the following measurements:

1 2 3 4 5

/s 5.0 4.9 4.7 5.3 5.2

𝑡= 1𝑛∑𝑖=1

𝑛

𝑡 ­𝑖=15(5.0+4.9+4.7+5.3+5.2)¿

25.15

=5.02=5.0 s

Standard deviation, 𝜎 𝑛−1=√ 1𝑛−1

∑𝑖=1

𝑛

(𝑡 𝑖− 𝑡 )2

s

Standard error in the mean, ¿0.23

√5=0.11=0.1 s

Therefore s

s

Mean,

SEM=𝜎𝑛− 1

√𝑛

?

? ?

Recording DataWhere possible, make multiple measurements to determine the random error properly and pick up any mistakes.

Where this is not possible, regard any single measurement as one drawn from a Gaussian (normal) population. Use common sense and experience to estimate what the

uncertainty would be – you are looking to accommodate 2/3 of all possible measurements within the ± spread you quote.Usually this means ± half the…• last digit on a display,• division marking on a scale,• the range of a fluctuating needle

2.31

Reading = 2.310±0.005 s Reading = 39.0±0.5mm

0 2.121 2.112 2.133 2.114 2.145 3.116 2.137 2.128 2.139 2.1210 2.12

Reading = 6.5±0.5A

Summary of Random and SystematicRandom• Equally likely to give results that are above and below the true value,

cancelling each other out to give a zero expected value.• Random errors are present in all experiments and therefore the researcher

should be prepared for them.

Measured values

Systematic• Biases in measurement

resulting in the mean of the separate measurements differing significantly from the true value, e.g. calibration or zero error.

• They can be either constant, or related (e.g. proportional or a percentage) of the true value.

True Value

Summary of Random and Systematic

Not accurateNot precise

- large systematic error- large random error

Not accuratePrecise

- large systematic error- small random error

AccurateNot precise

- small systematic error- large random error

AccuratePrecise

- small systematic error- small random error

True Value

Measured Value

True Value

Measured Value

True Value

Measured Value

True Value

Measured Value

Combining UncertaintiesWe can now define the uncertainty for either a single measurement or for the average value of a set of measurements.

In both cases the • Standard uncertainty is,

• Relative uncertainty is, ∆ 𝑋𝑋

(×100 %)

When combining measurements in an equation these two ways of quoting uncertainties become very important…

Adding or Subtracting the uncertainty in the result is found from the,

standard uncertainties of the initial values.

standard uncertainties

Multiplying or Dividing the uncertainty in the result is found from the,

relative uncertainties of the initial values.

relative uncertainties

Combining Uncertainties: Addition and Subtraction

If and the uncertainties in and are and The uncertainty in , is given by,

or

If then is still the sum of and , an uncertainty can never be reduced through any kind of calculation but it can be increased.

If m m

m m

6 m m

Combining Uncertainties: Multiplication and DivisionIf or then the uncertainty in is given using the relative uncertainties, i.e.

If    𝐴=6.0±0.5 kg∧𝐵=12.5±0.5 kg

⇒ ∆ 𝑋𝑋

=√( 0.56.0 )

2

+( 0.512.5 )

2

From

In general for a relative uncertainty, , we can state the value with its standard uncertainty using

∆ 𝑋𝑋

=0.09⇒∆ 𝑋=0.09 𝑋 To quote substitute for

i.e. ¿ 𝑋 (1±0.09)

¿0.09(¿ 9 %)

Combining Uncertainties: Multiplication and Division cont.…

If X = A × B:

kg

If X = A ÷ B:

kg

If    𝐴=6.0±0.5 kg∧𝐵=12.5±0.5 kg

⇒ ∆ 𝑋𝑋

=√( 0.56.0 )

2

+( 0.512.5 )

2

=0.09(¿9 %)

Same relative uncertainty

Different standard uncertainty

Multiplication

Division

Combining Uncertainties: Power TermsIf , then

∆ 𝑋𝑋

=𝑛( ∆ 𝐴𝐴 ) and

And The Rest… (Used rarely so no need to learn these, look them up when required)

𝑋=ln 𝐴⇒∆ 𝑋=∆ 𝐴𝐴; 𝑋=𝑒𝐴⇒

∆ 𝑋𝑋

=∆ 𝐴;

𝑋=sin 𝐴⇒∆ 𝑋=cos 𝐴∆ 𝐴 ; 𝑋=cos 𝐴⇒∆ 𝑋=− sin 𝐴∆ 𝐴;

Standard error in = relative error in Relative error in = standard error in

must be in radians and must be small

⇒∆ 𝑋𝑋

=2×0.55.0

=0.2

m2

Combining Uncertainties: Insignificant UncertaintyOften one source of uncertainty will dominate and others may then be considered to be not significant.

For example,

Mass g (sensitive balance for small masses)

Mass g (less sensitive balance for larger masses)

If

Uncertainty in mass is not significant and can be ignored as the uncertainty in mass is so large.• Always carry through error calculations to the final result for all significant

uncertainties.

g

If Relative uncertainty in and in

¿ ∆𝐵

Practice Makes PerfectThe resistance of a wire is given by,

Given, Ωmcm

The diameter of the wire was measured at 5 different points spaced out along the length of the wire. The values of /mm are:

𝑅=4 𝜌 𝑙𝜋 𝑑2

1. Find the mean and SEM for in mm

Using Excel:

mm mm

mm

mm

Using a calculator:

=AVERAGE(B2:B6) 0.561 7

=STDEV.S(B2:B6) 0.011 8

=B8/SQRT(A6) 0.005 9

A B

/mm 1

1 0.560 2

2 0.575 3

3 0.565 4

4 0.560 5

5 0.545 6

Practice Makes PerfectThe resistance of a wire is given by,

Given, Ωmcm

The diameter of the wire was measured at 5 different points spaced out along the length of the wire. The values of /mm are:

𝑅=4 𝜌 𝑙𝜋 𝑑2

2. Make units consistent

cm → m for length, m.

mm → m for diameter, 3. Calculate .

m

m mm

/mm

1 0.560

2 0.575

3 0.565

4 0.560

5 0.545

0.561

0.011

0.005

𝑅=4 𝜌 𝑙𝜋 𝑑2¿

4×44.2×10−6×4.302

𝜋× ( 0.560×10− 3 )2 ¿772Ω

Practice Makes PerfectThe resistance of a wire is given by,

Given, Ωmcm

The diameter of the wire was measured at 5 different points spaced out along the length of the wire. The values of /mm are:

𝑅=4 𝜌 𝑙𝜋 𝑑2

4. Find the uncertainty in .

, 4 and are constants and have no uncertainty, therefore as and are divided their relative uncertainties can be used to find the relative uncertainty in ,

m

m mm

/mm

1 0.560

2 0.575

3 0.565

4 0.560

5 0.545

0.561

0.011

0.005

¿772Ω

⇒ ∆𝑅𝑅

=√( ∆ 𝑙𝑙 )2

+(2 ∆ 𝑑𝑑 )2

¿√( 0.0014.302 )

2

+(2 0.0050.560 )

2

If

⇒∆𝑅𝑅

=0.018⇒𝑅=772 (1±0.018 )⇒

¿0.018=0.02(2 %)

𝑅=772±14Ω

772±14Ω

Processing uncertainties - summary Explain how to produce the uncertainty in a set of multiple

readings using a Normal (Gaussian) Distribution and the Standard Deviation, calculate the mean, standard deviation and standard

error in the mean,

Understand the difference between, the standard deviation of a sample from a population and the standard deviation of the entire population.

Be able to contrast how uncertainty effects the accuracy and precision of the data collected, linking accuracy to systematic uncertainty, linking precision to random uncertainty.

Combine uncertainties during calculations to provide the overall uncertainty in the calculated value, when adding or subtracting data, when multiplying and dividing data, when power terms, etc. are involved.

𝜎 𝑛−1=√ 1𝑛−1

∑𝑖=1

𝑛

(𝑥 𝑖−𝑥 )2

𝑑= 1𝑛∑

𝑖=1

𝑛

𝑑 ­𝑖

SEM=𝜎𝑛− 1

√𝑛

𝜎 𝑛−1

𝜎 𝑛

∆ 𝑋=√∆ 𝐴2+∆𝐵2

∆ 𝑋𝑋

=√( ∆ 𝐴𝐴 )2

+( ∆𝐵𝐵 )2

∆ 𝑋𝑋

=𝑛( ∆ 𝐴𝐴 )