PHYS491: Research Skills

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PHYS491 Research Skills

• Dr Andy Boston
• Room 408 OLL ajb_at_ns.ph.liv.ac.uk
• Aim To develop theoretical and practical
  understanding of statistical principles.
• Data analysis
• Using appropriate recipes
• Using computational aids (Excel)
• Textbook A guide to the Use of Statistical
  Methods in the Physical Sciences, R.J. Barlow,
  Wiley.
• Lectures
• Tuesday 9.00, Thursday 12.00 lectures
• Friday 11.00
• 5 problem classes (each 5)
• Week 6 one hour class test 25

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Statistical Analysis

• Can be used
• Politically (lies, dammed lies and statistics)
• In Engineering (Quality Assurance)
• DOE
• Play an essential role in all sciences
• Medicine (Hypothesis testing)
• Physical Sciences (parameter estimation)
• Designing and planning experiments
• Consideration of tolerances required of apparatus
• Measurement times as a function of required
  precision of results
• Materials, Time and Cost constraints can be
  estimated

3
Errors and Uncertainties

• Strictly speaking
• An error implies a mistake has been made
• An uncertainty refers to a lack of knowledge
• An error which is a mistake is classified as an
  illegitimate error and is corrected by repeating
  the measurement
• If we only know the result of an experiment but
  do not know the error then we are completely
  unable to judge the significance of the result.
• Whenever you determine a parameter, you must
  estimate the error, otherwise the experiment is
  useless.

4
Systematic and Random

• Systematic An uncertainty in the bias of the
  data
• Calibration offset on an energy spectrum
• Zero offset of a voltmeter
• Random
• Instrument imprecisions
• Intrinsic statistical nature of a process
• Repeated measurements provide solution

5
Accuracy and Precision

• Accuracy
• A measure of close a result is to the true value
  the correctness
• Precision
• How well the result has been determined with no
  reference to the true value
• Consistency
• Two measurements are consistent if

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Probability

• Statistics deals with random processes
• P(0) event never occurs
• P(1) event always occurs
• P(AB) at least one of events A and B
• Applies if A and B are exclusive
• P(AB) both A and B
• More details

Conditional probability B given A
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Probability

• If the occurrence of B does not affect whether or
  not A occurs then
• A and B are independent.
• Eg A is Sunday B is raining.
• However for
• Constrained by energy conservation are
  non-independent.

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Coin tossing

• Two independent events
• Multiplication principle applies
• Sampling with replacement.
• A fair coin is thrown twice
• Sample space W hh, ht, th, tt
• A heads first throw
• B heads second throw
• Probability seeing a head
• P(A OR B) 0.75
• Probability was equally spread between various
  probabilities in sample space.
• Example

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Representation

• Frequency Distribution
• Number of times each possible outcome occurs
• Probability Distribution P(x)
• P(x) 1/6 for a true die example.
• Discrete P(x) or Continuous P(x)dx

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Histograms

• A common way to represent data utilises a
  histogram
• Quantity vs Frequency.
• With a large number of observations the bin size
  will approximate a continuous curve.
• Total number of counts in histogram
• Probability distribution follows from
• Probability distributions are Normalized

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Histograms
Typically this might be a voltage signal which
has been digitised to make a time sequence of
numbers to represent it. There is always some
noise which arises from many processes and
therefore forms a fluctuating background with a
Gaussian spread.
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Sampling and Parameter estimation

• As experimentalists, we take data to try and
  deduce what rule or laws are relevant to our
  experiment.
• We need to express the significance of our
  measurements efficiently.
• This may require repeated measurements to check
  reproducibility and precision of the result -
  Parameter Estimation.
• It may then be important to test to see if the
  data are consistent with a given hypothesis -
  Hypothesis Testing.

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Theory and Experiment

• Measurement and theory will not be exactly the
  same.
• Use statistics and probability theory to
  determine the meaning and significance of the
  comparison our confidence.
• In parameter estimation, we try to determine x
  and Dx. In order to do this we experimentally
  sample the data.

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Sampling

• We must chose an unbiased sample
• If we try to measure the Maxwellian distribution
  of molecule velocities coming out of small hole
  in a hot oven. Faster molecules have a greater
  probability of getting out, the beam is a biased.
• We must not reject data that does not look
  right. Need overpowering reason.
• Best value from data minimises the variance
  between the estimate and the true value, known as
  estimation.
• Must determine best estimate and uncertainty of
  this.

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Experimental and theoretical quantities

• Data set x1, x2, x3 xN
• Sample mean (finite N)
• Theoretical mean
• Sample variance
• Theoretical variance
• s and m do not come from the data (s and ).

Notice true mean
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Expectation value Moments

• Expectation value (mean value) of f(x)
• A probability distribution is characterized by
  its moments.
• The nth moment of x about x0 defined as
  expectation value of
• Where n is an integer
• Only the first (centre) and second (spread) used.

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Moments

• The first moment n1 about zero x00
• Recognise the mean or average of x.
• The second moment n2 about mean x0m
• Second moment is the variance.
• Square root of variance is the Standard deviation
  a measure of dispersion or width.
• Third moment is the skewness a measure of the
  symmetry or asymmetry.

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Covariance

• For multivariant distributions mean and variance
  as before
• The covariance of x and y
• A measure of the linear correlation between two
  variables.
• Expressed as a correlation coefficient r.
• r 1 (anticorrelated) and 1 (correlated)

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Summary of Lecture 1

• Rule of probability and examples
• Expectation values
• Mean
• Variance
• Standard deviation
• Covariance