Statistics and Detection Flashcards
What is accuracy?
How close is the result to the true value?
What is precision?
how close together are repeat measurements
What is a systematic or determinate error?
Affects all readings in the same way –> caused by low accuracy
What is a random or indeterminate error?
Cause scatter about the true value –> caused by low precision
What are systematic errors caused by?
Sloppy practice, incorrect instrument settings, wrong calibrations
How can systematic errors be detected by?
Systematic errors can be detected by analysing standard reference materials available from recognised labs, alternative approach is to analyse by 2 or more independent methods, some errors are detected by running blank samples
What are random errors caused by?
Random errors in experiments, if all errors are truly random the results should follows a normal (or Gaussian) distribution around the mean
What is behind the Gaussian distribution - random errors: energy?
Imagine a harmonic oscillators eg an atom with a spring connected to a wall, finding the atom in the centre is most likely (at least at atomic level) and deviation from the central position is lined to energy
Force = -kx and energy energy = 1/2kx^2
With Boltzmann n/n0 = exp(-kx^2/2RT) we obtain a Gaussian population
Random distribution of energy leads to symmetry and Gaussian distribution or Gaussian peaks
What is behind the Gaussian distribution - random errors: entropy?
Imagine a two compartment system with a number of particles that are allowed to move. Snapshots are taken of the particle number left and right. Probability
W = N!/(Nleft!*Nright!)
Boltzmann: S = k lnW
We obtain a Gaussian like distribution of probabilities for particle numbers
How are the results characterised?
Mean and standard deviation
What confidence limit is normally used?
95% confidence limit, results mean ± 2 standard deviations, based on the assumption that the Gaussian is correct (note 1 in 20 is still ‘wrong’)
What is the standard error of the mean?
Standard deviation refers to the probable error in an individual measurement, if lots of repeat measurements are taken, the error in the mean will be lower than that of each individual result. It is better to reduce standard deviation than increase N to improve the precision, to get 10 x better precision need 100 x the number of measurements
What is the range?
Difference between highest and lowest values
What is the variance?
Square of the standard deviation
What is the median?
Value at the middle of the distribution
What is the absolute error?
(deviation) = xi - x true
What is the relative error?
((xi - x true)/x true) x 100%
What is the proportional error?
Uncertainty depends on the size of the sample
What is the constant error?
Does not depend on amount of sample
Error propagation?
Once a statistical error has merged it can propagate through a calculation, for multi parameter equation several errors can add up, say for area A calculated from length x length y: A = x*y
Systematic error propagation?
Use first order approximate error propagation
Statistical (random) error propagation?
Use second order approximate error propagation
How can errors differ?
Errors can be statistical (random) or systematic in nature, systematic errors needs to be addressed by multiple methods and standard sample. Statistical (random) error can be evaluated and minimise
Difference between accuracy and precision?
The terms accuracy and precision are linked to the presence of systematic and statistical (random) errors in the measurement
What does standard deviation allow?
The standard deviation allows confidence limits to be assigned as a measure of the precision of the measurement
What does error propagation allow?
Error propagation analysis can help tracking errors from a measurement through a calculation, for multi parameter error analysis often graphical methods for estimating error can be more practical
Five golden rules of mean and standard deviation?
The best estimate of a parameter is the mean
The errors is the standard error in the mean
Round up the error to the appropriate number of significant figures
Match the number of decimal places in the mean to the standard error
Include units
What is the Q test?
Allows us to judge whether an individual results should be ignored, simples and widely used compares the difference between the suspect result and its nearest value with the spread of all the results
Equation for Q test?
Q = d/w
Q = (‘outlier’ nearest)/(‘outlier’ furthest)
What does the result of the Q test mean?
The Q value is then compared with a table of ‘critical’ values, if Q > Q critical then the result can be rejected with the given degree of confidence (this methods takes into account the individual deviation as well as the general spread of data)
What is the T test?
Example of a hypothesis test “lets assume that two measurements are different - what’s the level of confidence that we can have in the assumption?” this is the null hypothesis ie the values are different within the range of errors that can be explained by random error
How can we use the T test?
If we know the ‘true’ value X true, we can compare our measurement with it.
The t value is related to the confidence in the measurements, if t is larger than the t table value then the measured mean is different from the true value. A large t suggests a big and significant difference between X true and the mean
Degree of freedom for two measurements?
The number of degrees of freedom = (N1 - 1) + (N2 - 1)
Again if t is larger than the t table value then the two means are different at the significance level being considered
Example of a ‘hypothesis test’?
Assume:
The measurement is not an outlier and could occur by chance
There is no difference in the averages/methods
Tables show the value needed to reject this hypothesis, if Q > Q table then the point is an outlier at the standard confidence limit, if t > t table then there is a difference at the standard confidence limit
