Statistics and Detection

Question 1

What is accuracy?

Answer 1

How close is the result to the true value?

Question 2

What is precision?

Answer 2

how close together are repeat measurements

Question 3

What is a systematic or determinate error?

Answer 3

Affects all readings in the same way –> caused by low accuracy

Question 4

What is a random or indeterminate error?

Answer 4

Cause scatter about the true value –> caused by low precision

Question 5

What are systematic errors caused by?

Answer 5

Sloppy practice, incorrect instrument settings, wrong calibrations

Question 6

How can systematic errors be detected by?

Answer 6

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

Question 7

What are random errors caused by?

Answer 7

Random errors in experiments, if all errors are truly random the results should follows a normal (or Gaussian) distribution around the mean

Question 8

What is behind the Gaussian distribution - random errors: energy?

Answer 8

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

Question 9

What is behind the Gaussian distribution - random errors: entropy?

Answer 9

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

Question 10

How are the results characterised?

Answer 10

Mean and standard deviation

Question 11

What confidence limit is normally used?

Answer 11

95% confidence limit, results mean ± 2 standard deviations, based on the assumption that the Gaussian is correct (note 1 in 20 is still ‘wrong’)

Question 12

What is the standard error of the mean?

Answer 12

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

Question 13

What is the range?

Answer 13

Difference between highest and lowest values

Question 14

What is the variance?

Answer 14

Square of the standard deviation

Question 15

What is the median?

Answer 15

Value at the middle of the distribution

Question 16

What is the absolute error?

Answer 16

(deviation) = xi - x true

Question 17

What is the relative error?

Answer 17

((xi - x true)/x true) x 100%

Question 18

What is the proportional error?

Answer 18

Uncertainty depends on the size of the sample

Question 19

What is the constant error?

Answer 19

Does not depend on amount of sample

Question 20

Error propagation?

Answer 20

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

Question 21

Systematic error propagation?

Answer 21

Use first order approximate error propagation

Question 22

Statistical (random) error propagation?

Answer 22

Use second order approximate error propagation

Question 23

How can errors differ?

Answer 23

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

Question 24

Difference between accuracy and precision?

Answer 24

The terms accuracy and precision are linked to the presence of systematic and statistical (random) errors in the measurement

Question 25

What does standard deviation allow?

Answer 25

The standard deviation allows confidence limits to be assigned as a measure of the precision of the measurement

Question 26

What does error propagation allow?

Answer 26

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

Question 27

Five golden rules of mean and standard deviation?

Answer 27

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

Question 28

What is the Q test?

Answer 28

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

Question 29

Equation for Q test?

Answer 29

Q = d/w

Q = (‘outlier’ nearest)/(‘outlier’ furthest)

Question 30

What does the result of the Q test mean?

Answer 30

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)

Question 31

What is the T test?

Answer 31

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

Question 32

How can we use the T test?

Answer 32

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

Question 33

Degree of freedom for two measurements?

Answer 33

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

Question 34

Example of a ‘hypothesis test’?

Answer 34

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