CHEN10011 Slides 1

Statistics

Models, Errors, Statistical Analysis, Experimental Design and Optimisation

N.J. Goddard

CHEN10011 Engineering Maths - Statistics 2

Recommended Reading• Statistics and Chemometrics for Analytical Chemistry, 6th

Edition, J.N. Miller and J.C. Miller, Prentice-Hall, ISBN 0273730428

• Chemometrics: Statistics and Computer Applications in Analytical Chemistry, M. Otto, Wiley-VCH, ISBN 3527314180

• Statistical Procedures for Analysis of Environmental Monitoring Data and Risk Assessment, E.A. McBean and F.A. Rovers, Prentice-Hall, ISBN 0136750184.

• Statistical Tables, J. Murdoch and J.A. Barnes, 4th Edition, McMillan, ISBN 0333558596


Also see:

• http://www.itl.nist.gov/div898/handbook/index.htm• This is a very comprehensive on-line handbook of

statistics with examples


Classification of Data• Two types of variable:• Discrete: also known as step variables

– can be counted– described by a fixed set of values– each value of the variable can be associated with an integer index

• Continuous– continuous range of values– dependent upon the precision of the measurement or the accuracy of

the observer


Classification of Errors• Two types of error:• Systematic: also known as determinate errors

– built into the observation– affect the accuracy of measurement– caused by imperfections in the instruments– may be quantified and corrected for

• Random: also known as indeterminate errors– due to random and uncontrolled variations– affect the precision of the observation

• Statistical methods may be used to assess the effect of a random or indeterminate error.


An example of a source of systematic error in a measurement

• How can we remove the systematic error?

V

Power SupplyTransducer

Stimulus

Copper AluminiumT


Summarising Data• We need single numbers that summarise large

data sets• We need a measure of central tendency – a

“middle” value of the data• We also need a measure of dispersion – how

widely spread the data are from the central value


Summarising Data – Central Tendency• The Mean• The mean is defined by:

– is the sample mean– xi is a member of the sample– n is the size of the sample

• Also referred to as the first moment of a sample

xx

n

ii

n

1

x


Summarising Data – Central Tendency

• The mean is not a robust measure of central tendency

• Outliers pull the mean away from the “true” central value



• The Mode– Is the value of the most frequent observation in a set

of data– May be evaluated using graphical techniques– There may not be a unique value of the mode– Data may be bimodal, trimodal etc– Is also not very robust


Summarising Data – Central Tendency• The Mode

x

-4 -2 0 2 4

Fre

quen

cy

0.00

0.05

0.10

0.15

0.20

0.25

x

-4 -2 0 2 4

Fre

quen

cy

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Bimodal Trimodal



• The Median – Is the mid-point in a sorted set of data– If the number of observations is odd:– the median is the centre value in a sorted list of

observations– Otherwise, is the average of the two observations on

either side of the centre


Measurements of Dispersion• The Variance (S2)• The variance of a sample is given by:

• Using the square of the difference between a sample value and the mean ensures that the variance is always positive

• Also referred to as the second moment of a sample

Sx x

n

ii

n

2

2

1

1

( )


Measurements of Dispersion• This equation can be re-written to make calculation easier:

• In this case, all that needs to be calculated are the sums of x and x2

• This is the preferred form for calculating the variance of a sample

Sx

x

nn

ii

n ii

n

2

2

1

1

2

1


Measurements of Dispersion

• Standard deviation (S)• Standard deviation is the square root of the

variance and so may be described simply by:

Sx x

n

ii

n

( )2

1

1


Measurements of Dispersion

• Or:

• This is the preferred form for performing calculations of standard deviation

Sx

x

nn

ii

n ii

n

2

1

1

2

1


Measurements of Dispersion• Relative Standard Deviation

– Ratio of the standard deviation to the mean as a percentage

• Coefficient of variation– Ratio of the standard deviation over the sample mean

RSDS

x 100%

Coefficient of VariationS

x


Measurements of Dispersion • Average Absolute Deviation• The average absolute deviation is defined by:

• m(X) is the measure of central tendency against which the absolute deviations are measured

• The mean, median or mode can be used

n

XmxAAD

n

ii

1

)(


Measurements of Dispersion • Mean Absolute Deviation (MAD)• The mean is used as the measure of central

tendency:

• For a Gaussian (normal) distribution, the MAD is about 0.8 times the standard deviation

n

xxMAD

n

ii

1


Measurements of Dispersion • The choice of central tendency has a significant effect on

the AAD• For the dataset 1, 1, 2, 3, 13:

Central Value Type

Central Value AAD

Mean 4 3.6

Median 2 2.8

Mode 1 3.0


Measurements of Dispersion • Median Absolute Deviation (MAD)• This is the median of the absolute deviations from

the median of the data:

• This is a robust measure of dispersion – that is, outliers have little effect

)(XmedianxmedianMAD i


Measurements of Dispersion • Median Absolute Deviation (MAD)• MAD can be used to make a robust estimate of the

population standard deviation (σ)• In this case, MAD/0.6745 is used as the estimate of

σ


Example 1

• 20 mass measurements were made of a flask of reagent

• The units and measurements were in grams:–12.475 12.469 12.481 12.466–12.474 12.465 12.475 12.473–12.481 12.472 12.482 12.475–12.485 12.473 12.465 12.485–12.468 12.477 12.450 12.513


Example 1

• Sample mean:

• Rounded to the same number of decimal places as the original data:

g

x

x ii

4752.1220

504.249

20

20

1

gx 475.12


Rounding• Do not forget to round properly• Do not round too soon

– Only round when presenting results• Round to the same number of decimal places as

original data• Remember the units!• Remember the units!• Remember the units!


Calculation of standard deviationxi

2 x xi x xi 2

Observation xi

1 12.513 156.575169 0.038 0.001444

2 12.485 155.875225 0.010 0.000100

3 12.485 155.875225 0.010 0.000100

4 12.482 155.800324 0.007 0.000049

5 12.481 155.775361 0.006 0.000036

6 12.481 155.775361 0.006 0.000036

7 12.477 155.675529 0.002 0.000004

8 12.475 155.625625 0.000 0.000000

9 12.475 155.625625 0.000 0.000000

10 12.475 155.625625 0.000 0.000000

11 12.474 155.600676 -0.001 0.000001

12 12.473 155.575729 -0.002 0.000004

13 12.473 155.575729 -0.002 0.000004

14 12.472 155.550784 -0.003 0.000009

15 12.469 155.475961 -0.006 0.000036

16 12.468 155.451024 -0.007 0.000049

17 12.466 155.401156 -0.009 0.000081

18 12.465 155.376225 -0.010 0.000100

19 12.465 155.376225 -0.010 0.000100

20 12.450 155.002500 -0.025 0.000625

Sums: 249.504 3112.615078 0.0 0.002778


Calculation of standard deviationx xi x xi

2

ix 3 dp 4 dp 5 dp 6 dp

1 12.513 0.038 0.001 0.0014 0.00144 0.001444

2 12.485 0.01 0 0.0001 0.00010 0.000100

3 12.485 0.01 0 0.0001 0.00010 0.000100

4 12.482 0.007 0 0 0.00004 0.000049

5 12.481 0.006 0 0 0.00003 0.000036

6 12.481 0.006 0 0 0.00003 0.000036

7 12.477 0.002 0 0 0.00000 0.000004

8 12.475 0 0 0 0.00000 0.000000

9 12.475 0 0 0 0.00000 0.000000

10 12.475 0 0 0 0.00000 0.000000

11 12.474 -0.001 0 0 0.00000 0.000001

12 12.473 -0.002 0 0 0.00000 0.000004

13 12.473 -0.002 0 0 0.00000 0.000004

14 12.472 -0.003 0 0 0.00000 0.000009

15 12.469 -0.006 0 0 0.00003 0.000036

16 12.468 -0.007 0 0 0.00004 0.000049

17 12.466 -0.009 0 0 0.00008 0.000081

18 12.465 -0.01 0 0.0001 0.00010 0.000100

19 12.465 -0.01 0 0.0001 0.00010 0.000100

20 12.45 -0.025 0 0.0006 0.00062 0.000625

Sums: 249.504 0 0.001 0.0024 0.00271 0.002778

S 0.007255 0.011239 0.011943 0.012092


Calculation of standard deviation (1)

• To calculate standard deviation:

232 10778.2 gxxi

24

3

20

1

2

2 10462.119

10778.2

19g

xxS i

i

S g 1462 10 0 0124. .


Calculation of standard deviation (2)

• Using the alternate form:

S

g

3112 615078249 504

20

19

3112 615078 3112 612301

190 012

2

..

. ..


RSD, Mode and Median

• Relative standard deviation:

• Mode = 12.475g (most frequent measurement)• Median:

RSDS

x 100%

0 012

12 475100% 0 096%

.

..

)(475.124745.122

474.12475.12roundedgg


Quick Check of Standard Deviation

• If your calculated S is greater than the span of the data (that is, the difference between the maximum and minimum value), then your S is WRONG

• Consider two observations, x1 and x2

222

2

22

2

2

122

1

122

212221

21

22

21

21

2221

212

221

2212

221

2

2

12

xxxxxxxxxxx

x

xxxxxx

xxxx

nn

xx

S

xxSpan


Quick Check of Standard Deviation

• In general, the maximum and minimum possible standard deviations as a function of the span of the observations are given by:– Odd number of observations:

– Even number of observations:

22

1 ,

4

1minmax

n

SpanSn

nSpanS

22

1 ,

44 minmax

nSpanS

n

nSpanS


Quick Check of Standard Deviation• Example:


Absolute Average Deviations

• From mean: 0.00776 g• From median: 0.0077 g• From mode: 0.0077 g


Median Absolute Deviation

• MAD = 0.0065 g• Robust estimate of population standard deviation

(σ) is MAD/0.6745• This is 0.00964 g (rounded to 3 significant figures)• Which is lower than our sample standard deviation

of 0.012 g• Is there an outlier?


Outlier Value (g)

12.38 12.40 12.42 12.44 12.46 12.48 12.50 12.52 12.54

Cen

tral

Val

ue (

g)

12.46

12.47

12.48

12.49

Mode

Median

Mean

Robustness of Measures• Effect of an outlier on the mean, mode and median:

• The mode is least robust and the median the most robust indicator of central tendency


Frequency Distributions

(a) Frequency Distribution of Observed Masses

Observed Mass (g)

12.44 12.46 12.48 12.50 12.52

Num

ber

of

Ob

serv

atio

ns

0

1

2

3

4

(b) Cumulative Frequency Distribution of Observed Masses

Observed Mass

12.44 12.46 12.48 12.50 12.52

Cum

ula

tive

Nu

mb

er o

f O

bse

rva

tion

s

0

5

10

15

20

25


Propagation of Errors• Most measurements are composite• That is, they are a function of two or more simple variables• Example: measurement of mass of material

• Given the errors in the simple variables, what is the error in the composite value?

vesselmaterialvesselmaterial mmm


Propagation of Errors• A small error ∆x in a length measurement x:

• Leads to increased errors in area estimates:

x ∆x

x ∆x

x

∆x

Error is 2xx+x2

x∆x

x∆x ∆x2


Propagation of Errors• And even larger errors in estimates of volume:

x ∆x

x2∆x

x∆x2

∆x3

Error is 3x2x+3xx2+x3


Propagation of Errors• Random errors tend to cancel each other out (consider

the drunkard’s walk)• Systematic errors are vectors which do not cancel out• Thus, the propagation of systematic and random errors

are undertaken in a slightly different ways • Consider x =a +b

– If a and b each have a systematic error of +1 then the whole systematic error of x is +2.

– If a and b have an indeterminate error of 1: – The random error in x is not 2– Sometimes a will have an error of +1 while b has an error of -1,

cancelling out.


Propagation of Determinate Errors• For a two factor system:

• where the uncertainty in x and z is x and z• If

• Then

y f x z ( , )

y x z .

y y x x z z xz x z z x x z ( )( )


Propagation of Determinate Errors• So,

• But ∆x ∆z will be small, so

• Differentiating with respect to x and z gives us:

y x z z x x z

y x z z x

xy

zx

zy

xz


Propagation of Determinate Errors

• So,

• for a multi-variate system:

yy

zz

y

xx

x z

y f x x xn ( , ),1 2

yy

xx

y

xx

y

xx

x xn x xn n x xn

n

1 2

12 1

2

1 1, , ,


Propagation of Determinate Errors

• The contribution of a determinate error in a variable to the total error is given by the magnitude of the error multiplied by the sensitivity of the derived variable to changes in that variable


Resolution• We can use this to calculate the resolution of a

composite measurement:

• Resolution of an individual variable is normally taken as half the scale division of the instrument used

• Can be taken as a determinate error in the measurement

yy

xx

y

xx

y

xx

x xn x xn n x xn

n

1 2

12 1

2

1 1, , ,


Example 2

• Determination of the density, , of the material weighed in Example 1

• The volume of the material was measured five times

• The resulting volume measurements were:– 6.0, 6.0, 5.8, 5.7, and 6.3 cm3


Example 2

• Density is given by:

• So the measurement resolution is given by:

m

V

mm

VV

V m


Example 2• But

• So,

• Note: each term is dimensionally correct

m V

V

1 V

m

Vm

2

m

V

m V

V 2


Example 2• Since

m = 510-4 g V = 510-2 cm3

– m = 12.475 g – V = 6.0 cm3

• Then

• The resolution of the density determination experiment was 0.017 g cm-3

322

24

cm g1074.10.6

105475.12

0.6

105


Propagation of Indeterminate Errors

• For functions of a variable:

• In some cases variance is used in the place of standard deviation:

dx

ydSSxfy xy ),(

2

22 ),(

xd

dySSxfy xy


Propagation of Indeterminate Errors• For linear combinations of variables:

y f x x xn ( , , )1 2

2

2

2

3

2

2

2

2

2

1

2

2

2

2

3

2

2

2

2

2

1

22

321

321

nxxxxy

nxxxxy

x

yS

x

yS

x

yS

x

ySS

x

yS

x

yS

x

yS

x

ySS

n

n


Propagation of Indeterminate Errors• For linear combinations

• Since

2233

222

211

3322110

nny

nn

SkSkSkSkS

xkxkxkxkky

etc , 22

11

kx

yk

x

y


Propagation of Indeterminate Errors• For multiplicative combinations

• Or, in general

2

4

4

2

3

3

2

2

2

2

1

1

43

21

x

S

x

S

x

S

x

S

y

S

xx

xkxy

y

2222

21 nxxxy CCCC


Propagation of Indeterminate Errors• For powers:

• Naively, you might think that xn = xxx…• But the errors in each x are not independent – the

error in x is the same for each occurrence

x

nS

y

S

xy

xy

n


Propagation of Indeterminate Errors• Going back to our original question:• What is the indeterminate error in the composite

measurement?

• This is a linear combination of the form

• So in this case, k0 = 0, k1 = 1, k2 = -1, k3..n = 0 and x = m

vesselmaterialvesselmaterial mmm

nn xkxkxkxkky 3322110



• So, given that:

• Then since

• So

2233

222

211 nny SkSkSkSkS

22

21

22

21

222

211

.1.1

mm

mm

mmm

SS

SS

SkSkS

21 mmm


Propagation of Indeterminate Errors• In general, if using the same instrument to perform the

measurements, then:

• So

• That is, the error in the final mass measurement is larger than the individual errors in the mass measurements

212 mm SS

21 mm SS



• In our example, the mass measurement had an indeterminate error of 0.012g

• If we perform a subtractive measurement of mass, the indeterminate error in the result will be:

• Result is rounded to the same precision as the original data

g

SS mm

017.0

012.02

2

2

21

CHEN10011 Engineering Maths - Statistics

Paired/Unpaired Observations

• Repeated weighings of the same object allow us to determine the errors in the measuring instrument

• Repeated determinations allow us to determine the errors in the entire operation, including the measuring instrument

60


Paired/Unpaired Observations

• Repeated weighings: observations are unpaired• We cannot estimate the standard deviation by

subtracting pairs of empty and full observations, then calculate the standard deviation from these differences

61


Paired/Unpaired Observations• Example:

62

Observation Full weight (g)

Empty weight (g)

Difference (g)

1 12.465 9.965 2.500

2 12.466 9.966 2.500

3 12.482 9.982 2.500

4 12.468 9.968 2.500

5 12.481 9.981 2.500

6 12.481 9.981 2.500

7 12.469 9.969 2.500

8 12.472 9.972 2.500

9 12.477 9.977 2.500

10 12.473 9.973 2.500

Mean 12.473 9.973 2.500

Standard deviation 0.006 0.006 0.000


Paired/Unpaired Observations• We can see that the standard deviation of the

measuring instrument is 0.006 g• The standard deviation of the paired differences is

zero.• This cannot be correct• This is because of the false pairing

63


Paired/Unpaired Observations• If we take a different order of pairings:

64

Observation Full weight (g)

Empty weight (g)

Difference (g)

1 12.465 9.965 2.500

2 12.466 9.966 2.500

3 12.482 9.968 2.514

4 12.468 9.969 2.499

5 12.481 9.972 2.509

6 12.481 9.973 2.508

7 12.469 9.969 2.500

8 12.472 9.977 2.495

9 12.477 9.981 2.496

10 12.473 9.981 2.492

Mean 12.473 9.982 2.501

Standard deviation 0.006 0.006 0.007


Paired/Unpaired Observations• This indicates that the pairing can give us wildly

varying estimates of the standard deviation• We should use:

65

g

SSS mmm

0085.0

006.0006.0 22

22

21


Distributions

• The set of all possible observations is the population

• The measurements we take are a sample of the population

• If there are no determinate errors then the mean of the population is the true value of the mass

• The mean of the population is denoted by


Distributions• Similarly the standard deviation of the population

would be a measure of the true distribution• The population standard deviation is denoted by

may be zero – for example, the concentration of

an analyte in a given material has a single true value

• Other populations will have non-zero , for example the height of chemical engineering undergraduates


Distributions

• The true mean of a distribution is given by the symbol • Is the maximum of the probability density function• Formal definition is given by:

• A measurement is an estimate of the probability density function of the observed variable.

dxxFx )(.


Distributions

• The position of the measurement is given by an estimate of the mean:– is an estimate for .

• The shape of the distribution is provided by an estimate for the standard deviation:– S is an estimate for .

x


Distributions• As more observations are made of a variable so a

continuous distribution begins to be defined• When an infinite number of observations are

available, the distribution is completely defined• The probability density function (PDF) is the

normalised distribution curve• The area under the PDF is equal to 1

P x F x dx( ) ( ).

1


Distributions

• That is, there is a probability of 1 that the variable will lie between ±∞

• The probability that the variable will lie between x and x+∆x is given by:

P x x F x dxx

x x

( ) ( ).


Distributions

x

-4 -2 0 2 4

Fre

qu

en

cy

0.0

0.1

0.2

0.3

0.4

0.5

x x+x

Area under the curve

is the probability that

the variable is in the range x to x+∆x


Distributions• There are many types of probability density function, but

the three most important are:– Normal distribution, also known as a Gaussian distribution– Poisson distribution, also known as a Stochastic distribution– Binomial distribution

• Normal distributions are common in physical measurements


Normal Distribution

• A normal distribution with mean μ and standard deviation σ is defined by:

f x e

x( )

1

2 2

2

2


Normal Distribution

x

-4 -2 0 2 4

Fre

qu

en

cy

0.0

0.1

0.2

0.3

0.4

0.5


Normal Distribution

• For the normal distribution, the area under the curve bounded by: 1 will contain approximately 68.3% of the population – 2 will contain approximately 95.5% of the population– 3 will contain approximately 99.7% of the population


Normal Distribution

x

-4 -2 0 2 4

Fre

quen

cy

0.0

0.1

0.2

0.3

0.4

0.5

x

-4 -2 0 2 4

Fre

quen

cy

0.0

0.1

0.2

0.3

0.4

0.5

x

-4 -2 0 2 4

Fre

quen

cy

0.0

0.1

0.2

0.3

0.4

0.5

~68.3% ~95.5% ~99.7%

1

2

3


Log-normal Distribution

• Encountered in many different situations• Examples include:

– Antibody concentration in blood serum– Aerosol particle size distribution

• In these cases, the variable being measured cannot go below zero

• Can be converted to a normal distribution by taking the log of the variable


Log-normal Distribution

Log(x)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

y

0.0

0.1

0.2

0.3

0.4

0.5

x

0 2 4 6 8 10 12

y

0.0

0.1

0.2

0.3

0.4

0.5

log(x)


Student’s t-distribution• The derivation of the t-distribution was first published in

1908 by William Sealy Gosset• He worked at the Guinness Brewery in Dublin• He was not allowed to publish under his own name, so the

paper was written under the pseudonym Student.• The t-test and the associated theory became well-known

through the work of R.A. Fisher, who called the distribution "Student's distribution".


Student’s t-distribution• Gossett studied the distribution of:

• And showed it was of the form:

• Which is independent of μ and σ• No estimate of σ is required, S can be used

nS

xT

n

n

2/)1(2 )/1()2/(

)2/)1(()(

xxf


Properties of the t Distribution• It tends to the normal distribution for high numbers

of degrees of freedom• At low degrees of freedom the tails of the t

distribution are higher than the normal distribution


Properties of the t Distribution

x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 1



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 2



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 3



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 4



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 5



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 6



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 7



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 8



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 9



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 10



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 15



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 20



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 25



x

-4 -2 0 2 4

Pro

bab

ility

0.0

0.1

0.2

0.3

0.4

= 30


Sampling Distribution• If a limited set of observations is made on a variable:• A range of values is obtained and from these the

sample mean and standard distribution can be obtained. – It is unlikely that the sample mean is equal to the true

value for the mean – It is unlikely that the sample standard deviation S is equal

to the true standard deviation .


Sampling Distribution

• If another set of readings is taken:• This will give new values for the sample mean and

standard deviation • It is unlikely that these new values obtained would

agree with those obtained from the first set of observations.


Sampling Distribution• If this process is repeated a distribution for x is

obtained • This is the sampling distribution of the mean• It has a mean equal to that of the original population• Its standard deviation is different and is referred to

as the standard error of the mean (SEM) and is defined by:

SEMn


Central Limit Theorem

• The Central Limit Theorem states that if the sum of many independent random variables has a finite variance, then it will be approximately normally distributed

• That is, it will have a normal or Gaussian distribution).



• Since measured variables are often the sum of many independent random variables, normal distributions are common


0.0

0.2

0.4

0.6

0.8

1.0

1.2

24

6

8

100

2

4

6

8

10

x

n




• See also:– http://www.vias.org/simulations/simusoft_cenlimit.html– http://www.socr.ucla.edu/htmls/SOCR_Experiments.html– http://www.statisticalengineering.com/central_limit_theor

em.htm


Accuracy and Precision• Accuracy is defined by the trueness of a

measurement. • An accurate measurement is one which produces

a value for equal to , without any systematic error.

x


Accuracy and Precision• A precise measurement produces a value for S

that is as close to zero as possible• Indeterminate error may arise from the variable

under observation as well as the measurement• In this case, a precise measurement would

produce a standard deviation (S) as close as possible to the standard deviation () of the observed variable


Accuracy and Precision• Precision and accuracy describe two different

properties of a measurement• Accuracy refers to measurements of central

tendency• Precision refers to measurements of dispersion• ACCURACY DOES NOT IMPLY PRECISION• PRECISION DOES NOT IMPLY ACCURACY


Accuracy and Precision

Observed Value

-15 -10 -5 0 5 10 15

No

rma

lise

d N

um

be

r o

f O

bse

rva

tio

ns

0.0

0.1

0.2

0.3

0.4

0.5

A

B

C

D

A: Precise and accurateB: Imprecise and accurateC: Precise and inaccurateD: Imprecise and inaccurate


Accuracy and Precision• Accuracy is tested by calibration and validation

methods• E.g. through the analysis of certified national and

international standards• Precision (indeterminate error) is best reported

through the use of confidence limits at a specified level of probability


Reporting of Measurements• Precision is best reported through the use of

confidence limits at a specified level of probability• Explicitly states that indeterminate errors are being

reported• A clear and straightforward format for reporting

confidence limits would be:• at the P% confidence limit for n measurementsx E

• Precision is best reported through the use of confidence limits at a specified level of probability

• Explicitly states that indeterminate errors are being reported

• A clear and straightforward format for reporting confidence limits would be:

• at the P% confidence limit for n measurements


Confidence Limits• Definition of a range within which we have a certain

confidence we will find the true value• This assumes an absence of systematic errors• For large samples we can assume that the sample

standard deviation S is a good estimate for the population standard deviation, σ


Confidence Limits

• Using the sampling distribution of the mean, we find that for 95% confidence level:

• Or,

n

Sx

n

Sx 96.196.1

n

Sx 96.1


Confidence Limits• For small samples, S is unlikely to be a good

estimate for σ• We use Student’s t distribution to allow us to use S

in place of σ:

n

Stx nP 1,


Confidence Limits• For our previous measurements of mass,

• We use the two-tailed value of t at the 95% confidence level to estimate our confidence limits:

g

g

n

Stx nP

056.0475.124721.4

012.0093.2475.12

1,

20 ,012.0 ,475.12 ngSgx

tsmeasuremen 20for level confidence 95% at the


T Tableα 0.05 0.025 0.005 0.0025 0.0005 0.00025

2α 0.1 0.05 0.01 0.005 0.001 0.0005ν 1 6.314 12.706 63.657 127.321 636.619 1273.2392 2.920 4.303 9.925 14.089 31.599 44.7053 2.353 3.182 5.841 7.453 12.924 16.3264 2.132 2.776 4.604 5.598 8.610 10.3065 2.015 2.571 4.032 4.773 6.869 7.9766 1.943 2.447 3.707 4.317 5.959 6.7887 1.895 2.365 3.499 4.029 5.408 6.0828 1.860 2.306 3.355 3.833 5.041 5.6179 1.833 2.262 3.250 3.690 4.781 5.291

10 1.812 2.228 3.169 3.581 4.587 5.04911 1.796 2.201 3.106 3.497 4.437 4.86312 1.782 2.179 3.055 3.428 4.318 4.71613 1.771 2.160 3.012 3.372 4.221 4.59714 1.761 2.145 2.977 3.326 4.140 4.49915 1.753 2.131 2.947 3.286 4.073 4.41716 1.746 2.120 2.921 3.252 4.015 4.34617 1.740 2.110 2.898 3.222 3.965 4.28618 1.734 2.101 2.878 3.197 3.922 4.23319 1.729 2.093 2.861 3.174 3.883 4.18720 1.725 2.086 2.845 3.153 3.850 4.14625 1.708 2.060 2.787 3.078 3.725 3.99630 1.697 2.042 2.750 3.030 3.646 3.90240 1.684 2.021 2.704 2.971 3.551 3.78850 1.676 2.009 2.678 2.937 3.496 3.723∞ 1.645 1.960 2.576 2.807 3.291 3.481


T Table

• See also:– Statistical Tables, J. Murdoch and J.A. Barnes, 4th Edition,

McMillan, ISBN 0333558596– Table 7, page 17


T Table

• Generated using the Excel TINV function:– TINV(p, ν)– p is the probability associated with the two-tailed Student's t-

distribution– ν is the number of degrees of freedom– Use 2p to obtain the one-tailed value


Statistical Tests• Presence of indeterminate error means it is unlikely

that:– The mean of one sample will agree exactly with the

mean of a second sample– Means generated using different methods will agree– Variances of two samples will be the same

• We can use statistical tests to establish the probability that these differences are a result of indeterminate error


Statistical Tests• Null Hypothesis (H0)• An exact statement of something we initially

suppose to be true• For example, we may propose that the means of two

samples are the same, and that any observed difference is a result of random error:

210 : xxH


Statistical Tests• The alternate hypothesis not simply the opposite of

H0

• In this case, there are three possibilities:

• The first two alternate hypotheses are one-tailed• The third alternate hypothesis is two-tailed

211

211

211

:

:

:

xxH

xxH

xxH


Statistical Tests• One-tailed tests:

– Direction of difference is important• Two-tailed tests:

– Direction of difference is unimportant• Example:

– Drug trials – H0 is typically that the drug has no effect

– H1 is that the drug produces an improvement in the condition treated (does not make it worse!)

– This is a one-tailed test


Statistical Tests

• Example:– Validation of one measurement method against another– H0 is typically that the two means are the same

– H1 is that the means are different– This is a two-tailed test


Statistical Tests

• Confidence level (P)– Usually expressed as a percentage– Typically 95%, 99%, 99.9%– Is a measure of how likely it is that the observed

differences are NOT a result of indeterminate error– For 95% confidence level, there is a 1 in 20 chance that

the difference is a result of indeterminate error


Statistical Tests• Confidence level (P)

– Related to the probability (α)– For a one tailed test

– For a two tailed test

%1001or %100)1(

PP

2%100

1or %100)21(

P

P


Statistical Tests

P (%) α90 0.1095 0.0599 0.01

99.9 0.001

P (%) α90 0.0595 0.02599 0.005

99.9 0.0005

One-tailed Two-tailed


Statistical Tests• Possible errors:

– Type 1: rejection of the null hypothesis even though it is in fact true

– Type 2: acceptance of the null hypothesis even though it is false


Statistical Tests

Type I errorType II error

xc

Sampling distribution of mean if null

hypothesis is true

Sampling distribution of

mean if alternate hypothesis is true


Statistical Tests• Reducing the chance of a Type I error increases the

chance of a Type II error• Reducing the chance of a Type II error increases the

chance of a Type I error• Only by increasing the sample size can we reduce

the chances of both types of error• This is because we then reduce the standard error

of the mean


Statistical Tests


xc


xc

Increased number of observations


Testing for Normality• Parametric statistical tests assume that the data are

described by a particular distribution• This is most often the normal (or Gaussian) distribution• The Central Limit Theorem shows that normal

distributions are common• We may need to test if our data are normally distributed

before performing further statistical tests


Testing for Normality

• We can use the Kolmogorov test (also called the Kolmogorov-Smirnov test)

• This test can be used for any distribution that has a known cumulative distribution function

• The null hypothesis is that the sample comes from the hypothesized distribution (the normal distribution in this case)



• To do this, we need to know the population mean µ and the population standard deviation σ

• We cannot use the sample mean and standard deviation S in this case

• The null hypothesis in this case is that all the observations are drawn from a normal distribution with mean µ and standard deviation σ

x


Testing for Normality• For the normal distribution, the cumulative distribution function

is given by:

• where

2

12

1

x

erfC

97531

)12(

0

16

105

8

15

4

3

2

11

2

!!1211

2

2

xxxxxe

xne

xerf

x

n

nn

nx


Testing for Normality• Tests if the data sample is compatible with being a

random sampling from a given distribution• In this case, the assumed distribution is normal• First, we calculate the fractional cumulative

frequency distribution for our sample

1

n

frequencycumulativeNS


Testing for Normality• Note we use n+1 as the divisor, not n to ensure that the

50% cumulative frequency falls in the middle of the sample• This is because the mid-point with the divisor n is:

• If we use the divisor n+1:

n

nnn

ncentre2

1

2

1

5.0)1(2

1

211

1

n

nnn

ncentre


Testing for Normality• To make the test easier, we normalise the variable

we are sampling:

• This has the effect of converting the variable to a normal distribution with:– Mean = 0 – Standard deviation = 1

• This is referred to as N(0,1)

Standard normal value x x

Si


Testing for Normality• We then find:

• for all x• Where F(x) is the expected cumulative frequency for x• We compare DN against critical values from tables

• If the DN value is greater than the critical value, we can reject the null hypothesis

• We can say that our sample does not have the same distribution as F(x)

)()(max xFxSD NN


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.46410.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.42470.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.38590.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.34830.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.31210.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.27760.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.24510.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.21480.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.18670.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.16111.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.13791.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.11701.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.09851.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.08231.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.06811.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.05591.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.04551.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.03671.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.02941.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.02332.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.01832.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.01432.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.01102.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.00842.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.00642.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.00482.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.00362.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.00262.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.00192.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.00143.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.00103.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.00073.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.00053.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.00033.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002

Areas in the upper tail of the Normal distribution


Using Table 3• If the SNV is negative:

– Ignore the sign– Look up the row corresponding to the first two digits X.X– Look up the column corresponding to the third digit 0.0X– The value at the intersection is the correct value

• If the SNV is positive:– Look up the value as before– Subtract the table value from 1– This is the correct value


Using Table 3• This is because we want the area in the lower tail,

but the values tabulated are the upper tail• Since the total area is 1, the area in the lower tail is

1 – area in lower tail

x

-4 -2 0 2 4

Fre

quen

cy

0.0

0.1

0.2

0.3

0.4

0.5

Tabulated value

1 - Tabulated value


Cumulative Frequency for the Normal Distribution• Table generated using Excel• Using the NORMDIST function:

– NORMDIST(x, mean, s, cumulative)– x is the value for which you want the distribution– mean is the arithmetic mean of the distribution– s is the standard deviation of the distribution– cumulative is true (1) if the cumulative value is required,

otherwise false (0) if the probability is required


Cumulative Frequency for the Normal Distribution• Can also use NORMSDIST:

– NORMSDIST(x)– x is the value for which you want the distribution– Returns the cumulative frequency at x for a standard normal

distribution (mean = 0, standard deviation = 1)

• This is the same as:– NORMDIST(x, 0, 1, 1)


Cumulative Frequency for the Normal Distribution• Firstly, we will test to see if the observations are consistent

with being drawn from a normal distribution with a mean of 12.480 g and a standard deviation of 0.010 g

• In this case, the null hypothesis is that the observations are drawn from a normal distribution with µ = 12.480 g and σ = 0.010 g

• We calculate the SNV as:010.0

480.12ix


Testing for NormalityValue Standard

Normal Value (normalised to N(0,1))

Number of occurrences

Cumulative frequency

Fractional cumulative frequency

Expected cumulative frequency

|expected-actual| cumulative frequency

12.450 -3 1 1 0.04760 0.00135 0.04625

12.465 -1.5 2 3 0.14290 0.06681 0.07609

12.466 -1.4 1 4 0.19050 0.08076 0.10974

12.468 -1.2 1 5 0.23810 0.11507 0.12303

12.469 -1.1 1 6 0.28570 0.13567 0.15003

12.472 -0.8 1 7 0.33330 0.21186 0.12144

12.473 -0.7 2 9 0.42860 0.24196 0.18664

12.474 -0.6 1 10 0.47620 0.27425 0.20195

12.475 -0.5 3 13 0.61900 0.30854 0.31046

12.477 -0.3 1 14 0.66670 0.38209 0.28461

12.481 0.1 2 16 0.76190 0.53983 0.22207

12.482 0.2 1 17 0.80950 0.57926 0.23024

12.485 0.5 2 19 0.90480 0.69146 0.21334

12.513 3.3 1 20 0.95240 0.99952 0.04712


Weight (g)

12.42 12.44 12.46 12.48 12.50 12.52 12.54

Fra

ctio

nal C

umul

ativ

e F

requ

ency

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Observed Fractional Cumulative FrequencyExpected Cumulative Frequency

146


DN


Table of Kologorov-Smirnov Critical Valuesn α=0.20 α=0.15 α=0.10 α=0.05 α=0.01

1 0.900 0.925 0.950 0.975 0.995

2 0.684 0.726 0.776 0.842 0.929

3 0.565 0.597 0.642 0.708 0.828

4 0.494 0.525 0.564 0.624 0.733

5 0.446 0.474 0.510 0.565 0.669

6 0.410 0.436 0.470 0.521 0.618

7 0.381 0.405 0.438 0.486 0.577

8 0.358 0.381 0.411 0.457 0.543

9 0.339 0.360 0.388 0.432 0.514

10 0.322 0.342 0.368 0.410 0.490

11 0.307 0.326 0.352 0.391 0.468

12 0.295 0.313 0.338 0.375 0.450

13 0.284 0.302 0.325 0.361 0.433

14 0.274 0.292 0.314 0.349 0.418

15 0.266 0.283 0.304 0.338 0.404


Table of Kologorov-Smirnov Critical Valuesn α=0.20 α=0.15 α=0.10 α=0.05 α=0.01

16 0.258 0.274 0.295 0.328 0.392

17 0.250 0.266 0.286 0.318 0.381

18 0.244 0.259 0.278 0.309 0.371

19 0.237 0.252 0.272 0.301 0.363

20 0.231 0.246 0.264 0.294 0.356

25 0.210 0.220 0.240 0.270 0.320

30 0.190 0.200 0.220 0.240 0.290

35 0.180 0.190 0.210 0.230 0.270

16 0.258 0.274 0.295 0.328 0.392

17 0.250 0.266 0.286 0.318 0.381

18 0.244 0.259 0.278 0.309 0.371

19 0.237 0.252 0.272 0.301 0.363

20 0.231 0.246 0.264 0.294 0.356

25 0.210 0.220 0.240 0.270 0.320

30 0.190 0.200 0.220 0.240 0.290


Table of Kologorov-Smirnov Critical Values

n α=0.20 α=0.15 α=0.10 α=0.05 α=0.01

35 0.180 0.190 0.210 0.230 0.270

>35n

22.1

n

14.1

n

36.1

n

07.1

n

63.1


Testing for Normality• Our null hypothesis is that the observations are consistent

with being drawn from a normal distribution with a mean of 12.480 g and a standard deviation of 0.010 g

• For 20 samples and 95% confidence level, the critical value is:– 0.294– Our DN is 0.3105


Testing for Normality• We thus reject the null hypothesis and accept the alternate

hypothesis that the observations are not normally distributed with a mean of 12.480 g and a standard deviation of 0.010 g

• We cannot say from this if there are any other normal distributions that would be consistent with these observations


Testing for Normality• What do we do if we do not know the distribution

parameters in advance?• We can estimate them from the sample mean and sample

standard deviation• In this case, the null hypothesis is that the observations are

consistent with being drawn from a normal distribution with µ = and σ = S

• We perform the test in a similar way, but use the sample mean and standard deviation to calculate the SNV

x


Testing for Normality• This is Lilliefors’ variant of the K-S test• Because the distribution has been estimated from

the data:– The hypothesized distribution is closer to the original

data– The maximum difference will be smaller– A new table of critical values is required









12.450 -2.0679 1 1 0.0476 0.0193 0.0283

12.465 -0.8272 2 3 0.1429 0.2041 0.0612

12.466 -0.7444 1 4 0.1905 0.2283 0.0378

12.468 -0.5790 1 5 0.2381 0.2813 0.0432

12.469 -0.4963 1 6 0.2857 0.3098 0.0241

12.472 -0.2481 1 7 0.3333 0.4020 0.0688

12.473 -0.1654 2 9 0.4286 0.4343 0.0057

12.474 -0.0827 1 10 0.4762 0.4670 0.0092

12.475 0.0000 3 13 0.6190 0.5000 0.1190

12.477 0.1654 1 14 0.6667 0.5657 0.1010

12.481 0.4963 2 16 0.7619 0.6902 0.0717

12.482 0.5790 1 17 0.8095 0.7187 0.0908

12.485 0.8272 2 19 0.9048 0.7959 0.1089

12.513 3.1432 1 20 0.9524 0.9992 0.0468



Normalised Value

-4 -3 -2 -1 0 1 2 3 4

Fra

ctio

na

l Cu

mu

lativ

e F

req

ue

ncy

0.00

0.25

0.50

0.75

1.00

DN


Lilliefors Critical Valuesn α=0.20 α=0.15 α=0.10 α=0.05 α=0.01

4 0.3027 0.3216 0.3456 0.3754 0.4129

5 0.2893 0.3027 0.3188 0.3427 0.3959

6 0.2694 0.2816 0.2982 0.3245 0.3728

7 0.2521 0.2641 0.2802 0.3041 0.3504

8 0.2387 0.2502 0.2649 0.2875 0.3331

9 0.2273 0.2382 0.2522 0.2744 0.3162

10 0.2171 0.2273 0.2410 0.2616 0.3037

11 0.2080 0.2179 0.2306 0.2506 0.2905

12 0.2004 0.2101 0.2228 0.2426 0.2812

13 0.1932 0.2025 0.2147 0.2337 0.2714

14 0.1869 0.1959 0.2077 0.2257 0.2627

15 0.1811 0.1899 0.2016 0.2196 0.2545

16 0.1758 0.1843 0.1956 0.2128 0.2477

17 0.1711 0.1794 0.1902 0.2071 0.2408

18 0.1666 0.1747 0.1852 0.2018 0.2345

19 0.1624 0.1700 0.1803 0.1965 0.2285

20 0.1589 0.1666 0.1764 0.1920 0.2226


Lilliefors Critical Valuesn α=0.20 α=0.15 α=0.10 α=0.05 α=0.01

21 .1553 .1629 .1726 .1881 .2190

22 .1517 .1592 .1690 .1840 .2141

23 .1484 .1555 .1650 .1798 .2090

24 .1458 .1527 .1619 .1766 .2053

25 .1429 .1498 .1589 .1726 .2010

26 .1406 .1472 .1562 .1699 .1985

27 .1381 .1448 .1533 .1665 .1941

28 .1358 .1423 .1509 .1641 .1911

29 .1334 .1398 .1483 .1614 .1886

30 .1315 .1378 .1460 .1590 .1848

31 .1291 .1353 .1432 .1559 .1820

32 .1274 .1336 .1415 .1542 .1798

33 .1254 .1314 .1392 .1518 .1770

34 .1236 .1295 .1373 .1497 .1747

35 .1220 .1278 .1356 .1478 .1720


Testing for Normality• Our null hypothesis is that the observations are consistent

with a normal distribution having a mean of 12.475 g and a standard deviation of 0.012 g

• For 20 samples and 95% confidence level, the critical value is:– 0.192– Our DN is 0.119

• We conclude that, at the 95% confidence level, there is no significant deviation from a normal distribution with a mean of 12.475 g and standard deviation of 0.012 g.


Testing for Normality• Limitations of the K-S and Lilliefors’ test:

– Distribution must be continuous– Most sensitive near the centre of the distribution– Test is weaker if the distribution parameters are

estimated from the data (as in our example)


Testing for Normality• These tests do not tell us which normal distribution

fits the observations best• That is, which distribution gives the smallest sum of

squares of the residuals


Testing for Normality• In this case, the best fit (r = 0.9919) is obtained with

µ = 12.4741 g and σ = 0.0085613 g:

Weight (g)

12.42 12.44 12.46 12.48 12.50 12.52 12.54

Fra

ctio

nal C

umul

ativ

e F

requ

enc

y

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Observed Fractional Cumulative FrequencyExpected Cumulative Frequency









12.45 -2.8149931 1 1 0.04760 0.00244 0.04516

12.465 -1.0629227 2 3 0.14290 0.14391 0.00101

12.466 -0.946118 1 4 0.19050 0.17204 0.01846

12.468 -0.7125086 1 5 0.23810 0.23807 0.00003

12.469 -0.5957039 1 6 0.28570 0.27569 0.01001

12.472 -0.2452899 1 7 0.33330 0.40312 0.06982

12.473 -0.1284852 2 9 0.42860 0.44888 0.02028

12.474 -0.0116805 1 10 0.47620 0.49534 0.01914

12.475 0.10512422 3 13 0.61900 0.54186 0.07714

12.477 0.3387336 1 14 0.66670 0.63259 0.03411

12.481 0.80595237 2 16 0.76190 0.78986 0.02796

12.482 0.92275706 1 17 0.80950 0.82193 0.01243

12.485 1.27317113 2 19 0.90480 0.89852 0.00628

12.513 4.54370248 1 20 0.95240 1.00000 0.04760


Testing for Normality• In this case, the maximum DN is only 0.07714• This is way below the critical value for either the K-

S or Lilliefors’ test• This is what we would expect• The larger sample mean (12.475 g) and standard

deviation (0.012 g) are a result of the one possible outlier (12.513 g)


Testing for Normality• The best fit is obtained with:

– µ = 12.4741 g and σ = 0.0085613 g• The median and σ derived from the MAD are:

– Median = 12.475 g and σ = 0.00964 g• The σ from MAD is much closer to the best fit value

than S (0.012 g)• This is because it is not affected by outliers


Testing for Normality• We can see the space of all normal distributions

with this monte carlo approach:


Statistical Tests• F Tests

– Used for the comparison of standard deviations of samples– Used to determine:– If one set of data is more precise (a one- tailed test)– If two sets are different in their precision (a two-tailed test)– The null hypothesis is that there is no difference in precision

(H0: S1 = S2)


Statistical Tests

• The F-test looks at the ratio of two sample variances:

• S1 and S2 are chosen such that F 1• The resulting F statistic is compared with a critical

value from tables

FS

S 1

2

22


Statistical Tests

• The critical values for F are determined by– The numbers of observations in each of the two

samples, n1 and n2

– The confidence level – The type of test performed

• The degrees of freedom for an F-test are given by n1-1 and n2-1


Statistical Tests• The degrees of freedom are n-1, not n• This is because degrees of freedom refers to the

number of independent deviations used to calculate S

• Since we know S, when we have n-1 deviations we can deduce the final deviation

• This is because

xxi

01

n

ii xx


Statistical Tests

• Tables of the F distribution are quite cumbersome• This is because they require a separate table for

each confidence level


Statistical Tests• A different experimental worker repeated the

measurement in Example 1. The data obtained were:– x2=12.501g, S2 = 0.019g, n2 = 5

• The original data were:– x1=12.475g, S1 = 0.012g, n1 = 20

• The null hypothesis adopted is:– – That is, there is no significant difference in the variance

between the two samples at the 95% confidence level–

22

210 : SSH


Statistical Tests• We calculate the F statistic:

• At the 95% confidence level, degrees of freedom 4 and 19, two-tailed test the critical value is– Fcrit = 3.56

51.21044.1

1061.3

012.0

019.0

4

4

2

2

F


Statistical Tests

• Since our calculated F statistic is less than the critical value:– We accept the null hypothesis that the variances are the

same at the 95% confidence level


Statistical Tests• F Table, one-tailed, 95% confidence level (α = 0.05)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88 242.98 243.91 244.69 245.36 245.95 246.46 246.92 247.32 247.69 248.012 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.40 19.41 19.42 19.42 19.43 19.43 19.44 19.44 19.44 19.453 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.76 8.74 8.73 8.71 8.70 8.69 8.68 8.67 8.67 8.664 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.94 5.91 5.89 5.87 5.86 5.84 5.83 5.82 5.81 5.805 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.70 4.68 4.66 4.64 4.62 4.60 4.59 4.58 4.57 4.566 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.03 4.00 3.98 3.96 3.94 3.92 3.91 3.90 3.88 3.877 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.60 3.57 3.55 3.53 3.51 3.49 3.48 3.47 3.46 3.448 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.31 3.28 3.26 3.24 3.22 3.20 3.19 3.17 3.16 3.159 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94

10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.94 2.91 2.89 2.86 2.85 2.83 2.81 2.80 2.79 2.7711 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.82 2.79 2.76 2.74 2.72 2.70 2.69 2.67 2.66 2.6512 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.72 2.69 2.66 2.64 2.62 2.60 2.58 2.57 2.56 2.5413 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.63 2.60 2.58 2.55 2.53 2.51 2.50 2.48 2.47 2.4614 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.57 2.53 2.51 2.48 2.46 2.44 2.43 2.41 2.40 2.3915 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.51 2.48 2.45 2.42 2.40 2.38 2.37 2.35 2.34 2.3316 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.46 2.42 2.40 2.37 2.35 2.33 2.32 2.30 2.29 2.2817 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.41 2.38 2.35 2.33 2.31 2.29 2.27 2.26 2.24 2.2318 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.37 2.34 2.31 2.29 2.27 2.25 2.23 2.22 2.20 2.1919 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.34 2.31 2.28 2.26 2.23 2.21 2.20 2.18 2.17 2.1620 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.31 2.28 2.25 2.22 2.20 2.18 2.17 2.15 2.14 2.12

Numerator degrees of freedom (ν1)

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Statistical Tests• F Table, two-tailed, 95% confidence level (α = 0.05)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 647.79 799.50 864.16 899.58 921.85 937.11 948.22 956.66 963.28 968.63 973.03 976.71 979.84 982.53 984.87 986.92 988.73 990.35 991.80 993.102 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.41 39.41 39.42 39.43 39.43 39.44 39.44 39.44 39.45 39.453 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.37 14.34 14.30 14.28 14.25 14.23 14.21 14.20 14.18 14.174 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.79 8.75 8.71 8.68 8.66 8.63 8.61 8.59 8.58 8.565 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.57 6.52 6.49 6.46 6.43 6.40 6.38 6.36 6.34 6.336 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.41 5.37 5.33 5.30 5.27 5.24 5.22 5.20 5.18 5.177 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.71 4.67 4.63 4.60 4.57 4.54 4.52 4.50 4.48 4.478 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.24 4.20 4.16 4.13 4.10 4.08 4.05 4.03 4.02 4.009 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.91 3.87 3.83 3.80 3.77 3.74 3.72 3.70 3.68 3.67

10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.66 3.62 3.58 3.55 3.52 3.50 3.47 3.45 3.44 3.4211 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53 3.47 3.43 3.39 3.36 3.33 3.30 3.28 3.26 3.24 3.2312 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.32 3.28 3.24 3.21 3.18 3.15 3.13 3.11 3.09 3.0713 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31 3.25 3.20 3.15 3.12 3.08 3.05 3.03 3.00 2.98 2.96 2.9514 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21 3.15 3.09 3.05 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.8415 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06 3.01 2.96 2.92 2.89 2.86 2.84 2.81 2.79 2.77 2.7616 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05 2.99 2.93 2.89 2.85 2.82 2.79 2.76 2.74 2.72 2.70 2.6817 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98 2.92 2.87 2.82 2.79 2.75 2.72 2.70 2.67 2.65 2.63 2.6218 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93 2.87 2.81 2.77 2.73 2.70 2.67 2.64 2.62 2.60 2.58 2.5619 5.92 4.51 3.90 3.56 3.33 3.17 3.05 2.96 2.88 2.82 2.76 2.72 2.68 2.65 2.62 2.59 2.57 2.55 2.53 2.5120 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.72 2.68 2.64 2.60 2.57 2.55 2.52 2.50 2.48 2.46


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Statistical Tests• F Table, one-tailed, 99% confidence level (α = 0.01)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 4052.2 4999.5 5403.4 5624.6 5763.6 5859.0 5928.4 5981.1 6022.5 6055.8 6083.3 6106.3 6125.9 6142.7 6157.3 6170.1 6181.4 6191.5 6200.6 6208.72 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.41 99.42 99.42 99.43 99.43 99.44 99.44 99.44 99.45 99.453 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.13 27.05 26.98 26.92 26.87 26.83 26.79 26.75 26.72 26.694 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.45 14.37 14.31 14.25 14.20 14.15 14.11 14.08 14.05 14.025 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.96 9.89 9.82 9.77 9.72 9.68 9.64 9.61 9.58 9.556 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.79 7.72 7.66 7.60 7.56 7.52 7.48 7.45 7.42 7.407 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.54 6.47 6.41 6.36 6.31 6.28 6.24 6.21 6.18 6.168 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.73 5.67 5.61 5.56 5.52 5.48 5.44 5.41 5.38 5.369 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.18 5.11 5.05 5.01 4.96 4.92 4.89 4.86 4.83 4.81

10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.77 4.71 4.65 4.60 4.56 4.52 4.49 4.46 4.43 4.4111 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.46 4.40 4.34 4.29 4.25 4.21 4.18 4.15 4.12 4.1012 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.22 4.16 4.10 4.05 4.01 3.97 3.94 3.91 3.88 3.8613 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 4.02 3.96 3.91 3.86 3.82 3.78 3.75 3.72 3.69 3.6614 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03 3.94 3.86 3.80 3.75 3.70 3.66 3.62 3.59 3.56 3.53 3.5115 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.73 3.67 3.61 3.56 3.52 3.49 3.45 3.42 3.40 3.3716 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.62 3.55 3.50 3.45 3.41 3.37 3.34 3.31 3.28 3.2617 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.52 3.46 3.40 3.35 3.31 3.27 3.24 3.21 3.19 3.1618 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.43 3.37 3.32 3.27 3.23 3.19 3.16 3.13 3.10 3.0819 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.36 3.30 3.24 3.19 3.15 3.12 3.08 3.05 3.03 3.0020 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.29 3.23 3.18 3.13 3.09 3.05 3.02 2.99 2.96 2.94


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Statistical Tests• F Table, two-tailed, 99% confidence level (α = 0.01)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 16211 19999 21615 22500 23056 23437 23715 23925 24091 24224 24334 24426 24505 24572 24630 24681 24727 24767 24803 248362 198.50 199.00 199.17 199.25 199.30 199.33 199.36 199.37 199.39 199.40 199.41 199.42 199.42 199.43 199.43 199.44 199.44 199.44 199.45 199.453 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.88 43.69 43.52 43.39 43.27 43.17 43.08 43.01 42.94 42.88 42.83 42.784 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 21.14 20.97 20.82 20.70 20.60 20.51 20.44 20.37 20.31 20.26 20.21 20.175 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.77 13.62 13.49 13.38 13.29 13.21 13.15 13.09 13.03 12.98 12.94 12.906 18.63 14.54 12.92 12.03 11.46 11.07 10.79 10.57 10.39 10.25 10.13 10.03 9.95 9.88 9.81 9.76 9.71 9.66 9.62 9.597 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.51 8.38 8.27 8.18 8.10 8.03 7.97 7.91 7.87 7.83 7.79 7.758 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.34 7.21 7.10 7.01 6.94 6.87 6.81 6.76 6.72 6.68 6.64 6.619 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.54 6.42 6.31 6.23 6.15 6.09 6.03 5.98 5.94 5.90 5.86 5.83

10 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.97 5.85 5.75 5.66 5.59 5.53 5.47 5.42 5.38 5.34 5.31 5.2711 12.23 8.91 7.60 6.88 6.42 6.10 5.86 5.68 5.54 5.42 5.32 5.24 5.16 5.10 5.05 5.00 4.96 4.92 4.89 4.8612 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 5.20 5.09 4.99 4.91 4.84 4.77 4.72 4.67 4.63 4.59 4.56 4.5313 11.37 8.19 6.93 6.23 5.79 5.48 5.25 5.08 4.94 4.82 4.72 4.64 4.57 4.51 4.46 4.41 4.37 4.33 4.30 4.2714 11.06 7.92 6.68 6.00 5.56 5.26 5.03 4.86 4.72 4.60 4.51 4.43 4.36 4.30 4.25 4.20 4.16 4.12 4.09 4.0615 10.80 7.70 6.48 5.80 5.37 5.07 4.85 4.67 4.54 4.42 4.33 4.25 4.18 4.12 4.07 4.02 3.98 3.95 3.91 3.8816 10.58 7.51 6.30 5.64 5.21 4.91 4.69 4.52 4.38 4.27 4.18 4.10 4.03 3.97 3.92 3.87 3.83 3.80 3.76 3.7317 10.38 7.35 6.16 5.50 5.07 4.78 4.56 4.39 4.25 4.14 4.05 3.97 3.90 3.84 3.79 3.75 3.71 3.67 3.64 3.6118 10.22 7.21 6.03 5.37 4.96 4.66 4.44 4.28 4.14 4.03 3.94 3.86 3.79 3.73 3.68 3.64 3.60 3.56 3.53 3.5019 10.07 7.09 5.92 5.27 4.85 4.56 4.34 4.18 4.04 3.93 3.84 3.76 3.70 3.64 3.59 3.54 3.50 3.46 3.43 3.4020 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.96 3.85 3.76 3.68 3.61 3.55 3.50 3.46 3.42 3.38 3.35 3.32


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F Tables

• See also:– Statistical Tables, J. Murdoch and J.A. Barnes, 4th Edition,

McMillan, ISBN 0333558596– Table 9, pages 20-21– This is the one-tailed table where each entry contains the α=0.05,

0.025, 0.01 and 0.001 values


Statistical Tests

• Tables generated using Excel• Three functions used for F tests• FTEST(array1, array2)

– Returns the probability that the observed difference in standard deviations of array1 and array2 are a result of indeterminate error.

– Is always a two-tailed test in Excel


Statistical Tests• FDIST(x, v1, v2)

– For a variable x, returns the probability that a random variable with an F distribution and degrees of freedom v1 and v2 will be greater than x.

• FINV(α, v1, v2)– Inverse function to FDIST– Returns the value x corresponding to the probability α that a

random variable with an F distribution and degrees of freedom v1 and v2 will be greater than x


Statistical Tests• Both FDIST and FINV are one-tailed, so the

probability has to be changed if a two-tailed result is require

• Use half the required probability for the two-tailed value


Statistical Tests• t Tests

– Used to compare means– Three types of test:

• Comparison of a mean to reference value• Comparison of two means• Comparison of more than two means


Statistical Tests• Comparison of a mean to reference value• Remember that:

• So, rearranging:

n

Stx nP 1,

S

nxt nP 1,


Statistical Tests• Comparison of a mean to reference value

– Our null hypothesis is:–– We compare the calculated t value against a critical value from

tables– If the modulus of t is greater than the critical value then the null

hypothesis is rejected– Since we are only interested in determining if the means are

different, we use a two-tailed t-test– If we wanted to determine if one mean was greater than the other,

we would use a one-tailed test

xH :0


Statistical Tests

• The material weighed in Example 1 was obtained from a machine set to deliver 12.45g per operation – Is the unit operating outside its specification?– In this case the null hypothesis is that =12.45g, = 12.475g, n = 20, S = 0.012g

x x

t 12 475 12 4520

0 0129 317. .

..


Statistical Tests

• The critical value for t at the 95% confidence limit with 19 degrees of freedom is 2.093

• The calculated t-value is much greater, so:– The null hypothesis (that the means are equal) is

rejected– The alternative hypothesis (that the means are

different) is accepted


Statistical Tests

• Comparison of the Means of Two Samples• The t-test is used to test if the means of two

samples is significant• The null hypothesis is that the two means are not

significantly different• Firstly, we perform an F-test to see if the variances

are significantly different (two-tailed F-test)


Statistical Tests

• If the variances are not significantly different, the t value is calculated using:

• The degrees of freedom are given by:

t

x x

Sn n

1 2

1 2

1 1

n n1 2 2


Statistical Tests

• A “pooled” value for the standard deviation is used• This is a weighted average of the two sample

variances:

Sn S n S

n n2 1 1

22 2

2

1 2

1 1

2


Statistical Tests

• If the standard deviations are significantly different, a different calculation is used:

t

x x

S

n

S

n

1 2

12

1

22

2


Statistical Tests• The degrees of freedom are given by:

• Rounded to the nearest integer

2

11 2

2

2

22

1

2

1

21

2

2

22

1

21

n

nS

n

nS

nS

nS


Statistical Tests

• From the earlier mass measurements, the means were noticeably different

• Is this difference significant?• The null hypothesis is:• We have already performed the F-test and shown

that the standard deviations are not significantly different, so we use the first set of equations

210 : xxH


Statistical Tests

• Pooled standard deviation:

• T value:

g 0.01349

1082.1

2520

019.04012.019

24

222

S

g

S

854.3

51

201

01349.0

501.12475.12

t


Statistical Tests• (20 + 5 - 2 ) = 23 degrees of freedom• At the 95% confidence level (P = 0.05)• A two-tailed test (we have no a priori reason to

suppose that one mean will be greater than the other)

• The critical t value is is 2.069


Statistical Tests• Our calculated t statistic is larger

– We therefore reject the null hypothesis– We accept the alternate hypothesis that the means are

significantly different


Rejection of Data

• A set of data may contain outliers• These are points far away from the bulk of the

data• They may be a result of determinate error• Removing “inconvenient” results is wrong• Care must be taken when removing outliers


Rejection of Data

• If an outlier exists:• Check the the experimental procedures operating

at the time the datum was obtained.• Check records and observations to try to identify

the cause of the outlier• You may identify weaknesses in record keeping

and experimental procedures


Rejection of Data• If no determinate error is identified:• It is nevertheless highly unlikely that the outlier lies

on the probability density function• The expected frequency of such an observation is

so low that it would require a very large sample size to observe it

• Consequently such an outlier is more likely to contain determinate error, so the outlier may be rejected


Rejection of Data

• Note: The most important data are the ones that don't conform to existing models


Criteria for the Rejection of Data • Chauvanet’s criterion

– Null hypothesis is that all observations are from the same Normal distribution

– Remove outlier from set of observations– Recalculate mean and standard deviation– Calculate confidence limits at an appropriate

confidence level– If the presumed outlier is outside the confidence limits,

then we reject the null hypothesis


Criteria for the Rejection of Data • For our weight data:

– Removing the suspected outlier from the set leaves 19 observations with a mean of 12.473 g and a standard deviation of 0.00841 g.

– The confidence limits are given by:

g

g

n

Stx np

477.12469.12

004054.0473.1219

00841.0101.2473.12

1,


Criteria for the Rejection of Data • Where n´= n – 1 and n is the original number of

observations• We have used the 95% confidence level• Our presumed outlier is outside the confidence

limits• We reject the null hypothesis and accept the

alternate hypothesis that the observation is from another population – it is an outlier

• This test should only be used once per sample


Criteria for the Rejection of Data• Dixon’s (or Q) test• Works for small sample sizes• Null hypothesis is that all the data are from the same

Normal distribution• A rank difference ratio (Q statistic) is calculated that

depends on the sample size• This is compared to a critical value from tables


Criteria for the Rejection of DataRank Difference Ratio (Q statistic) n α = 0.10 α = 0.05 α = 0.01

3 0.886 0.941 0.9884 0.679 0.765 0.8895 0.557 0.642 0.7806 0.482 0.560 0.6987 0.434 0.507 0.6378 0.650 0.710 0.8299 0.594 0.657 0.776

10 0.551 0.612 0.72611 0.517 0.576 0.67912 0.490 0.546 0.64213 0.467 0.521 0.61514 0.448 0.501 0.59315 0.472 0.525 0.61616 0.454 0.507 0.59517 0.438 0.490 0.57718 0.424 0.475 0.56119 0.412 0.462 0.54720 0.401 0.450 0.535

or 1

1

1

12

xx

xx

xx

xx

n

nn

n

or 2

2

11

13

xx

xx

xx

xx

n

nn

n

or 3

2

12

13

xx

xx

xx

xx

n

nn

n


Criteria for the Rejection of Data• If the calculated Q value is larger than the critical

value, then the null hypothesis is rejected• The alternate hypothesis is accepted that the outlier

comes from a different distribution


Rejection of Data• Using the data from Example 1:

• The critical value for 20 observations is 0.45 at the 95% confidence level

• Our value exceeds this, so we can reject the null hypothesis

• We accept the alternate hypothesis that the point is an outlier

583.0465.12513.12

485.12513.12

3

2

xx

xxQ

n

nn


Criteria for the Rejection of Data • Grubb’s test

– ISO recommended method– The suspect value is that furthest away from the mean– Null hypothesis is that all measurements are from the

same population– We calculate:

S

xG

luesuspect va


Criteria for the Rejection of Data • Presence of an outlier increases both the

numerator and the denominator• The G statistic therefore cannot increase

indefinitely• In fact, G cannot exceed:

n

nG

1


Criteria for the Rejection of Data Critical values for Grubb’s test

n Gcrit,

= 0.05

Gcrit,

= 0.01

n Gcrit,

= 0.05

Gcrit,

= 0.01

n Gcrit,

= 0.05

Gcrit,

= 0.01

3 1.1543 1.1547 15 2.5483 2.8061 80 3.3061 3.6729

4 1.4812 1.4962 16 2.5857 2.8521 90 3.3477 3.7163

5 1.7150 1.7637 17 2.6200 2.8940 100 3.3841 3.7540

6 1.8871 1.9728 18 2.6516 2.9325 120 3.4451 3.8167

7 2.0200 2.1391 19 2.6809 2.9680 140 3.4951 3.8673

8 2.1266 2.2744 20 2.7082 3.0008 160 3.5373 3.9097

9 2.2150 2.3868 25 2.8217 3.1353 180 3.5736 3.9460

10 2.2900 2.4821 30 2.9085 3.2361 200 3.6055 3.9777

11 2.3547 2.5641 40 3.0361 3.3807 300 3.7236 4.0935

12 2.4116 2.6357 50 3.1282 3.4825 400 3.8032 4.1707

13 2.4620 2.6990 60 3.1997 3.5599 500 3.8631 4.2283

14 2.5073 2.7554 70 3.2576 3.6217 600 3.9109 4.2740


Rejection of Data• From our data in Example 1, the most extreme

value is 12.513 g, which is just over 3 standard deviations from the mean

• Is this point an outlier at the 95% confidence level?• Our null hypothesis is that the data are all from the

same distribution (that is, the point is not an outlier)

3.1432012.0

475.12513.12

S

xluesuspect vaG


Rejection of Data• The calculated value of G is:

• The critical value for 20 observations is 2.7082• Our value exceeds this, so we can reject the null

hypothesis at the 95% confidence level• We accept the alternate hypothesis that the point is

an outlier

3.1432012.0

475.12513.12

S

xluesuspect vaG


Rejection of Data• Concluding Comments on the Rejection of Data

– It is important to always retain and report outliers– They may contain important information that you are

unaware of– Explain the basis for their exclusion from your data

analysis– Do not hide such data, even if it is highly inconvenient

and even embarrassing to report it– History may attach a great deal more importance to it

than you do.


Regression and Correlation• Instrumental analysis techniques are often used to

determine the concentration of an analyte over a wide range

• A calibration curve is obtained from the analysis of reference standards

• The unknown concentration of analyte in a sample yields a response

• The unknown analyte concentration can be interpolated using the calibration graph.


Regression and Correlation• Important questions are raised in adopting such an

approach.– Is the graph linear? If not, what is the form of the

curve?– As each calibration point is subject to indeterminate

error, then what is the best straight line through the data?

– What errors are present in the fitted curve?– What is the error in a determined concentration?– What is the limit of detection?


Regression and Correlation

• An important assumption is made in conventional linear regression analysis:– There is no error in x axis values

• That is, • Reduced Major Axis methods can produce a

regression line where

S Sx y

S Sx y


Regression and Correlation• The “goodness of fit” is measured by the product-moment

correlation coefficient, r• Also known as just the correlation coefficient

• It measures how much of the variance of the dependent (y) variable is accounted for by the variance of the independent (x) variable

rx x y y

x x y y

i ii

n

i ii

n

i

n

( )( )

( ) . ( )

1

2 2

11



• The alternative form just uses sums:

• The advantage of this form is that only the sums of x, y, xy, x2 and y2 need be accumulated.

rn x y x y

n x x n y y

i ii

n

ii

n

ii

n

i ii

n

i

n

i ii

n

i

n

1 1 1

2

1

2

1

2

1

2

1


Regression and Correlation• A "perfect" straight line fit will result in r = 1• The sign of r is determined by the sign of the

gradient of the line:

X

0 1 2 3 4 5 6 7 8

Y

0

1

2

3

4

5

6

7

8

r = +1

X

0 1 2 3 4 5 6 7 8

Y

0

1

2

3

4

5

6

7

8

r = -1


Regression and Correlation• It should be noted that the correlation coefficient

can give very low correlations when there is an obvious relationship between the dependent and independent variables:

X Data

0 2 4 6 8 10 12

Y D

ata

0

1

2

3

4

5

6

7

R=0


Regression and Correlation• Indeterminate errors may cause a large scatter of

points about the best-fit line.• In such cases a t-test may be used to determine if

a low value for r is significant• The null hypothesis adopted in such instances is

that y is not correlated to x• In fact, the null hypothesis is that r is not

significantly different from zero


Regression and Correlation• A two-tailed t-test is used and if t is greater than

the critical value at the adopted confidence level then the hypothesis is rejected:

• Where the number of degrees of freedom is given by:

tr n

r

2

1 2

n 2



• We use a two-tailed test because we are only interested if the correlation coefficient is significantly different from zero (no correlation)

• Unless we have a reason to suppose that there will be a specific direction of correlation, when we use a one-tailed test



• Alternatively, we can use tables of critical values for the correlation coefficient at given probabilities

• Example: Table 10 in Murdoch and Barnes• In this case, the probabilities at the head of the

table are the two-tailed values• They should be halved for a one-tailed test


Critical Values of the Correlation Coefficient α 0.05 0.025 0.005 0.0025 0.0005 0.00025 2α 0.1 0.05 0.01 0.005 0.001 0.0005

ν = 1 0.98769 0.99692 0.999877 0.999969 0.99999877 0.99999969 2 0.9000 0.9500 0.9900 0.995000 0.999000 0.999500 3 0.8054 0.8783 0.9587 0.97404 0.99114 0.99442 4 0.7293 0.8114 0.9172 0.9417 0.9741 0.98169 5 0.6694 0.7545 0.8745 0.9056 0.9509 0.96287 6 0.6215 0.7067 0.8343 0.8697 0.9249 0.9406 7 0.5822 0.6664 0.7977 0.8359 0.8983 0.9170 8 0.5494 0.6319 0.7646 0.8046 0.8721 0.8932 9 0.5214 0.6021 0.7348 0.7759 0.8470 0.8699 10 0.4973 0.5760 0.7079 0.7496 0.8233 0.8475 11 0.4762 0.5529 0.6835 0.7255 0.8010 0.8262 12 0.4575 0.5324 0.6614 0.7034 0.7800 0.8060 13 0.4409 0.5140 0.6411 0.6831 0.7604 0.7869 14 0.4259 0.4973 0.6226 0.6643 0.7419 0.7689 15 0.4124 0.4821 0.6055 0.6470 0.7247 0.7519 16 0.4000 0.4683 0.5897 0.6308 0.7084 0.7358 17 0.3887 0.4555 0.5751 0.6158 0.6932 0.7207 18 0.3783 0.4438 0.5614 0.6018 0.6788 0.7063 19 0.3687 0.4329 0.5487 0.5886 0.6652 0.6927 20 0.3598 0.4227 0.5368 0.5763 0.6524 0.6799 25 0.3233 0.3809 0.4869 0.5243 0.5974 0.6244 30 0.2960 0.3494 0.4487 0.4840 0.5541 0.5802 40 0.2573 0.3044 0.3932 0.4252 0.4896 0.5139 50 0.2306 0.2732 0.3542 0.3836 0.4432 0.4659


Regression and Correlation• Least Squares Fitting• A least squares fit is used to draw a straight line

through data that minimises the residuals in the y-axis:

X Data

0 2 4 6 8 10 12

Y D

ata

0

2

4

6

8

10

12

Residual



• We minimise the sum of the squares of the residuals

• The residual is the vertical difference between the actual y value and the y value calculated from the regression line



• For a straight line of form y = a+bx the coefficients b and a are given by:

b

x x y y

x x

i ii

n

ii

n

1

1

a y bx



• Or:

• Once again, these forms only require simple sums of x, y, xy and x2

bn x y x y

n x x

i ii

n

ii

n

ii

n

i ii

n

i

n

1 1 1

2

1

2

1

ay b x

n

ii

n

ii

n

1 1


Regression and Correlation• It is important to provide an estimate of the uncertainty in

the slope and intercept calculated through a least squares fit

• This is especially so when involved in the characterisation of a systems response to a proposed factor.

• The first stage is to calculate the y residuals• These are the differences between the calculated data

and the observed data for a given value of yi

yi xi



• Having done this a statistic is obtained:

• This is the standard deviation of the residuals between the data points and the best fit line

S

y y

ny

x

i ii

n

2

1

2



• It has the same units as the y values• This is used to estimate the standard deviation in

b, Sb, and a, Sa

• It can be considered an estimate of the random errors in the y measurement

• It may not be a good estimate as the errors may scale with the magnitude of y


Regression and Correlation• We can also define:

Q x x xx

n

Q y y yy

n

Q x x y y x yx y

n

x ii

n

ii

n ii

n

y ii

n

ii

n ii

n

xy i ii

n

i ii

n ii

n

ii

n

2

1

2

1

1

2

2

1

2

1

1

2

1 1

1 1


Regression and Correlation• Which leads to alternatives for calculating r and b:

x

xy

yx

xy

Q

Qb

QQ

Qr


Regression and Correlation• Which also leads to an alternative formulation for Sy/x:

• Which does not involve the calculation of the individual values and can be performed using only the sums of x, y, xy, x2 and y2.

S

QQ

Q

ny

x

yxy

x

2

2

yi


Regression and Correlation• Then Sb:

• And Sa:

S

S

x x

S

Qb

yx

ii

n

yx

x

2

1

S S

x

n x xS

x

na y

x

ii

n

ii

n b

ii

n

2

1

2

1

2

1



• These estimates for the standard deviation may be used in the normal way in t and F-tests

• We can then use t tests to compare slopes and intercepts

• Also may be used to provide estimates of appropriate confidence limits


Regression and Correlation• Some graphing packages can show confidence

limits on lines of best fit (e.g. Sigmaplot):Graph of ozone concentration versus time

Time (hours)

0 2 4 6 8 10 12

Ozo

ne

conc

ent

ratio

n (p

pm)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

95% confidence intervals

Best fit line

95% confidence intervals


Regression and Correlation• Confidence limits on regression lines: Interpolation is OK: Extrapolation is not:

Age of Plasma (s)

0 200 400 600 800 1000 1200

Coa

gula

tion

Tim

e (s

)

18

20

22

24

26

28

30

32

Age of Plasma (s)

0 2000 4000 6000 8000 10000

Coa

gula

tion

Tim

e (s

)

20

40

60

80

100



• These estimates of the standard deviation of the coefficients a and b may be used to determine the uncertainty that accompanies a value of x0 obtained from the interpolation of an unknown yielding of response of y0



• An alternative approach is to calculate the standard deviation of x0:

• Note: this is an approximation!

S

S

b n

y y

b x xx

yx

ii

n00

2

2 2

1

11



• This can be simplified further by substituting for b:

• This is the preferred form for calculation

2

20

0

11

xy

xxy

x Q

yyQ

nb

SS



• If y0 is the mean of m measurements then this is modified and becomes:

S

S

b m n

y y

b x xx

yx

ii

n00

2

2 2

1

1 1



• This is simplified further as before to:

2

20

0

11

xy

xxy

x Q

yyQ

nmb

SS


Regression and Correlation• These expressions are valid if:

• Where t is the value of the t statistic for the appropriate confidence level and n-2 degrees of freedom

• See http://www.rsc.org/images/Brief22_tcm18-51117.pdf for a fuller discussion

05.02

22

xy

xyx

Q

SQt


Regression and Correlation• How can we minimise the error Sx0?

– By making replicate measurements (m > 1)– By working in the middle of the line– By using a well-determined line (b >> 0)– Maximising

)( 0 yy

S

S

b m n

y y

b x xx

yx

ii

n00

2

2 2

1

1 1

2)( xxi


Regression and Correlation• The approach described above is widely used, but it is

flawed:– It assumes that x values are free of errors. This is not

necessarily true– It assumes that errors in y values are constant

• This is rarely true. All y values are given equal weighting regardless of the uncertainty associated with them.

• Nevertheless, used carefully linear regression analysis and least squares fit provides useful information.



• Non-linear Regression• The linear regression method works for any linear

relationship• Many relationships between variables do not

follow a simple linear relationship• May be linearised by the appropriate

transformation


Regression and Correlation• For example:

• May be linearised by taking logs of both sides:

• We can now plot log y against log x to determine the slope and intercept

y axb

log log logy a b x


Roach Data - Log data

log(length)

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

log

(ma

ss)

-1

0

1

2

3

Roach Data - Raw Data

Length (cm)

4 6 8 10 12 14 16 18

Ma

ss (

g)

-40

-20

0

20

40

60

Regression and Correlation• Example: Fish mass versus length



• What is wrong with the linear plot?– Below ~6 cm, the fish appear to have negative mass– This is not physically realistic

• From the log-log plot we obtain log a =-1.954 (a = 0.0111g) and b = 3.013, so:

y x0 0111 3 013. .


Regression and Correlation• Example - Is a correlation significant?• The correlation coefficient for the transformed data

is 0.8733 for 29 observations• Our null hypothesis is that there is no correlation

(H0: r = 0)• If we wish to see if this correlation is significant,

we can determine the t value:

tr n

r

2

1

0 8733 29 2

1 0 87339 314

2 2

.

..


Regression and Correlation• We use a one-tailed test because we expect a positive

correlation between fish mass and length.• The critical value for a one-tailed t-test with 27 degrees of

freedom at the 99% confidence level is 2.473• This is much less than our calculated t value. • Accordingly, we reject the null hypothesis (that there is no

correlation) • And accept the alternative hypothesis that there is a

positive correlation (H1: r > 0)



• We can also use tables of critical values of the correlation coefficient

• Example – Table 10 in Murdoch and Barnes• We can also generate our own table from the T

Table using the equation:

22

2

nt

tr

crit

critcrit


Critical Values of the Correlation Coefficientα 0.05 0.025 0.005 0.0025 0.0005 0.00025

2α 0.1 0.05 0.01 0.005 0.001 0.0005ν 1 0.98769 0.99692 0.999877 0.999969 0.99999877 0.999999692 0.9000 0.9500 0.9900 0.995000 0.999000 0.9995003 0.8054 0.8783 0.9587 0.97404 0.99114 0.994424 0.7293 0.8114 0.9172 0.9417 0.9741 0.981695 0.6694 0.7545 0.8745 0.9056 0.9509 0.962876 0.6215 0.7067 0.8343 0.8697 0.9249 0.94067 0.5822 0.6664 0.7977 0.8359 0.8983 0.91708 0.5494 0.6319 0.7646 0.8046 0.8721 0.89329 0.5214 0.6021 0.7348 0.7759 0.8470 0.8699

10 0.4973 0.5760 0.7079 0.7496 0.8233 0.847511 0.4762 0.5529 0.6835 0.7255 0.8010 0.826212 0.4575 0.5324 0.6614 0.7034 0.7800 0.806013 0.4409 0.5140 0.6411 0.6831 0.7604 0.786914 0.4259 0.4973 0.6226 0.6643 0.7419 0.768915 0.4124 0.4821 0.6055 0.6470 0.7247 0.751916 0.4000 0.4683 0.5897 0.6308 0.7084 0.735817 0.3887 0.4555 0.5751 0.6158 0.6932 0.720718 0.3783 0.4438 0.5614 0.6018 0.6788 0.706319 0.3687 0.4329 0.5487 0.5886 0.6652 0.692720 0.3598 0.4227 0.5368 0.5763 0.6524 0.679925 0.3233 0.3809 0.4869 0.5243 0.5974 0.624430 0.2960 0.3494 0.4487 0.4840 0.5541 0.580240 0.2573 0.3044 0.3932 0.4252 0.4896 0.513950 0.2306 0.2732 0.3542 0.3836 0.4432 0.4659


Regression and Correlation• Example - Testing the significance of the gradient • We see that the exponent in the relationship is

very close to 3 • This is what we would expect if the fish show

isometric growth• That is, the fish grow uniformly in all three

dimensions• Some fish show allometric growth, which is non

uniform in all three dimensions


Regression and Correlation• We can perform a t-test on the exponent against

the reference value (3)• Our null hypothesis is that there is no difference

between the observed and reference values• We first calculating the standard deviation of the

exponent (the slope in this case):

S yx01944. Sb

01944

0 6010 3235

.

..


Regression and Correlation• We calculate t using:

• The critical value for 27 degrees of freedom at the 99% confidence level for a two-tailed test is 2.771

• Our t value is well below this value• We accept the null hypothesis that the slope is not

significantly different from 3.000 at the 99% confidence level

t xn

S 3013 3000

29

0 32350 2164. .

..


Limits of Detection • IUPAC has proposed that the limit of detection be

defined as:– The response at zero analyte concentration plus three

standard deviations of the response at zero analyte – If a calibration curve is used then the limit of detection

is given by the y axis intercept plus three times the standard deviation associated with that value:

– Limit of detection (LOD) = a Sa 3


Limits of Detection• From calibration graph:

0

0

a

a+3Sa

LOD

Concentration

Re

spo

nse


Limits of Detection

• Miller and Miller use Sy/x as an estimate of the error in the intercept

• This may underestimate the LOD if the errors in y are constant


Limits of Detection

• It may be possible to record the presence of an analyte below this level

• The uncertainty associated with such an observation is such as to make it unreliable

CHEN10011 Slides 1

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