Pentti MinkkinenLappeenranta University of Technology

e-mail: Pentti.Minkkinen@lut.fi

WSC 5

Weighting Error – the Often Neglected Component of the Total

Sampling Error

Error components of analytical determination according to P.Gy

Global Estimation Error GEE

Total Sampling ErrorTSE

Point Selection ErrorPSE

Total Analytical ErrorTAE

Point Materialization ErrorPME

Weighting ErrorSWE

Increment Delimi-tation Error

IDE

Long Range Point Selection Error

PSE1

Periodic Point Selection Error

PSE2

Fundamental Sampling Error

FSE

Grouping and Segregation Error

GSE

Increment Extraction ErrorIXE

Increment and SamplePreparation Error

IPE

GEE=TSE +TAETSE= (PSE+FSE+GSE)+(IDE+IXE+IPE)+SWE

Weighting error

Weighting error is made in the estimation of simple arithmetic mean, when• the sampling target consists of several sub-strata of different sizes• in process analysis, when the flow-rate varies

Lot Consisting of Strata of Different Sizes and Heterogeneities

LOT

N1n11

N2n22

Nknkk

x11 x12 x21 xk1 xk2 xk3

ML1 ML2

MLk

LOT

N1n11

N2n22

Nknkk

x11 x12 x21 xk1 xk2 xk3

ML1 ML2

MLk

Lot consisting of k strata of different sizes and the quantities needed to optimize the sampling plan

i

ii M

MW

L

L = Relative size of the stratum i (1)

MLi = sizes of strata (e.g. as mass or volume), (i = 1,2, …, k)

Ni = Relative size of stratum i expressed as the number of potential samples that could be taken from a strata = , MSi is the size of samples taken from stratum i.

i

ii M

MN

S

L

i = Standard deviation of one sample taken from stratum ici = Cost of one sample analyzed from stratum i ct = Total cost of the estimation of the grand mean of the lot ni = No. of samples taken from stratum i

nt = Total number of samples analyzed = in

i

n

jij

in

xx

i

= Mean of stratum i (2)

k

iii xWx

1= Grand mean of the lot (2)

Variance of the lot mean =

i

i

i

iiix nN

nNW

222

1

(3a)

If the samples taken are small in comparison to the stratum size (as is usually the case) this equation simplifies to

i

iix n

W2

22 , if in all strata ni << Ni and Ni>> 1 (3b)

Total cost of the investigation in general case is

k

iiit cnc

1(4a)

Usually the costs are independent of strata, and

*cnc tt , if

*,...,21 cccc k (4b)

a) Optimal allocation of samples, if only the relative sizes Wi of the strata are

known

If only the sizes of strata are known and the total cost ct of the investigation is fixed, then the best strategy is to allocate the samples proportionally to the sizes of strata:

tii nWn , where*c

cn t

t (5)

Both nt and ni have to be rounded to integers so that the total cost will not be exceeded.

*1

cc

W

Wn T

i

k

ii

iii

(6)

Here, too, ni’s have to rounded into integers so that the target cost is not exceeded.

b) Optimal allocation of samples, when the target value, cT, is given to the total cost and the

variance of the lot mean has to be minimized

i

k

ii

T

iii W

Wn

1

2 (7)

c) Optimal allocation of samples, when the target value, T, is given to the standard

deviation of the lot mean and total cost has to be minimized

Again, ni’s have to be rounded to integers so that the required standard deviation of the lot mean will not be exceeded.

Sampling error in process analysis

• In process analysis the fluctuation of the flow-rate should be taken into account in estimating the mean over a time

Incorrect sample delimitation

Incorrect sample profileCutter movement

Correct sample delimitation

Correct sample profilesCutter movement

Correct design for proportional sampler:correct increment extraction

ba

c

v

v = constant 0.6 m/s

if d > 3 mm, b 3d = b0

if d < 3 mm, b 10 mm = b0

d = diameter of largest particlesb0 = minimum opening of the sample cutter

Proportional sampling

• Correctly executed proportional sampling eliminates the weighting error, if each sample is weighed and the mean is calculated as weighted mean by using the sample masses as weights.

• Subsamples have to be sampled proportionally, if they are combined into a composite sample

Effect of density

• If the density of the material varies within the lot and equal volumes are sampled the simple mean is erroneous

Total mass of the drill core: Mtot=152.8 kgMass of the valuable mineral = 47.5 kg Density of the valuable mineral = 5 kg/dm3 Density of the gangue = 2.6 kg/dm3 Average density = 3.056 kg/dm3  True mass fraction of the mineral = 47.5kg/(152.8 kg) = 0.3109 = 31.09 %

Example on weighting error: Drill core of stratified rock type

SAMPLING PLAN:

The drill core is divided into 100 slices of equal sizes,

volume = 0,5 dm3 and average mass, Ms =1.528 kg

Example on weighting error(cont.)

Each sample is analyzed separately. The mean concentration as mass fraction, cm = 0.190

Based on this result the average mass of the valuable mineral in the core is = cm · Ms ·100=29.03 kg

If every sample is weighed (mass Mi) and the weighted mean of the mass fraction is estimated the correct mean concentration is obtained:

and the total mass of the valuable mineral   Relative weighting error is thus:

(0.19-0,3109)/0.3109 = -0.389 = -39,9 %

3109,0

i

iiw M

Mcc

kg5,47100min McM w

The drill core is divided into 100 slices of equal sizes, volume = 0,5 dm3 and average mass, Ms =1.528 kg

Sample #

Mass fraction

Sample mass kg

Mineral content kg

1 1 5,0 5,0 2 0 2,6 0 3 0 2,6 0 4 0 2,6 0 5 0,6579 3,8 2,5 6 0 2,6 0 7 0 2,6 0 8 0 2,6 0 9 0 2,6 0

10 0,6579 3,8 2,5 11 0 2,6 0 12 0 2,6 0 13 0 2,6 0 14 0 2,6 0 15 0 2,6 0 16 0 2,6 0 17 0,6579 3,8 2,5 18 0,6579 3,8 2,5 19 0 2,6 0 20 0 2,6 0

MEAN 0,1816 2,96 SUM 59,2 15,0

Weighted mean= 15 kg/(59,2 kg) = 0,2534

20 samples, 1 dm3 by volume analysed

Correct mean =0.3109 = 31,09 %

Process sampling: Simulation study

Three processes with 1000 data points were generated with low medium and high correlation between concentration and flow-rate

Weighted mean Simple mean Relat. Error (%) All sampled 45.3669 45.3429 -0.053Every tenth sampled 45.5452 0.3929Every tenth sampled 45.4100 0.0948

25 30 35 40 45 50 55 60 650

2

4

6

8

Flow-rate

ai

r = 0.011

0 100 200 300 400 500 600 700 800 900 1000-4

-2

0

2

4

Sample No.

ai, V

i

Weighted mean Simple mean Relat. Error (%) All sampled 43.3210 45.3429 4.67Every tenth sampled 43.4187 0.225Every tenth sampled 45.4100 4.82

25 30 35 40 45 50 55 60 650

5

10

15

20

Flow-rate

ai

r = -0.558

0 100 200 300 400 500 600 700 800 900 1000-4

-2

0

2

Sample No.

ai, V

i

Weigted mean Simple mean Relat. Error (%) All sampled 47.1692 45.3429 -3.87 Every tenth sampled 47.2751 0.225Every tenth sampled 45.4100 -3.73

25 30 35 40 45 50 55 60 6520

40

60

80

Flow-rate

ai

r = 0.972

0 100 200 300 400 500 600 700 800 900 1000-4

-2

0

2

4

Sample No.

ai, V

i

Second simulationCorrel. Weighted Simple Rel. Errorcoefficient mean mean % 0.0248 36.1798 36.1346 -0.125

0.169 36.6219 36.1346 -1.33

0.971 37.5777 36.1346 -3.84

Weighting error: calculation of mean when flow-rate varies

0 100 200 300 400 500 600 700 800100

150

200

250

300

350N

Ox

(mg/

m3)

0 100 200 300 400 500 600 700 8001

1.5

2

2.5

3

3.5 x 10 5

Time (h)

FLO

WR

ATE

(m3/

h)

NOx concentrations and total gas flow-rate measured as one-hour averages from a power plant during one month

Weighting error of simple mean: in mean concentration = -7.97 mg/m3 in total monthly emission = -1400 kg

Mean of NOx concentrations: 229.5 mg/m3 Mean of gas flow-rate : 2.327 ·105 m3/h

Total gas flow: 1.718·108 m3

Total NOx emitted (unweighted): 39400 kg

Weighted mean of NOx concentration:

= 237.5 mg/m3 Vi

ciVi

Total NOx emitted (weighted): 40800 kg

CALCULATION OF TOTAL MONTHLY NOx EMISSION

Minimization of weighting error in Process analysis, when proportional

cross-steam sampling cannot be used

• Flow-rate is measured simultaneously with sampling and used as weight in calculating the mean.

• Sampling system is coupled to a flow meter so that a fixed volume is taken when the required total volume has passed the sampling point. In this case the simple average can be used as the mean concentration.

CONCLUSIONS

Weighting error is often a significant component of sampling errors and has to taken into account when the average value, mean concentration or total mass of analyte in the sampling target is estimated.

Increasing the No. of samples does not necessarily reduce the sampling error, if the flow-rate and concentration are correlated.

THANK YOU

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