SAMPLING

Fundamental Statistical Terminologies

A measurement is considered to be accurate if the difference between the measured value and the true value falls within an acceptable margin.

How do we know the true value? We rely on statistics therefore.

A random error (or variation) on average, over a period of time, tend to zero whereas integrated systematic errors result in a net positive or negative value

• The bias is the difference between the true value and the average of a

number of experimental values and hence is the same as the systematic error. • The variance between repeated samples is a measure of precision or

reproducibility. • The difference between the mean of a series of repeat samples and the true

value is a measure of accuracy

Mode and median are measures of the central value of a distribution. The mode forms the peak of the frequency distribution, while the median divides the total number of measurements into two equal sets of data. If the frequency distribution is symmetrical, then its mean, mode and median coincide.

Symmetrical Distribution Asymmetrical Distribution

The variance, and its derived parameters such as the standard deviation and the coefficient of variation, are the most important measures for variability between test results.

The classical formula for the calculation of the variance is:

where n = number of measurementsn-l = degrees of freedom

The standard deviation, a, is the square root of the variance.

Another important term is confidence interval

What is Sampling ?What is Sampling ?

• Procedure by which some members of a Procedure by which some members of a population are selected as representative population are selected as representative of the entire populationof the entire population

• The sub-group thus selected to represent The sub-group thus selected to represent the whole population is known as SAMPLEthe whole population is known as SAMPLE

Methods Of SamplingMethods Of Sampling• Several methods are used to ascertain a Several methods are used to ascertain a

particular aspect of the population, through particular aspect of the population, through an unbiased sample drawn from the an unbiased sample drawn from the populationpopulation

Sampling is divided in two categoriesSampling is divided in two categories• 1. Probability Sampling1. Probability Sampling• 2. Non-probability Sampling 2. Non-probability Sampling

Probability samplingProbability sampling

• It is any method of sampling that utilizes It is any method of sampling that utilizes some form of some form of random selectionrandom selection

• The procedure should assure that the The procedure should assure that the different units in the population have different units in the population have equal probabilities of being chosen.equal probabilities of being chosen.

Non probability samplingNon probability sampling

• It does not involve It does not involve randomrandom selection selection • May or may not represent the population May or may not represent the population

well well • Used when researcher lacks a sampling Used when researcher lacks a sampling

frame for the populationframe for the population

Probability samplingProbability sampling

Includes: Includes: • Simple Random MethodSimple Random Method• Systematic Sampling Systematic Sampling • Stratified SamplingStratified Sampling• Cluster SamplingCluster Sampling• Multistage SamplingMultistage Sampling

Nonprobability SamplingNonprobability Sampling

Includes :Includes :• Accidental SamplingAccidental Sampling• Voluntary SamplingVoluntary Sampling• Purposive SamplingPurposive Sampling• Quota SamplingQuota Sampling

Importance of Sampling in Metallurgical Accounting

x

LOT

Primary Secondary Analysis Resultsample sample

s1 s2 s3 sx

Propagation of errors:

Example:

GOAL: x =

2ix ss

%5.5(%)30%)1(%)2(%)5( 2222 xs

Analytical process usually contains several sampling and sample preparation steps

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 Sample No.

Concentration mg/l

Volume m 3

c·V g

1 6.25 4.58 28.6

2 4.36 3.71 16.2 3 5.58 5.20 28.99

4 4.64 5.71 26.48

5 4.86 4 .54 22.08

6 3.65 6.78 24.75

7 3.73 7.12 26.55

8 5.98 5.81 34.76

9 4.96 5.86 29.05

Mean 4.89 5.479 26.39 Sum 44.01 49.3 237.47

ciVi

Total emission estimate (unweighted):

mmgVcM 241.13479.5/89.499 33

Total emission estimate (weighted):

mmgVcM ww 237.47 g479.5/86.499 33

Weighting error (in concentration): 0.03 mg/l

Weighting error (in total emission): 3.66 g

Weighted mean concentration: = 4.86 mg/l

Vi

Sample delimitation

Incorrect Correct

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

Incorrect Increment and Sample Preparation

Errors • Contamination (extraneous material in sample)• Losses (adsorption, condensation, precipitation, etc.)• Alteration of chemical composition (preservation)• Alteration of physical composition (agglomeration,

breaking of particles, moisture, etc.)• Involuntary mistakes (mixed sample numbers, lack

of knowledge, negligence)• Deliberate faults (salting of gold ores, deliberate

errors in increment delimitation, forgery, etc.)

Estimation of Fundamental Sampling Error by Using Poisson Distribution

• Poisson distribution describes the random distribution of rare events in a given interval.

• If n is the number of critical particles in sample, the relative standard deviation expressed as the number of particles is

nr

1

Example

Plant Manager: I am producing fine-ground limestone that is used in paper mills for coating printing paper. According to their speci-fication my product must not contain more than 5 particles/tonne particles larger than 5 m. How should I sample my product?

Sampling Expert: That is a bit too general a question. Let’s first define our goal. Would 20 % relative standard deviation for the coarse particles be sufficient?

Plant Manager: Yes.

Sampling Expert: Well, let’s consider the problem. We could use the Poisson distribution to estimate the required sample size. Let’s see:

The maximum relative standard deviation sr = 20 % = 0.2. From equation 2 we can estimate how many coarse particles there should be in the sample to have this standard deviation

252.011

22 rs

n

If 1 tonne contains 5 coarse particles this result means that the primary sample should be 25 tonnes. This is a good example of an impossible sampling problem. Even though you could take a 25 tonne sample there is no feasible technology to separate and count the coarse particles from it. You shouldn’t try the traditional analytical approach in con-trolling the quality of your product. Instead, if the specification is really sensible, you forget the particle size analyzers and maintain the quality of your product by process technological means, that is, you take care that all equipment are regularly serviced and their high performance maintained to guarantee the product quality.

Plant Manager: Thank you

d d d d

f= 1 f= 0,524 f= 0,5 f= 0,1

default in most cases

l = (L/d)0.5 l = 1

Limitations of Gy’s Equation

m

l l = 0.4

C = f.g.l.m = 0.5*0.25*0.4*19.3 = 0.97

Example 2

PLANNING OF SAMPLING

1. GATHERING OF INFORMATION

What are the analytes to be determined? What kind of estimates are needed?

Average (hour, day, shift, batch, shipment, etc.) Distribution (heterogeneity) of the determinand in the lot Highest or lowest values

Is there available useful a priory information (variance estimates, unit costs)?

Is all the necessary personnel and equipment available? What is the maximum cost or uncertainty level of the investigation?

PLANNING OF SAMPLING

2. DECISIONS TO BE MADE

Manual vs. automatic sampling Sampling frequency Sample sizes Sampling locations Individual vs. composite samples Sampling strategy

Random selection Stratified random selection Systematic stratified selection

Training

Plant Sampling – Moving Stream

Incorrect Sampling

All these do not collect samples from the full stream

Reverse Spoon Cutter – Incorrect because of Preferential Losses of Coarse Particles

Sampling of smelter products, residues and wastes

Sampling from Stationary Situations

Sampling from Stockpiles

Sampling from the side of a stockpile will not provide a representative sample

Sampling from Ships, Trucks and Wagons

Moisture Sampling

Substantial moisture loss can occur when primary increments are conveyed to the secondary cutter on small open conveyors in sample stations

Moisture loss can occur when samples are stored in containers for long periods prior to being taken back to the laboratory for moisture determination

Laboratory Sampling

Grab Sampling

Step 1

Step 2

Coning and Quartering

Sample Splitters

Rotary Riffle

SUMMARY

ISO 13909-2 Guideline for Coal & Coke Samples

Thanks