SAMPLING
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