STATISTICAL ASSESSMENT OF QUANTITATIVE QUALITY PERFORMANCE OF BUILDING PRODUCTS DURING CONFORMITY ASSESSMENT AND CERTIFICATION IN THE FIRE PROTECTION FIELD

This paper proposes statistical methods for evaluation of quantitative performance of building products determined by tests for objective decision during conformity assessment/certification in the fire protection field. The procedure is applicable even for other types of products, and for research / development of new material products.


INTRODUCTION
On the Czech market the construction products are subject to Law No. 22/1997 Coll.[1] and the Government regulation (GR) No. 163/2002 [2] among other things.GR also obligates to examine the so-called "specified products".Every manufacturer shall demonstrate their compliance with the relevant standards.The reason is consumer protection from the point of view of health protection, fire safety, hygiene, energy saving and environmental protection.Annex 2 of the abovementioned regulation shows the following main groups of products specified for conformity assessment:  construction products for concrete and reinforced concrete construction parts,  construction products for masonry constructions,  building wood products and wood constructions,  construction products for metal constructions,  protective, thermal insulation materials and products, waterproof materials, roof covering and adhesives,  glass building products,  construction products for sewer systems and distribution of liquids and gases,  construction products for hole fillings,  special materials, products, structures and equipment,  the technical equipment of buildings,  construction products for internal and external finishes of walls, ceilings, floors,  construction products for sanitary facilities and other special products.
The European Parliament and Council Regulation (EU) No. 305/2011 (CPR) [3] also apply to the European market of construction products.According to this regulation the assessment obligation occurs when the products are determined by the requirements of the harmonized Czech (European) technical standards and/or European Technical Assessments (ETA).Manufacturers are entitled to label their products with the CE mark (CCZ is the Czech mark of conformity) after fulfilling the prescribed requirements.
The statistical assessment of the product's prescribed quantitative quality performance test values determined according to the prescribed test standards is an important tool for the objective determination whether the product meets/does not meet it.

STATISTICAL ASSESSMENT OF THE QUANTITATIVE RESULTS OF TEST DETERMINATIONS
The quality of the material construction products may be characterized by the test of a quality feature e.g.M, test-determined which has a specific value e.g.M0 in a quality product.If the value M = M0 ± U is found in the supplied sample of product in verifying the property M during certification/conformity assessment, its quality is confirmed and a certificate can be granted.U is permissible tolerance's result.
Otherwise, if M < M0 -Δ, or if M > M0 + Δ, and Δ is > U its quality is not confirmed and a certificate cannot be granted to it.Standardized test method usually requires measurements of M product's property repeatedly n times with particular numerical results of X1, X2,…, Xn.
Results of Xi (i = 1, 2, ..., n) are not equal to M, but they have values of Xi = M + ui where ui are the individual measurement uncertainties.We assume that the measured results were/are statistically evaluated to remoteness and outliers and that the remote values were /are excluded [5].If these uncertainties are mutually independent and have normal distribution with the mean value Θ and the variance σ 2 or more precisely the standard deviation σ, then the following procedure can be used to evaluate the results of tests: -the so-called confidence interval (CI) is determined for the M value so that there applies: (2) The probability that the interval does not contain the actual value is approximately α.
Number (1-α), selected according to the severity of the consequences of errors (usually 0.90 or 0.95 or 0.99) is the so-called coefficient of reliability (confidence coefficient).Its complement to 1, equal to α, is the so-called Risk of mistake.
Under the assumptions (normality of errors distribution) the confidence interval has the following formula Where x is the sample average of the results and -α/2 is the so-called quantile of the standard normal distribution, defined by the formula For the most common values of α (0.1 or 0.05 or 0.01) the quantilesʼ sizes are listed in the following table [6] Tab. 1: Quantiles of standard normal distribution for the selected values of errors' risks α Decision-making about the product's ability to the given purpose/ compliance could be as follows: this is the possibility that M > MH is not excluded, therefore the product does not meet the quality criterion and cannot be approved, this is the possibility M < MD is not excluded.Again, the product does not meet quality criterion and cannot be approved, indicates with a high probability that M is among the permissible limits MD and MH and so the product can be approved.

Degree of protection against possible error can be expressed as follows:
-If in fact then with the preselected high probability of (1-α)/2 will be M > MH and if M > MH, the product is needed to be disapproved with a high probability , then with the probability of (1 -α)/2 will be M < MD and the product must also be disapproved, (M equals to the correct value M0), then with a high probability (1 -α) will be

MD < M < M0 <
M < MH and the product can be approved (justified correct decision).An unpleasant/indecisive situation can occur when confidence interval (M, M ) contains both the value of M0, and one of the values MD, MH.In this case, the possibility M = M0 (correct value) cannot be excluded, nor M < MD (M is too low) possibly M > MH (M is too high).This situation can be avoided if the number of measurements n is chosen so that the length of the confidence interval A simple calculation can convince us that in this case it is not possible that the interval (M, M ) contains the correct value M0, and simultaneously one of already unacceptable levels of MD or MH.
However, if the value of σ, characterizing the variability of the results of errors of measurement is not known, it should be estimated from the measurement results of where tn-1,α/2 is the so-called 100 α/2 %-critical value of the Studentʼs t-distribution with (n-1) degrees of freedom [6].The length of the confidence interval of course, is not known.It depends on the value s that is determined by measuring/test.Therefore we cannot determine the required number n of repeated measurements.
The way out of this situation is the following two-step procedure for determining a confidence interval of predetermined length: 1. n1 repeated measurements of monitored variable X will be realized and then basic statistical indicators of the result will be calculated: -sample arithmetic average   x is the sample arithmetic mean of all n = n1+ n2 measurements.
d1 ≤ Δ, one can proceed as in the case where σ is known with the value s1 instead of the σ value.

3 .
If in the second step d1 > Δ further n2 measurements , , ,…,xn1+ xn1+n2 will be realized and one will calculate the new arithmetic mean Confidence interval (M, M ) shall be determined as needed n2 measurements is determined so that .