BS 6200-5:1997

Committees responsible
Foreword
Section 1. General
1.1 Scope
1.2 References
1.3 Definitions
Section 2. Theoretical considerations
2.1 Accuracy, trueness and precision
2.2 Distribution of errors
Section 3. Principles of statistical evaluation
3.1 Degrees of freedom
3.2 The t-test
3.3 Repeatability and reproducibility limits r and R
3.4 Within-laboratory reproducibility limit Rw
3.5 Standardization testing and precision testing
3.6 Regression analysis
Section 4. Experimental design
4.1 Planning
4.2 Standardisation tests
4.3 Precision tests
4.4 Reporting of results
4.5 Data truncation
4.6 Rounding of results
4.7 Choice of test samples
Section 5. Analysis of variance: method 1
5.1 Effect of replication
5.2 Nomenclature
5.3 Analysis of variance applied to an example
5.4 Calculation of variance and standard deviation from
      accepted results
5.5 Tolerance
Section 6. Analysis of variance: method 2
6.1 Direct mathematical method
6.2 Summary of results from a test programme
6.3 Calculation of sums of squares: alternative procedure
6.4 Degrees of freedom
6.5 Table of variances
6.6 Calculation of repeatability and reproducibility
6.7 Statement of precision
Section 7. Analysis of variance: method 3
7.1 Simplified method using data transformation
7.2 Example using transformed data
7.3 Recovery of statistical results from transformed data
Section 8. Intermediate precision
8.1 Introduction
8.2 Intermediate precision conditions
8.3 Estimation of time-different intermediate precision
8.4 Formation of the data table
8.5 Calculation of sums and squares
8.6 Separating the sums of squares
8.7 Preparation of analysis of variance table, ANOVA
8.8 Summary of analysis of variance
Section 9. F-variance ratio test
9.1 Theoretical considerations
9.2 Applications for the F-statistic
9.3 Examples of the use of the F-statistic
Section 10. The t-test
10.1 General introduction
10.2 Comparison of two means
10.3 Confidence limits, confidence interval
10.4 Comparison of two methods
10.5 Method of checking the correct application of a
       procedure
10.6 Application to regression analysis
Section 11. Tests for outliers and stragglers
11.1 Theoretical considerations
11.2 Cochran's test for excessive variance
11.3 Grubbs' test for outlying single results or
      laboratory means
Section 12. Regression and correlation
12.1 Regression analysis
12.2 Examples of regression calculations
12.3 Correlation coefficient
12.4 Confidence limits of regression lines
12.5 Probability limits for a regression line
Section 13. Graphical presentation
13.1 95% confidence limits of regression lines
13.2 New statistical tests
13.3 Youden diagrams
Section 14. Analytical limits
14.1 Uncertainty factors for estimated standard deviations
14.2 Practical measurements of the r/R ratio
14.3 Limit of detection
14.4 Choice of factor k for limit of detection
14.5 Limit of determination
14.6 Precision of determination
Section 15. Tests for a normal distribution
15.1 Introduction
15.2 Non-normal distributions
15.3 Use of normal probability plot
15.4 Example of a normal probability plot
Annexes
A (normative) Summary of statistical terms and symbols
B (informative) Statistical tables
Tables
1 Nickel content determinations
2 Theoretical probabilities
3 Convergence of standard deviation
4 Distribution of 1000 normal standard deviates
5 Relationship between degrees of freedom and t
6 Example of rounding rules
7 Results for variance analysis
8 Laboratory averages, standard deviations and totals
9 Collated results for 11 laboratories (p=11)
10 Evaluation from collated results
11 Statistical summary
12 Manganese content of five alloys
13 Sample C manganese
14 ANOVA analysis of variance
15 Precision data for five manganese alloys
16 The use of data transformations
17 Comparison of transformed data with original unmodified
    results
18 Data and statistical totals
19 ANOVA table (offset vanadium content)
20 F-ratios
21 Mean, repeatability and reproducibility from two
    standard methods
22 Variances and numbers of degrees of freedom
23 Ratios for F-test
24 Manganese content of steel bars
25 Offset equivalents for manganese content results
26 Determination of vanadium content by flame atomic
    absorption spectrometry
27 Significance of differences in pairs of means
28 Standard deviations and squares for Cochran's and
    Grubbs' tests
29 Cobalt content in steel in ascending order
30 Repeatability limits for vanadium content
31 Statistical parameters for regression analysis
32 Repeatability limits, predicted from regression
    equations
33 Minimum correlation against probability
34 Repeatability limits at 95% probability based on
    logarithmic regression
35 Values of the uncertainty factors for estimates of
    repeatability and reproducibility standard deviations
    (and of the corresponding repeatability and
    reproducibility limits)
36 Probability statistics for transformed results
B.1 Critical values for the t-test (two-tailed)
B.2 Critical values for the F-variance ratio test (one-
    tailed) at the 0.05 significance level
B.3 Critical values for the F-variance ratio test
    (one-tailed) at the 0.01 significance level
B.4 Critical values for the F-variance ratio test
    (two-tailed) at the 0.05 significance level
B.5 Critical values for the F-variance ratio test
    (two-tailed) at the 0.01 significance level
B.6 Critical values for Cochran's test
B.7 Critical values for Grubbs' test
B.8 Indicators for Mandel's h and k statistics at the
    1% significance level
B.9 Indicators for Mandel's h and k statistics at the
    5% significance level
Figures
1 Results of 57 determinations of 3.19% nickel
2 Illustration of tendency of histogram to approach
    normal curve as intervals decrease and replicates
    increase
3 Ratio of area between limits y1 and y2 to total area
4 Distribution of 1000 random normal deviates
5 Uncertainty of precision measure sr at 0.95 probability
6 Uncertainty of precision measure sR at 0.95
    probability
7 Linear and logarithmic regression lines; confidence
    limits
8 Logarithmic relationship between vanadium content
    (%(m/m)) and repeatability, r, or reproducibility, R and
    Rw
9 Mandel's h and k statistics for precision data for
    vanadium
10 Vanadium analysis data
11 Youden diagram, two-sample plot
12 Frequency distribution for means 3.29 standard deviations
    apart
13 Frequency distribution for means 3.29 standard deviations
    apart
14 The calibration curve, decision limit, detection limit
    and capability of detection
15 Density functions and normal probability graphs of
    cumulative distributions
16 Vanadium analyses plotted on normal probability graph
    paper
List of references

Gives guidelines on the statistical procedures used in developing analysis methods for iron and steel, particularly BS 6200:Part 3 and associated BS ENs. Also gives guidance on the planning of precision tests and aspects of the preparation and use of reference materials.

This Part of BS 6200 gives guidance on the statistical procedures used during the development of methods for the analysis of iron and steel, in particular those methods described in BS 6200: Part 3 and associated BS ENs. It explains the significance of the statistical data given in these standards. This Part of BS 6200 also gives guidance on the planning of standardization and precision tests and covers some aspects of the preparation and use of certified reference materials.