## Repeatability conditions

In many practical situations, there is no time or resources to repeat measurements more than just a few times. This section provides methods to check if a number (one, two, number, …) of measurement results, obtained under repeatability conditions, are acceptable or if further testing is needed.

An example is also given of the method applied to the results of an expensive chemical analysis.

## One test result

It is not usual to have only one result. No statistics can be applied in that case, and we recommend that at least one further measurement is performed and that the results are treated as below:

## Two measurements

Start with two results, *x*_{1} and *x*_{2}. Comparing the difference in measurements with repeatability limit, *r*

If |*x*_{1} – *x*_{2}| <*r*, then the final result is (*x*_{1} + *x*_{2}) / 2.

If |*x*_{1} – *x*_{2}|> *r*, two additional measurements need to be made. Then one compares the difference between the minimum and maximum results with the so-called critical range, *CR*_{0,95} (4), for 4 measurements:

If *X*_{max} – *X*_{min} <*CR*_{0,95} (4) the final result is:

(*x*_{1} + *x*_{2} + *x*_{3} + *x*_{4}) / 4

If *X*_{max} – *X*_{min}> *CR*_{0,95} (4) the final result is:

(*x*(2) + *x*(3)) / 2

where *x*(2) and *x*(3), respectively, are the second and third smallest results.

## More than two measurements

Start with *n* outcomes. Compare the difference between the minimum and maximum test results with the critical range, *CR*_{0,95} (*n*), for *n* measurements

For a range obtained with *n* results <*CR*_{0,95} (*n*) the end result can be calculated directly as the arithmetic mean of all *n* results.

For a range obtained with *n* results > *CR*_{0,95} (*n*), one must obtain *m* additional test results (if possible) (where *m* is an integer *n* / 3 <*m* <*n* / 2)

For a range obtained with (_{n} + _{m}) results <*CR*_{0,95} (_{n} + _{m}), the end result is the arithmetic mean of all _{n} + _{m} results.

For a range obtained with (_{n} + _{m}) results > *CR*_{0,95} (_{n} + _{m}), the end result the median of all _{n} + _{m} results.

### For example, a costly chemical analysis

[ISO 5725-6]

In determining the gold and silver content of the ore with an acid test, there are several methods, but all require expensive equipment, skilled operators and much time, usually 2 days and even several days more if the ore contains platinum group metals or other co-existing elements.

Four test results of the gold content were obtained under repeatability conditions:

Au (in g / t) 10.5 11.0 10.8 11.0

The repeatability standard deviation (precision) of this standardized method of measurement has been determined previously to be *s*_{r} = 0.12 g / t.

The first two measurements differ by more than the repeatability limit (*r *= 2.8 x 0.12 g / t = 0.34 g / t). Therefore two further measurements were conducted:

4 measurements of gold content:

The critical range *CR*_{0,95} (4) = 3.6 x 0.12 = 0.43 g / t

where *f *(4) = 3.6.

Since the range of the four results is 11.0 to 10.5 = 0.5 g / t, which is greater than the critical range *CR*_{0,95} (4), the final result is the median of all 4 outcomes, i.e:

(11.0 + 10.8) / 2 = 10.9 g / t

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### Critical Range, CR

For a number (*n*) of measurements:

*CR*_{0,95} (n) = *f*(*n*) *s*_{r}

n |
f(n) |
n |
f(n) |

2 | 2,8 | 17 | 4,9 |

3 | 3,3 | 18 | 4,9 |

4 | 3,6 | 19 | 5,0 |

5 | 3,9 | 20 | 5,0 |

6 | 4,0 | 25 | 5,2 |

7 | 4,2 | 30 | 5,3 |

8 | 4,3 | 35 | 5,4 |

9 | 4,4 | 40 | 5,5 |

10 | 4,5 | 45 | 5,6 |

11 | 4,6 | 50 | 5,6 |

12 | 4,6 | 60 | 5,8 |

13 | 4,7 | 70 | 5,9 |

14 | 4,7 | 80 | 5,9 |

15 | 4,8 | 90 | 6,0 |

16 | 4,8 | 100 | 6,1 |

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ISO 5725-6: 1994 Use in practice of accuracy values

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