Statistics for Maintenance Modelling: Continuous Lifetime Distributions 1

# Using the Gumbel Distribution to Model Breaking Strengths of Wires¤

The breaking strengths of 15 equal length wires are 48, 52, 58, 61, 62, 63, 64, 65, 69, 69, 72, 72.5, 75, 75 and 76 Newtons. Assigning a cumulative probability of 1/15 to 48, 2/15 to 52, etc., and then plotting the reliability function gives the solid line in this figure. The reliability function for a Gumbel distribution with = 69.5 and = 7.8 is shown as the dotted line.

This data is taken from Practical Reliability Engineering by O'Connor (1995) p. 82.

Note: Assigning multiples of 1/15 for the cumulative probabilities is slightly inaccurate as we want to find the mid-points of the probability bands and these bands are narrower nearer the mean. More accurate cumulative probabilities are given by Bérnard's median rank value of , where i is the ranked order of occurrence (i.e. 48 is 1, 52 is 2, etc.) and n is the number of observations. Hence 48 would receive a value of 0.045, 52 a value of 0.110, etc.