If we have a list of numbers, saw cycles to failure for a test. None are negative. And, we do not have time for a complete analysis before being asked about the results.

What can we do?

In this case, the Markov Inequality may prove useful for a quick assessment of the results.

Markov’s Inequality

If the mean of a list of numbers is M, and the list contains no negative number, then the fraction of numbers in the list that are greater than or equal to x is less than or equal to M divided by x.

[fraction of numbers in the list that are greater than or equal to x] ≤ M / x

Example Problem

So, let’s say we have 200 sample run to failure during a test and we have just found the average time to failure is 15 months. How many of the samples could have lasted 12 months or more?

Solution

fraction of samples that survived 75 months or longer ≤ 15 / 75 = 0.2 or 20%

Which doesn’t mean 20% have survived 75 months, it means that at most 20% survived that long. Thus we can look for up to 40 units to conduct further testing to learn why they survived longer than the rest.

This provides a quick and rough way to bound the data without having anything more than the mean and number of items in the list.

If we have the standard deviation, we can use the Chebychev’s inequality for additional accuracy pending further analysis. I like having these two concepts available when time is short and we need to make claims about the data quickly. While not the most accurate, it does provide a bound that has a solid basis in statistics.

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Related:

Chebychev Inequalities (article)

Extended bogey testing (article)

Hypergeometric Distribution (article)