We propose a new measure of reliability, called the cumulative mean intervals, that assesses the mean behaviour of a process by computing the probability that the cumulative sample mean will remain below its long-term sample mean with a given tolerance over a period of time. We further derive a lower bound for the measure when the underlying data is independent and identically distributed with a normal distribution. This deduction provides a preliminary basis for parallel extensions to the two limiting case when we compute the probability that the sample mean stays within a given distance from the true mean with no assumptions made on independence and normality.

KEYWORDS: cumulative mean, probability, reliability, intervals, sample mean.