# Sidebar: Into the Bathtub

The definition of MTBF uses the word mean, an arithmetic average. This has led some people to interpret MTBF as the time (on average) when half the items will fail. That might be the case if failure occurs at a constant rate during an item's lifetime, and in fact MTBF makes just this assumption, even though it's rarely the case in the real world.

For example, many electronic components may experience a relatively high failure rate in their first few hours of operation and then operate essentially trouble-free for very long periods thereafter.

Thus, the failure rate at any point in time depends on the overall failure profile for that system, which we can express as the probability of failure prior to a specified time. If we calculate the failure rate for ever-smaller time intervals during the projected life span, we can determine what's called a hazard function, the instantaneous failure rate at any point in time.

It turns out that there's one failure profile shared by many mechanical devices, including especially complex systems - think of an automobile, with its thousands of parts. This hazard function is called the "bathtub" curve because of its shape, and it's characterized by three distinct phases:

• Early on, a high but decreasing failure rate, sometimes called infant mortality.

• A relatively constant failure rate, basically representing random failures.

• Near the end of life, an increasing failure rate as the product wears out.

Think of a new car. When you first get it, there may be a number of items that need fixing either because of improper installation or a hidden defect in a part. (Since these are usually covered by warranty, they're often an annoyance but perhaps not a major problem, but that's a different issue.) After the newness wears off, while the car is still relatively young and doesn't have too many miles on it, it's rarely out of commission; repair bills are few and far between, and problems are likely to be caused by random events, such as an accident or a part failure. As the years and mileage mount, however, a number of parts wear out—many of them, such as brakes and tires, are designed to last only for a limited period and then fail or wear out. At this stage, the car spends more and more time in the repair shop. If we graph the incidence of failures (repairs) against time, we get the characteristic bathtub-shaped curve.

FAILURES AGAINST TIME
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