# Hazard Rate: Definition, How To Calculate, and Example

## What Is the Hazard Rate?

The hazard rate refers to the rate of death for an item of a given age (x). It is part of a larger equation called the hazard function, which analyzes the likelihood that an item will survive to a certain point in time, based on its survival to an earlier time (t). In other words, it is the likelihood that if something survives to one moment, it will also survive to the next.

The hazard rate only applies to items that cannot be repaired and is sometimes referred to as the failure rate. It is fundamental to the design of safe systems in applications and is often relied on in commerce, engineering, finance, insurance, and regulatory industries.

### Key Takeaways

• The hazard rate refers to the rate of death for an item of a given age (x).
• It is part of a larger equation called the hazard function, which analyzes the likelihood that an item will survive to a certain point in time based on its survival to an earlier time (t).
• The hazard rate cannot be negative, and it requires a set "lifetime" on which to model the equation.

## Understanding the Hazard Rate

The hazard rate measures the propensity of an item to fail or die depending on the age it has reached. It is part of a wider branch of statistics called survival analysis, which is a set of methods for predicting the amount of time until a certain event occurs, such as the death or failure of an engineering system or component.

The concept is applied to other branches of research under slightly different names, including reliability analysis (engineering), duration analysis (economics), and event history analysis (sociology).

## Calculating the Hazard Rate

The hazard rate for any time can be determined using the following equation:

$h(t) = f(t) / R(t)$

F(t) is the probability density function (PDF), or the probability that the value (failure or death) will fall in a specified interval, such as a specific year. R(t), on the other hand, represents the survival function, or the probability that something will survive past a certain time (t).

The hazard rate cannot be negative, and it is necessary to have a set "lifetime" on which to model the equation.

## Example of the Hazard Rate

The probability density calculates the probability of failure at any given time. For instance, a person is certain to die eventually. As you get older, you have a greater chance of dying at a specific age, since the average failure rate is calculated as a fraction of the number of units that exist in a specific interval, divided by the number of total units at the beginning of the interval.

If we were to calculate a person's chances of dying at a certain age, we would divide one year by the number of years that person potentially has left to live. This number would grow larger each year.

A person aged 60 would have a higher probability of dying at age 65 than a person aged 30 because the person aged 30 still has many more units of time (years) left in their life, and the probability that the person will die during one specific unit of time is lower.

## The Shape of the Hazard Rate Curve

In many instances, the hazard rate can resemble the shape of a bathtub. The curve slopes downwards at the beginning, indicating a decreasing hazard rate, then levels out to be constant, before moving upwards as the item in question ages.

Think of it this way: when an auto manufacturer puts together a car, its components are not expected to fail in its first few years of service. However, as the car ages, the probability of malfunction increases. By the time the curve slopes upwards, the useful life period of the product has expired and the chance of non-random issues suddenly occurring becomes much more likely.

## What Is the Difference Between the Failure Rate and Hazard Rate?

The hazard rate and failure rate are essentially the same thing. Failure rate is basically another way to say hazard rate.

## What Is the Hazard Rate Used For?

The hazard rate seeks to determine the chances of survival of its subject at a certain time point. It can effectively be applied to any item with a set lifetime and is commonly used in engineering, medicine, and insurance.

## What Is the Bathtub Hazard Rate Curve?

The bathtub curve is a visual representation of the typical failure rate of a product or group of products over time. The bathtub image maps out three distinct periods. The initial downward slope, also known as “infant mortality,” represents the timeframe when the product is first used and goes from a high failure occurrence to a low one. Next, the line levels out and becomes flat, depicting the useful life period when assets are expected to have a relatively constant failure risk. Then finally the slope moves upward, taking the form of a bathtub, as the product in question enters its wear-out period and degrades at an accelerated pace.

## The Bottom Line

The hazard rate is a simple yet effective way to determine the likelihood that an item will survive to a given time point. That might not sound particularly useful or groundbreaking. However, in certain industries, the concept is used to make key decisions.

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1. Graph Pad. "Hazards and Hazard Rates."

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