Understanding Data Distribution in Reliability Analysis

Having a knowledge of how the data is distributed is critical to model failure times and life in reliability analysis. Every distribution is unique and suitable for different types of reliability data. 

Important Definitions

Data Type: Categorical and Continuous. Examples of categorical data include pass-fail, in spec-out of spec, failure reasons etc. Categorical data could be ordinal as well. Examples of continuous data include time, voltage, temperature etc. 

Population: Entire set or collection of observations/measurements of interest.

Sample: A subset of data from the population, typically randomly sampled [each observation having equal chance of being selected]. Random sampling reduces ‘bias’ and incorrect inference about the population as much as possible.

Histogram: Useful data visualization tool to see how the data is distributed in terms of frequency bar chart (link to histogram article). Sample frequency distribution represented by histogram is an estimate of the theoretical model or equation for population data distribution.

Probability Density Function (pdf): Sample frequency distribution used to estimate population model denoted by f(x) to describe continuous data distribution. 

Cumulative Distribution Function (cdf): Cumulative frequency table that accumulates number of observations <= given value. Population model representing sample cumulative frequency distribution and denoted by F(x). Represent cumulative values of pdf till that point. In reliability, we use cdf to measure probability of failure before associated time t. Ranges from 0 to 1.

Mean Life (MTTF): Average time to failure

$$ \displaystyle \bar{T}=m=\int_{0}^{\infty}t\cdot f\left(t\right)dt $$

Median Life: Value of random variable having equal area under pdf to the left and right. 

$$ \displaystyle \int_{-\infty}^{\breve{T}}f\left(t\right)dt=0.5 $$

Some commonly used life distributions in reliability analysis

Exponential Distribution

Constant Failure Rate. Example: life modeling of electronic components having constant failure rate.

pdf is given by:

$$ \displaystyle f\left(t\right)=\lambda e^{-\lambda t} $$ $$ \displaystyle \lambda=\frac{1}{m} $$

Where l = constant failure rate

m = mean time between failures

cdf is given by:

$$ \displaystyle F\left(t\right)=1-R\left(t\right)=1-e^{-\lambda t} $$

Weibull Distribution

Flexible distribution modeling increasing, decreasing or constant failure rates (bathtub curve). Example life data modeling in industry for mechanical and industrial components (wear out failure). Characterized by shape (β) and scale (h) parameter.\

 

pdf given by:

$$ \displaystyle f\left(t\right)=\frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta=1}e^{-\left(\frac{t}{\eta}\right)^{\beta}} $$

cdf given by:

$$ \displaystyle F\left(t\right)=1-e^{-\left(\frac{t}{\eta}\right)^{\beta}} $$

Normal (Gaussian) Distribution

Symmetric bell-shaped curve and failure times are normally distributed. Useful for modeling symmetric life distribution and where central limit theorem applies.

pdf given by

$$ \displaystyle f\left(t\right)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{t-\mu}{\sigma}\right)^{2}} $$

cdf given by

$$ \displaystyle F\left(t\right)=\frac{1}{2}\left[1+erf\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right] $$

Where key parameters µ = mean and s = standard deviation

There are many practical software packages such as JMP and Python scipy libraries for modeling reliability life distributions.

Source: Reliawiki