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New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/accelerated_life_testing_data_analysis

Chapter 3: Distributions Used in Accelerated Testing


ALTAbox.png

Chapter 3  
Distributions Used in Accelerated Testing  

Synthesis-icon.png

Available Software:
ALTA

Examples icon.png

More Resources:
ALTA Examples


Life Distributions


In this section we will briefly present three lifetime distributions commonly used in accelerated life test analysis, namely the 1-parameter exponential, the 2-parameter Weibull and the lognormal distributions. Readers who are interested in a more rigorous overview or in different forms of these and other life distributions can refer to ReliaSoft's Life Data Analysis Reference.

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/accelerated_life_testing_data_analysis

Chapter 3: Distributions Used in Accelerated Testing


ALTAbox.png

Chapter 3  
Distributions Used in Accelerated Testing  

Synthesis-icon.png

Available Software:
ALTA

Examples icon.png

More Resources:
ALTA Examples


Life Distributions


In this section we will briefly present three lifetime distributions commonly used in accelerated life test analysis, namely the 1-parameter exponential, the 2-parameter Weibull and the lognormal distributions. Readers who are interested in a more rigorous overview or in different forms of these and other life distributions can refer to ReliaSoft's Life Data Analysis Reference.

Template loop detected: Template:Alta exponential distribution

Parameter Estimation

The parameter of the exponential distribution can be estimated graphically on probability plotting paper or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in Appendix B.)


Let's assume six identical units are reliability tested at the same application and operation stress levels. All of these units fail during the test after operating for the following times (in hours): 96, 257, 498, 763, 1051 and 1744.

The steps for using the probability plotting method to determine the parameters of the exponential pdf representing the data are as follows:

Rank the times-to-failure in ascending order as shown next.

[math]\displaystyle{ \begin{matrix} \text{Failure} & \text{Failure Order Number} \\ \text{Time (Hr)} & \text{out of a Sample Size of 6} \\ \text{96} & \text{1} \\ \text{257} & \text{2} \\ \text{498} & \text{3} \\ \text{763} & \text{4} \\ \text{1,051} & \text{5} \\ \text{1,744} & \text{6} \\ \end{matrix}\,\! }[/math]

Obtain their median rank plotting positions. Median rank positions are used instead of other ranking methods because median ranks are at a specific confidence level (50%).

The times-to-failure, with their corresponding median ranks, are shown next:

[math]\displaystyle{ \begin{matrix} \text{Failure} & \text{Median} \\ \text{Time (Hr)} & \text{Rank, }% \\ \text{96} & \text{10}\text{.91} \\ \text{257} & \text{26}\text{.44} \\ \text{498} & \text{42}\text{.14} \\ \text{763} & \text{57}\text{.86} \\ \text{1,051} & \text{73}\text{.56} \\ \text{1,744} & \text{89}\text{.10} \\ \end{matrix}\,\! }[/math]

On an exponential probability paper, plot the times on the x-axis and their corresponding rank value on the y-axis. The next figure displays an example of an exponential probability paper. The paper is simply a log-linear paper.

ALTA4.1.png

Draw the best possible straight line that goes through the [math]\displaystyle{ t=0\,\! }[/math] and [math]\displaystyle{ (t)=100%\,\! }[/math] point and through the plotted points (as shown in the plot below).

At the [math]\displaystyle{ Q(t)=63.2%\,\! }[/math] or [math]\displaystyle{ R(t)=36.8%\,\! }[/math] ordinate point, draw a straight horizontal line until this line intersects the fitted straight line. Draw a vertical line through this intersection until it crosses the abscissa. The value at the intersection of the abscissa is the estimate of the mean. For this case, [math]\displaystyle{ \widehat{\mu }=833\,\! }[/math] hours which means that [math]\displaystyle{ \lambda =\tfrac{1}{\mu }=0.0012\,\! }[/math] (This is always at 63.2% because [math]\displaystyle{ (T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%)\,\! }[/math].

ALTA4.2.png

Now any reliability value for any mission time [math]\displaystyle{ t\,\! }[/math] can be obtained. For example, the reliability for a mission of 15 hours, or any other time, can now be obtained either from the plot or analytically.

To obtain the value from the plot, draw a vertical line from the abscissa, at [math]\displaystyle{ t=15\,\! }[/math] hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read [math]\displaystyle{ R(t)\,\! }[/math]. In this case, [math]\displaystyle{ R(t=15)=98.15%\,\! }[/math]. This can also be obtained analytically, from the exponential reliability function.

Template loop detected: Template:Alta weibull distribution

Template loop detected: Template:Alta ld

Parameter Estimation

The parameter of the exponential distribution can be estimated graphically on probability plotting paper or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in Appendix B.)


Let's assume six identical units are reliability tested at the same application and operation stress levels. All of these units fail during the test after operating for the following times (in hours): 96, 257, 498, 763, 1051 and 1744.

The steps for using the probability plotting method to determine the parameters of the exponential pdf representing the data are as follows:

Rank the times-to-failure in ascending order as shown next.

[math]\displaystyle{ \begin{matrix} \text{Failure} & \text{Failure Order Number} \\ \text{Time (Hr)} & \text{out of a Sample Size of 6} \\ \text{96} & \text{1} \\ \text{257} & \text{2} \\ \text{498} & \text{3} \\ \text{763} & \text{4} \\ \text{1,051} & \text{5} \\ \text{1,744} & \text{6} \\ \end{matrix}\,\! }[/math]

Obtain their median rank plotting positions. Median rank positions are used instead of other ranking methods because median ranks are at a specific confidence level (50%).

The times-to-failure, with their corresponding median ranks, are shown next:

[math]\displaystyle{ \begin{matrix} \text{Failure} & \text{Median} \\ \text{Time (Hr)} & \text{Rank, }% \\ \text{96} & \text{10}\text{.91} \\ \text{257} & \text{26}\text{.44} \\ \text{498} & \text{42}\text{.14} \\ \text{763} & \text{57}\text{.86} \\ \text{1,051} & \text{73}\text{.56} \\ \text{1,744} & \text{89}\text{.10} \\ \end{matrix}\,\! }[/math]

On an exponential probability paper, plot the times on the x-axis and their corresponding rank value on the y-axis. The next figure displays an example of an exponential probability paper. The paper is simply a log-linear paper.

ALTA4.1.png

Draw the best possible straight line that goes through the [math]\displaystyle{ t=0\,\! }[/math] and [math]\displaystyle{ (t)=100%\,\! }[/math] point and through the plotted points (as shown in the plot below).

At the [math]\displaystyle{ Q(t)=63.2%\,\! }[/math] or [math]\displaystyle{ R(t)=36.8%\,\! }[/math] ordinate point, draw a straight horizontal line until this line intersects the fitted straight line. Draw a vertical line through this intersection until it crosses the abscissa. The value at the intersection of the abscissa is the estimate of the mean. For this case, [math]\displaystyle{ \widehat{\mu }=833\,\! }[/math] hours which means that [math]\displaystyle{ \lambda =\tfrac{1}{\mu }=0.0012\,\! }[/math] (This is always at 63.2% because [math]\displaystyle{ (T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%)\,\! }[/math].

ALTA4.2.png

Now any reliability value for any mission time [math]\displaystyle{ t\,\! }[/math] can be obtained. For example, the reliability for a mission of 15 hours, or any other time, can now be obtained either from the plot or analytically.

To obtain the value from the plot, draw a vertical line from the abscissa, at [math]\displaystyle{ t=15\,\! }[/math] hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read [math]\displaystyle{ R(t)\,\! }[/math]. In this case, [math]\displaystyle{ R(t=15)=98.15%\,\! }[/math]. This can also be obtained analytically, from the exponential reliability function.

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/accelerated_life_testing_data_analysis

Chapter 3: Distributions Used in Accelerated Testing


ALTAbox.png

Chapter 3  
Distributions Used in Accelerated Testing  

Synthesis-icon.png

Available Software:
ALTA

Examples icon.png

More Resources:
ALTA Examples


Life Distributions


In this section we will briefly present three lifetime distributions commonly used in accelerated life test analysis, namely the 1-parameter exponential, the 2-parameter Weibull and the lognormal distributions. Readers who are interested in a more rigorous overview or in different forms of these and other life distributions can refer to ReliaSoft's Life Data Analysis Reference.

Template loop detected: Template:Alta exponential distribution

Parameter Estimation

The parameter of the exponential distribution can be estimated graphically on probability plotting paper or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in Appendix B.)


Let's assume six identical units are reliability tested at the same application and operation stress levels. All of these units fail during the test after operating for the following times (in hours): 96, 257, 498, 763, 1051 and 1744.

The steps for using the probability plotting method to determine the parameters of the exponential pdf representing the data are as follows:

Rank the times-to-failure in ascending order as shown next.

[math]\displaystyle{ \begin{matrix} \text{Failure} & \text{Failure Order Number} \\ \text{Time (Hr)} & \text{out of a Sample Size of 6} \\ \text{96} & \text{1} \\ \text{257} & \text{2} \\ \text{498} & \text{3} \\ \text{763} & \text{4} \\ \text{1,051} & \text{5} \\ \text{1,744} & \text{6} \\ \end{matrix}\,\! }[/math]

Obtain their median rank plotting positions. Median rank positions are used instead of other ranking methods because median ranks are at a specific confidence level (50%).

The times-to-failure, with their corresponding median ranks, are shown next:

[math]\displaystyle{ \begin{matrix} \text{Failure} & \text{Median} \\ \text{Time (Hr)} & \text{Rank, }% \\ \text{96} & \text{10}\text{.91} \\ \text{257} & \text{26}\text{.44} \\ \text{498} & \text{42}\text{.14} \\ \text{763} & \text{57}\text{.86} \\ \text{1,051} & \text{73}\text{.56} \\ \text{1,744} & \text{89}\text{.10} \\ \end{matrix}\,\! }[/math]

On an exponential probability paper, plot the times on the x-axis and their corresponding rank value on the y-axis. The next figure displays an example of an exponential probability paper. The paper is simply a log-linear paper.

ALTA4.1.png

Draw the best possible straight line that goes through the [math]\displaystyle{ t=0\,\! }[/math] and [math]\displaystyle{ (t)=100%\,\! }[/math] point and through the plotted points (as shown in the plot below).

At the [math]\displaystyle{ Q(t)=63.2%\,\! }[/math] or [math]\displaystyle{ R(t)=36.8%\,\! }[/math] ordinate point, draw a straight horizontal line until this line intersects the fitted straight line. Draw a vertical line through this intersection until it crosses the abscissa. The value at the intersection of the abscissa is the estimate of the mean. For this case, [math]\displaystyle{ \widehat{\mu }=833\,\! }[/math] hours which means that [math]\displaystyle{ \lambda =\tfrac{1}{\mu }=0.0012\,\! }[/math] (This is always at 63.2% because [math]\displaystyle{ (T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%)\,\! }[/math].

ALTA4.2.png

Now any reliability value for any mission time [math]\displaystyle{ t\,\! }[/math] can be obtained. For example, the reliability for a mission of 15 hours, or any other time, can now be obtained either from the plot or analytically.

To obtain the value from the plot, draw a vertical line from the abscissa, at [math]\displaystyle{ t=15\,\! }[/math] hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read [math]\displaystyle{ R(t)\,\! }[/math]. In this case, [math]\displaystyle{ R(t=15)=98.15%\,\! }[/math]. This can also be obtained analytically, from the exponential reliability function.

Template loop detected: Template:Alta weibull distribution

Template loop detected: Template:Alta ld

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/accelerated_life_testing_data_analysis

Chapter 3: Distributions Used in Accelerated Testing


ALTAbox.png

Chapter 3  
Distributions Used in Accelerated Testing  

Synthesis-icon.png

Available Software:
ALTA

Examples icon.png

More Resources:
ALTA Examples


Life Distributions


In this section we will briefly present three lifetime distributions commonly used in accelerated life test analysis, namely the 1-parameter exponential, the 2-parameter Weibull and the lognormal distributions. Readers who are interested in a more rigorous overview or in different forms of these and other life distributions can refer to ReliaSoft's Life Data Analysis Reference.

Template loop detected: Template:Alta exponential distribution

Parameter Estimation

The parameter of the exponential distribution can be estimated graphically on probability plotting paper or analytically using either least squares or maximum likelihood. (Parameter estimation methods are presented in detail in Appendix B.)


Let's assume six identical units are reliability tested at the same application and operation stress levels. All of these units fail during the test after operating for the following times (in hours): 96, 257, 498, 763, 1051 and 1744.

The steps for using the probability plotting method to determine the parameters of the exponential pdf representing the data are as follows:

Rank the times-to-failure in ascending order as shown next.

[math]\displaystyle{ \begin{matrix} \text{Failure} & \text{Failure Order Number} \\ \text{Time (Hr)} & \text{out of a Sample Size of 6} \\ \text{96} & \text{1} \\ \text{257} & \text{2} \\ \text{498} & \text{3} \\ \text{763} & \text{4} \\ \text{1,051} & \text{5} \\ \text{1,744} & \text{6} \\ \end{matrix}\,\! }[/math]

Obtain their median rank plotting positions. Median rank positions are used instead of other ranking methods because median ranks are at a specific confidence level (50%).

The times-to-failure, with their corresponding median ranks, are shown next:

[math]\displaystyle{ \begin{matrix} \text{Failure} & \text{Median} \\ \text{Time (Hr)} & \text{Rank, }% \\ \text{96} & \text{10}\text{.91} \\ \text{257} & \text{26}\text{.44} \\ \text{498} & \text{42}\text{.14} \\ \text{763} & \text{57}\text{.86} \\ \text{1,051} & \text{73}\text{.56} \\ \text{1,744} & \text{89}\text{.10} \\ \end{matrix}\,\! }[/math]

On an exponential probability paper, plot the times on the x-axis and their corresponding rank value on the y-axis. The next figure displays an example of an exponential probability paper. The paper is simply a log-linear paper.

ALTA4.1.png

Draw the best possible straight line that goes through the [math]\displaystyle{ t=0\,\! }[/math] and [math]\displaystyle{ (t)=100%\,\! }[/math] point and through the plotted points (as shown in the plot below).

At the [math]\displaystyle{ Q(t)=63.2%\,\! }[/math] or [math]\displaystyle{ R(t)=36.8%\,\! }[/math] ordinate point, draw a straight horizontal line until this line intersects the fitted straight line. Draw a vertical line through this intersection until it crosses the abscissa. The value at the intersection of the abscissa is the estimate of the mean. For this case, [math]\displaystyle{ \widehat{\mu }=833\,\! }[/math] hours which means that [math]\displaystyle{ \lambda =\tfrac{1}{\mu }=0.0012\,\! }[/math] (This is always at 63.2% because [math]\displaystyle{ (T)=1-{{e}^{-\tfrac{\mu }{\mu }}}=1-{{e}^{-1}}=0.632=63.2%)\,\! }[/math].

ALTA4.2.png

Now any reliability value for any mission time [math]\displaystyle{ t\,\! }[/math] can be obtained. For example, the reliability for a mission of 15 hours, or any other time, can now be obtained either from the plot or analytically.

To obtain the value from the plot, draw a vertical line from the abscissa, at [math]\displaystyle{ t=15\,\! }[/math] hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read [math]\displaystyle{ R(t)\,\! }[/math]. In this case, [math]\displaystyle{ R(t=15)=98.15%\,\! }[/math]. This can also be obtained analytically, from the exponential reliability function.

Template loop detected: Template:Alta weibull distribution

Template loop detected: Template:Alta ld