Template:Aw probability plotting - Revision history

Lisa Hacker: Redirected page to Distributions Used in Accelerated Testing

2012-08-09T10:17:27Z

Redirected page to Distributions Used in Accelerated Testing

← Older revision		Revision as of 10:17, 9 August 2012
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	~~====Probability Plotting====~~		#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]

	~~One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example~~ [[~~Reference Appendix D: References\|[18]]].~~

	~~===== Example 3 =====~~

	Let's assume six identical units are being reliability tested at the same application and operation stress levels. All of these units fail during the test after operating the following times (in hours), <span class="texhtml">''T''<sub>''i''</sub></span> : 93, 34, 16, 120, 53 and 75. The steps for using probability plotting to determine the parameters of the Weibull <span class="texhtml">''pdf''</span> are as follows: <br>

	~~:*Rank the failure times in ascending order as shown next.~~

	~~[[Image:failureordernumber.png\|center\|300px]]~~

	~~:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.~~

	~~[[Image:median rank.png\|center\|300px]]~~

	~~:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.~~

	~~<br> [[Image:ALTA4.8.png\|center\|400px]] <br>~~

	~~:*Draw the best possible straight line through the plotted points.~~
	:*Obtain the slope of this line by drawing a line, parallel to the one just obtained, through the slope indicator. This value is the estimate of the shape parameter <math>\widehat{\beta }</math>. In this case <math>\widehat{\beta }=1.4</math>.
	:*At the <span class="texhtml">''Q''(''t'') = 63.2%</span> 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 <math>\widehat{\eta }</math> . For this case <math>\widehat{\eta }=76</math> hours. (This is always at 63.2% since <math>Q(T)=1-{{e}^{-{{(\tfrac{\eta }{\eta })}^{\beta }}}}=1-{{e}^{-1}}=0.632=63.2%</math>.)

	~~<br> [[Image:ALTA4.9.png\|center\|400px~~]] <br> Now any reliability value for any mission time <span class="texhtml">''t''</span> 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 <span class="texhtml">''t'' = 15</span> hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read <span class="texhtml">''Q''(''t'')</span>, in this case <span class="texhtml">''Q''(''t'' = 15) = 9.8%</span>. Thus, <span class="texhtml">''R''(''t'' = 15) = 1 − ''Q''(''t'') = 90.2%</span>. This can also be obtained analytically from the Weibull reliability function since both of the parameters are known. <br>

	~~::<math>R(t=15)={{e}^{-{{\left( \tfrac{15}{\eta } \right)}^{\beta }}}}={{e}^{-{{\left( \tfrac{15}{76} \right)}^{1.4}}}}=90.2%</math>~~

	~~<br>~~

Chris Kahn: /* Example 3 */

2012-06-25T10:08:48Z

Example 3

← Older revision		Revision as of 10:08, 25 June 2012
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	:*Rank the failure times in ascending order as shown next.		:*Rank the failure times in ascending order as shown next.

	[[Image:failureordernumber.png\|center\|~~400px~~]]		[[Image:failureordernumber.png\|center\|300px]]

	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.png\|center\|~~400px~~]]		[[Image:median rank.png\|center\|300px]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.
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	:*At the <span class="texhtml">''Q''(''t'') = 63.2%</span> 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 <math>\widehat{\eta }</math> . For this case <math>\widehat{\eta }=76</math> hours. (This is always at 63.2% since <math>Q(T)=1-{{e}^{-{{(\tfrac{\eta }{\eta })}^{\beta }}}}=1-{{e}^{-1}}=0.632=63.2%</math>.)		:*At the <span class="texhtml">''Q''(''t'') = 63.2%</span> 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 <math>\widehat{\eta }</math> . For this case <math>\widehat{\eta }=76</math> hours. (This is always at 63.2% since <math>Q(T)=1-{{e}^{-{{(\tfrac{\eta }{\eta })}^{\beta }}}}=1-{{e}^{-1}}=0.632=63.2%</math>.)

	<br> [[Image:ALTA4.9.png\|center\|~~200px~~]] <br> Now any reliability value for any mission time <span class="texhtml">''t''</span> 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.		<br> [[Image:ALTA4.9.png\|center\|400px]] <br> Now any reliability value for any mission time <span class="texhtml">''t''</span> 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 <span class="texhtml">''t'' = 15</span> hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read <span class="texhtml">''Q''(''t'')</span>, in this case <span class="texhtml">''Q''(''t'' = 15) = 9.8%</span>. Thus, <span class="texhtml">''R''(''t'' = 15) = 1 − ''Q''(''t'') = 90.2%</span>. This can also be obtained analytically from the Weibull reliability function since both of the parameters are known. <br>		To obtain the value from the plot, draw a vertical line from the abscissa, at <span class="texhtml">''t'' = 15</span> hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read <span class="texhtml">''Q''(''t'')</span>, in this case <span class="texhtml">''Q''(''t'' = 15) = 9.8%</span>. Thus, <span class="texhtml">''R''(''t'' = 15) = 1 − ''Q''(''t'') = 90.2%</span>. This can also be obtained analytically from the Weibull reliability function since both of the parameters are known. <br>

Chris Kahn: /* Example 3 */

2012-06-25T10:08:16Z

Example 3

← Older revision		Revision as of 10:08, 25 June 2012
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	:*Rank the failure times in ascending order as shown next.		:*Rank the failure times in ascending order as shown next.

	[[Image:failureordernumber.png\|center\|~~200px~~]]		[[Image:failureordernumber.png\|center\|400px]]

	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.png\|center\|~~200px~~]]		[[Image:median rank.png\|center\|400px]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.

	<br> [[Image:ALTA4.8.png\|center\|~~200px~~]] <br>		<br> [[Image:ALTA4.8.png\|center\|400px]] <br>

	:*Draw the best possible straight line through the plotted points.		:*Draw the best possible straight line through the plotted points.

Chuck Smith: /* Example 3 */

2012-04-26T16:06:31Z

Example 3

← Older revision		Revision as of 16:06, 26 April 2012
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	:*Rank the failure times in ascending order as shown next.		:*Rank the failure times in ascending order as shown next.

	[[Image:failureordernumber.png\|center\|~~250px~~]]		[[Image:failureordernumber.png\|center\|200px]]

	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.png\|center\|~~250px~~]]		[[Image:median rank.png\|center\|200px]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.

	<br> [[Image:ALTA4.8.png\|center\|~~400px~~]] <br>		<br> [[Image:ALTA4.8.png\|center\|200px]] <br>

	:*Draw the best possible straight line through the plotted points.		:*Draw the best possible straight line through the plotted points.
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	:*At the <span class="texhtml">''Q''(''t'') = 63.2%</span> 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 <math>\widehat{\eta }</math> . For this case <math>\widehat{\eta }=76</math> hours. (This is always at 63.2% since <math>Q(T)=1-{{e}^{-{{(\tfrac{\eta }{\eta })}^{\beta }}}}=1-{{e}^{-1}}=0.632=63.2%</math>.)		:*At the <span class="texhtml">''Q''(''t'') = 63.2%</span> 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 <math>\widehat{\eta }</math> . For this case <math>\widehat{\eta }=76</math> hours. (This is always at 63.2% since <math>Q(T)=1-{{e}^{-{{(\tfrac{\eta }{\eta })}^{\beta }}}}=1-{{e}^{-1}}=0.632=63.2%</math>.)

	<br> [[Image:ALTA4.9.png\|center\|~~400px~~]] <br> Now any reliability value for any mission time <span class="texhtml">''t''</span> 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.		<br> [[Image:ALTA4.9.png\|center\|200px]] <br> Now any reliability value for any mission time <span class="texhtml">''t''</span> 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 <span class="texhtml">''t'' = 15</span> hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read <span class="texhtml">''Q''(''t'')</span>, in this case <span class="texhtml">''Q''(''t'' = 15) = 9.8%</span>. Thus, <span class="texhtml">''R''(''t'' = 15) = 1 − ''Q''(''t'') = 90.2%</span>. This can also be obtained analytically from the Weibull reliability function since both of the parameters are known. <br>		To obtain the value from the plot, draw a vertical line from the abscissa, at <span class="texhtml">''t'' = 15</span> hours, to the fitted line. Draw a horizontal line from this intersection to the ordinate and read <span class="texhtml">''Q''(''t'')</span>, in this case <span class="texhtml">''Q''(''t'' = 15) = 9.8%</span>. Thus, <span class="texhtml">''R''(''t'' = 15) = 1 − ''Q''(''t'') = 90.2%</span>. This can also be obtained analytically from the Weibull reliability function since both of the parameters are known. <br>

Chuck Smith: /* Example 3 */

2012-04-03T17:12:19Z

Example 3

← Older revision		Revision as of 17:12, 3 April 2012
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	:*Rank the failure times in ascending order as shown next.		:*Rank the failure times in ascending order as shown next.

	[[Image:failureordernumber.png\|center]]		[[Image:failureordernumber.png\|center\|250px]]

	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.png\|center]]		[[Image:median rank.png\|center\|250px]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.

Nicolette Young: /* Example 3 */

2012-03-20T18:32:43Z

Example 3

← Older revision		Revision as of 18:32, 20 March 2012
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	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.~~jpg~~\|center]]		[[Image:median rank.png\|center]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.

	<br> [[Image:ALTA4.8.~~jpg~~\|center\|400px]] <br>		<br> [[Image:ALTA4.8.png\|center\|400px]] <br>

	:*Draw the best possible straight line through the plotted points.		:*Draw the best possible straight line through the plotted points.

Nicolette Young: /* Example 3 */

2012-03-20T18:32:16Z

Example 3

← Older revision		Revision as of 18:32, 20 March 2012
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	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.~~png~~\|center]]		[[Image:median rank.jpg\|center]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.

	<br> [[Image:ALTA4.8.~~png~~\|center\|400px]] <br>		<br> [[Image:ALTA4.8.jpg\|center\|400px]] <br>

	:*Draw the best possible straight line through the plotted points.		:*Draw the best possible straight line through the plotted points.

Nicolette Young: /* Example 3 */

2012-03-20T18:27:42Z

Example 3

← Older revision		Revision as of 18:27, 20 March 2012
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	:*Rank the failure times in ascending order as shown next.		:*Rank the failure times in ascending order as shown next.

	[[Image:failureordernumber.png\|center~~\|300px~~]]		[[Image:failureordernumber.png\|center]]

	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.png\|center~~\|300px~~]]		[[Image:median rank.png\|center]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.

Nicolette Young: /* Example 3 */

2012-03-20T17:10:01Z

Example 3

← Older revision		Revision as of 17:10, 20 March 2012
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	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.png\|center\|~~170px~~]]		[[Image:median rank.png\|center\|300px]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.

Nicolette Young: /* Example 3 */

2012-03-20T16:56:10Z

Example 3

← Older revision		Revision as of 16:56, 20 March 2012
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	:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.		:*Obtain their median rank plotting positions. The failure, with their corresponding median ranks, are shown next.

	[[Image:median rank.png\|center\|~~200px~~]]		[[Image:median rank.png\|center\|170px]]

	:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.		:*On a Weibull probability paper, plot the times and their corresponding ranks. The next figure displays an example of a Weibull probability paper.