Template:Weibull parameters probability plotting - Revision history

Lisa Hacker: Redirected page to The Weibull Distribution

2012-08-08T08:25:29Z

Redirected page to The Weibull Distribution

← Older revision		Revision as of 08:25, 8 August 2012
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	~~=== Probability Plotting ===~~		#REDIRECT [[The Weibull Distribution]]

	~~One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from~~ [[~~Appendix:~~ Weibull ~~References\|Kececioglu [20]~~]].


	~~'''Example 1:'''~~
	~~{{Example: Weibull Probability Plot}}~~


	~~'''Probability Plotting for the Location Parameter, <span class="texhtml">γ</span>'''~~

	The third parameter of the Weibull distribution is utilized when the data do not fall on a straight line, but fall on either a concave up or down curve. The following statements can be made regarding the value of <span class="texhtml">γ:</span>

	:''Case 1'': If the curve for MR versus <span class="texhtml">''t''<sub>''j''</sub></span> is concave down and the curve for MR versus <span class="texhtml">(''t''<sub>''j''</sub> − ''t''<sub>1</sub>)</span> is concave up, then there exists a <span class="texhtml">γ</span> such that <span class="texhtml">0 < γ < ''t''<sub>1</sub></span>, or <span class="texhtml">γ</span> has a positive value.

	:''Case 2'': If the curves for MR versus <span class="texhtml">''t''<sub>''j''</sub></span> and MR versus <span class="texhtml">(''t''<sub>''j''</sub> − ''t''<sub>1</sub>)</span> are both concave up, then there exists a negative <span class="texhtml">γ</span> which will straighten out the curve of MR versus <span class="texhtml">''t''<sub>''j''</sub></span>.

	:''Case 3'': If neither one of the previous two cases prevails, then either reject the Weibull as one capable of representing the data, or proceed with the multiple population (mixed Weibull) analysis. To obtain the location parameter, <span class="texhtml">γ:</span>

	~~::*Subtract the same arbitrary value, <span class="texhtml">γ</span>, from all the times to failure and replot the data.~~
	~~::*If the initial curve is concave up, subtract a negative <span class="texhtml">γ</span> from each failure time.~~
	~~::*If the initial curve is concave down, subtract a positive <span class="texhtml">γ</span> from each failure time.~~
	~~::*Repeat until the data plots on an acceptable straight line.~~
	~~::*The value of <span class="texhtml">γ</span> is the subtracted (positive or negative) value that places the points in an acceptable straight line.~~


	The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''t'' − γ).</span>


	~~'''Example 2:'''~~
	~~{{Example: 3P Weibull Distribution}}~~

Harry Guo: /* Probability Plotting */

2012-02-09T22:15:53Z

Probability Plotting

Harry Guo: /* Probability Plotting */

2012-02-09T22:08:07Z

Probability Plotting

← Older revision		Revision as of 22:08, 9 February 2012
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	=== Probability Plotting ===		=== Probability Plotting ===

	One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from Kececioglu [20].		One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from [[Appendix: Weibull References\|Kececioglu [20]]].

Harry Guo: /* Probability Plotting for the Location Parameter, γ */

2012-02-09T21:03:39Z

Probability Plotting for the Location Parameter, γ

← Older revision		Revision as of 21:03, 9 February 2012
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	{{Example: Weibull Probability Plot}}		{{Example: Weibull Probability Plot}}

	~~====~~ Probability Plotting for the Location Parameter, <span class="texhtml">γ</span> ~~====~~		'''Probability Plotting for the Location Parameter, <span class="texhtml">γ</span>'''

	The third parameter of the Weibull distribution is utilized when the data do not fall on a straight line, but fall on either a concave up or down curve. The following statements can be made regarding the value of <span class="texhtml">γ:</span>		The third parameter of the Weibull distribution is utilized when the data do not fall on a straight line, but fall on either a concave up or down curve. The following statements can be made regarding the value of <span class="texhtml">γ:</span>

Harry Guo: /* Probability Plotting */

2012-02-09T19:05:57Z

Probability Plotting

← Older revision		Revision as of 19:05, 9 February 2012
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	One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from Kececioglu [20].		One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from Kececioglu [20].


	'''Example 1:'''		'''Example 1:'''
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	The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''T'' − γ).</span>		The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''T'' − γ).</span>


	'''Example 2:'''		'''Example 2:'''
	{{Example: 3P Weibull Distribution}}		{{Example: 3P Weibull Distribution}}

Harry Guo at 19:05, 9 February 2012

2012-02-09T19:05:35Z

← Older revision		Revision as of 19:05, 9 February 2012
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	'''Example 2:'''		'''Example 2:'''
	{{Example:3P Weibull Distribution}}		{{Example: 3P Weibull Distribution}}

Harry Guo: /* Probability Plotting */

2012-02-09T19:03:59Z

Probability Plotting

← Older revision		Revision as of 19:03, 9 February 2012
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	One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from Kececioglu [20].		One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from Kececioglu [20].

			'''Example 1:'''
	{{Example: Weibull Probability Plot}}		{{Example: Weibull Probability Plot}}

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	The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''T'' − γ).</span>		The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''T'' − γ).</span>

			'''Example 2:'''
	{{Example:3P Weibull Distribution}}		{{Example:3P Weibull Distribution}}

Harry Guo at 17:23, 9 February 2012

2012-02-09T17:23:12Z

@@ Line 3: / Line 3: @@
 One method of calculating the parameters of the Weibull distribution is by using probability plotting. To better illustrate this procedure, consider the following example from Kececioglu [20].
-==== Probability Plotting Example ====
+{{Example: Weibull Probability Plot}}
 ==== Probability Plotting for the Location Parameter, <span class="texhtml">γ</span> ====

Harry Guo: /* Probability Plotting Example Solution */

2012-02-09T17:16:42Z

Probability Plotting Example Solution

← Older revision		Revision as of 17:16, 9 February 2012
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	Assume that 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 number of hours, <span class="texhtml">''T''<sub>''i''</sub></span>: 93, 34, 16, 120, 53 and 75. Estimate the values of the parameters for a two-parameter Weibull distribution and determine the reliability of the units at a time of 15 hours.		Assume that 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 number of hours, <span class="texhtml">''T''<sub>''i''</sub></span>: 93, 34, 16, 120, 53 and 75. Estimate the values of the parameters for a two-parameter Weibull distribution and determine the reliability of the units at a time of 15 hours.

	~~===== Probability Plotting Example~~ Solution ~~=====~~		'''Solution'''

	The steps for determining the parameters of the Weibull representing the data, using probability plotting, are outlined in the following instructions. First, rank the times-to-failure in ascending order as shown next.		The steps for determining the parameters of the Weibull representing the data, using probability plotting, are outlined in the following instructions. First, rank the times-to-failure in ascending order as shown next.

Nicolette Young: /* Three-parameter Weibull distribution example */

2012-02-08T19:47:24Z

Three-parameter Weibull distribution example

← Older revision		Revision as of 19:47, 8 February 2012
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	The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''T'' − γ).</span>		The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''T'' − γ).</span>
	~~===Three-parameter Weibull distribution example===~~		{{Example:3P Weibull Distribution}}

	~~Six identical units are reliability tested under the same stresses and conditions. All units are tested to failure and the following times-to-failure are recorded~~: ~~48, 66, 85, 107, 125 and 152 hours. Find the parameters of the three-parameter~~ Weibull ~~distribution using probability plotting.~~

	~~==== Solution ====~~

	The following figure shows the results. Note that since the original data set was concave down, 17.26 was subtracted from all the times-to-failure and replotted, resulting in a straight line, thus <span class="texhtml">γ = 17.26</span>. (We used Weibull++ to get the results. To perform this by hand, one would attempt different values of <span class="texhtml">γ,</span> using a trial and error methodology, until an acceptable straight line is found. When performed manually, you do not expect decimal accuracy.)
	~~<br>~~
	~~[[Image:lda6.9.gif\|thumb\|center\|500px\|]]~~

← Older revision		Revision as of 22:15, 9 February 2012
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	'''Example 1:'''		'''Example 1:'''
	{{Example: Weibull Probability Plot}}		{{Example: Weibull Probability Plot}}


	'''Probability Plotting for the Location Parameter, <span class="texhtml">γ</span>'''		'''Probability Plotting for the Location Parameter, <span class="texhtml">γ</span>'''
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	The third parameter of the Weibull distribution is utilized when the data do not fall on a straight line, but fall on either a concave up or down curve. The following statements can be made regarding the value of <span class="texhtml">γ:</span>		The third parameter of the Weibull distribution is utilized when the data do not fall on a straight line, but fall on either a concave up or down curve. The following statements can be made regarding the value of <span class="texhtml">γ:</span>

	:''Case 1'': If the curve for MR versus <span class="texhtml">''T''<sub>''j''</sub></span> is concave down and the curve for MR versus <span class="texhtml">(''T''<sub>''j''</sub> − ''T''<sub>1</sub>)</span> is concave up, then there exists a <span class="texhtml">γ</span> such that <span class="texhtml">0 < γ < ''T''<sub>1</sub></span>, or <span class="texhtml">γ</span> has a positive value.		:''Case 1'': If the curve for MR versus <span class="texhtml">''t''<sub>''j''</sub></span> is concave down and the curve for MR versus <span class="texhtml">(''t''<sub>''j''</sub> − ''t''<sub>1</sub>)</span> is concave up, then there exists a <span class="texhtml">γ</span> such that <span class="texhtml">0 < γ < ''t''<sub>1</sub></span>, or <span class="texhtml">γ</span> has a positive value.

	:''Case 2'': If the curves for MR versus <span class="texhtml">''T''<sub>''j''</sub></span> and MR versus <span class="texhtml">(''T''<sub>''j''</sub> − ''T''<sub>1</sub>)</span> are both concave up, then there exists a negative <span class="texhtml">γ</span> which will straighten out the curve of MR versus <span class="texhtml">''T''<sub>''j''</sub></span>.		:''Case 2'': If the curves for MR versus <span class="texhtml">''t''<sub>''j''</sub></span> and MR versus <span class="texhtml">(''t''<sub>''j''</sub> − ''t''<sub>1</sub>)</span> are both concave up, then there exists a negative <span class="texhtml">γ</span> which will straighten out the curve of MR versus <span class="texhtml">''t''<sub>''j''</sub></span>.

	:''Case 3'': If neither one of the previous two cases prevails, then either reject the Weibull as one capable of representing the data, or proceed with the multiple population (mixed Weibull) analysis. To obtain the location parameter, <span class="texhtml">γ:</span>		:''Case 3'': If neither one of the previous two cases prevails, then either reject the Weibull as one capable of representing the data, or proceed with the multiple population (mixed Weibull) analysis. To obtain the location parameter, <span class="texhtml">γ:</span>


	::*Subtract the same arbitrary value, <span class="texhtml">γ</span>, from all the times to failure and replot the data.		::*Subtract the same arbitrary value, <span class="texhtml">γ</span>, from all the times to failure and replot the data.
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	The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''T'' − γ).</span>		The other two parameters are then obtained using the techniques previously described. Also, it is important to note that we used the term subtract a positive or negative gamma, where subtracting a negative gamma is equivalent to adding it. Note that when adjusting for gamma, the x-axis scale for the straight line becomes <span class="texhtml">(''t'' − γ).</span>


	'''Example 2:'''		'''Example 2:'''
	{{Example: 3P Weibull Distribution}}		{{Example: 3P Weibull Distribution}}