Lognormal Distribution Characteristics - Revision history

Lisa Hacker at 21:40, 18 September 2023

2023-09-18T21:40:38Z

← Older revision		Revision as of 21:40, 18 September 2023
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	<noinclude>		<noinclude>
	''This article also appears in the [~~[The_Lognormal_Distribution\|~~Life ~~Data Analysis Reference]~~] and [~~[Distributions_Used_in_Accelerated_Testing\|~~Accelerated ~~Life Testing Data Analysis Reference]~~] ~~books~~.'' </noinclude>		''This article also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference] and [https://help.reliasoft.com/reference/accelerated_life_testing_data_analysis Accelerated life testing reference].'' </noinclude>
	[[Image:WB.10 effect of sigma.png\|center\|450px\| ]]		[[Image:WB.10 effect of sigma.png\|center\|450px\| ]]

Nicolette Young at 15:16, 17 December 2015

2015-12-17T15:16:30Z

← Older revision		Revision as of 15:16, 17 December 2015
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	<noinclude>		<noinclude>
	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	[[Image:WB.10 effect of sigma.png\|center\|~~400px~~\| ]]		[[Image:WB.10 effect of sigma.png\|center\|450px\| ]]

	*The lognormal distribution is a distribution skewed to the right.		*The lognormal distribution is a distribution skewed to the right.
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	*The degree of skewness increases as <math>{{\sigma'}}\,\!</math> increases, for a given <math>\mu'\,\!</math>		*The degree of skewness increases as <math>{{\sigma'}}\,\!</math> increases, for a given <math>\mu'\,\!</math>

	[[Image:WB.10 lognormal pdf.png\|center\|~~400px~~\| ]]		[[Image:WB.10 lognormal pdf.png\|center\|450px\| ]]

	*For the same <math>{{\sigma'}}\,\!</math>, the ''pdf'' 's skewness increases as <math>{\mu }'\,\!</math> increases.		*For the same <math>{{\sigma'}}\,\!</math>, the ''pdf'' 's skewness increases as <math>{\mu }'\,\!</math> increases.

Richard House at 18:08, 24 September 2012

2012-09-24T18:08:25Z

Richard House at 20:33, 29 August 2012

2012-08-29T20:33:10Z

← Older revision		Revision as of 20:33, 29 August 2012
Line 5:		Line 5:

	*The lognormal distribution is a distribution skewed to the right.		*The lognormal distribution is a distribution skewed to the right.
	*The ~~<math>~~pdf~~</math>~~ starts at zero, increases to its mode, and decreases thereafter.		*The ''pdf'' starts at zero, increases to its mode, and decreases thereafter.
	*The degree of skewness increases as <math>{{\sigma'}}\,\!</math> increases, for a given <math>\mu'\,\!</math>		*The degree of skewness increases as <math>{{\sigma'}}\,\!</math> increases, for a given <math>\mu'\,\!</math>

	[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]		[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]

	*For the same <math>{{\sigma'}}\,\!</math> , the ~~<math>~~pdf~~</math>~~ 's skewness increases as <math>{\mu }'\,\!</math> increases.		*For the same <math>{{\sigma'}}\,\!</math> , the ''pdf'' 's skewness increases as <math>{\mu }'\,\!</math> increases.
	*For <math>{{\sigma' }}\,\!</math> values significantly greater than 1, the ~~<math>~~pdf~~</math>~~ rises very sharply in the beginning, (i.e., for very small values of <math>T\,\!</math> near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential ~~<math>~~pdf~~</math>~~ or a Weibull ~~<math>~~pdf~~</math>~~ with <math>0<\beta <1\,\!</math> .		*For <math>{{\sigma' }}\,\!</math> values significantly greater than 1, the ''pdf'' rises very sharply in the beginning, (i.e., for very small values of <math>T\,\!</math> near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential ''pdf'' or a Weibull ''pdf'' with <math>0<\beta <1\,\!</math> .
	*The parameter, <math>{\mu }'\,\!</math>, in terms of the logarithm of the <math>{T}'s\,\!</math> is also the scale parameter, and not the location parameter as in the case of the normal ~~<math>~~pdf~~</math>~~ .		*The parameter, <math>{\mu }'\,\!</math>, in terms of the logarithm of the <math>{T}'s\,\!</math> is also the scale parameter, and not the location parameter as in the case of the normal ''pdf''.
	*The parameter <math>{{\sigma'}}\,\!</math>, or the standard deviation of the <math>{T}'s\,\!</math> in terms of their logarithm or of their <math>{T}'\,\!</math>, is also the shape parameter and not the scale parameter, as in the normal ~~<math>~~pdf~~</math>~~, and assumes only positive values.		*The parameter <math>{{\sigma'}}\,\!</math>, or the standard deviation of the <math>{T}'s\,\!</math> in terms of their logarithm or of their <math>{T}'\,\!</math>, is also the shape parameter and not the scale parameter, as in the normal ''pdf'', and assumes only positive values.

	'''Lognormal Distribution Parameters in ReliaSoft's Software'''		'''Lognormal Distribution Parameters in ReliaSoft's Software'''

	In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter <math>{\mu }'\,\!</math> represents the mean of the natural logarithms of the times-to-failure, while <math>{{\sigma' }}\,\!</math> represents the standard deviation of these data point logarithms. Specifically, the returned <math>{{\sigma' }}\,\!</math> is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.		In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter <math>{\mu }'\,\!</math> represents the mean of the natural logarithms of the times-to-failure, while <math>{{\sigma' }}\,\!</math> represents the standard deviation of these data point logarithms. Specifically, the returned <math>{{\sigma' }}\,\!</math> is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.

Richard House at 03:06, 22 August 2012

2012-08-22T03:06:49Z

← Older revision		Revision as of 03:06, 22 August 2012
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	*The lognormal distribution is a distribution skewed to the right.		*The lognormal distribution is a distribution skewed to the right.
	*The <math>pdf</math> starts at zero, increases to its mode, and decreases thereafter.		*The <math>pdf</math> starts at zero, increases to its mode, and decreases thereafter.
	*The degree of skewness increases as <math>{{\sigma'}}</math> increases, for a given <math>\mu'</math>		*The degree of skewness increases as <math>{{\sigma'}}\,\!</math> increases, for a given <math>\mu'\,\!</math>

	[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]		[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]

	*For the same <math>{{\sigma'}}</math> , the <math>pdf</math> 's skewness increases as <math>{\mu }'</math> increases.		*For the same <math>{{\sigma'}}\,\!</math> , the <math>pdf</math> 's skewness increases as <math>{\mu }'\,\!</math> increases.
	*For <math>{{\sigma' }}</math> values significantly greater than 1, the <math>pdf</math> rises very sharply in the beginning, (i.e., for very small values of <math>T</math> near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential <math>pdf</math> or a Weibull <math>pdf</math> with <math>0<\beta <1</math> .		*For <math>{{\sigma' }}\,\!</math> values significantly greater than 1, the <math>pdf</math> rises very sharply in the beginning, (i.e., for very small values of <math>T\,\!</math> near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential <math>pdf</math> or a Weibull <math>pdf</math> with <math>0<\beta <1\,\!</math> .
	*The parameter, <math>{\mu }'</math>, in terms of the logarithm of the <math>{T}'s</math> is also the scale parameter, and not the location parameter as in the case of the normal <math>pdf</math> .		*The parameter, <math>{\mu }'\,\!</math>, in terms of the logarithm of the <math>{T}'s\,\!</math> is also the scale parameter, and not the location parameter as in the case of the normal <math>pdf</math> .
	*The parameter <math>{{\sigma'}}</math>, or the standard deviation of the <math>{T}'s</math> in terms of their logarithm or of their <math>{T}'</math>, is also the shape parameter and not the scale parameter, as in the normal <math>pdf</math>, and assumes only positive values.		*The parameter <math>{{\sigma'}}\,\!</math>, or the standard deviation of the <math>{T}'s\,\!</math> in terms of their logarithm or of their <math>{T}'\,\!</math>, is also the shape parameter and not the scale parameter, as in the normal <math>pdf</math>, and assumes only positive values.

	'''Lognormal Distribution Parameters in ReliaSoft's Software'''		'''Lognormal Distribution Parameters in ReliaSoft's Software'''

	In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter <math>{\mu }'</math> represents the mean of the natural logarithms of the times-to-failure, while <math>{{\sigma' }}</math> represents the standard deviation of these data point logarithms. Specifically, the returned <math>{{\sigma' }}</math> is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.		In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter <math>{\mu }'\,\!</math> represents the mean of the natural logarithms of the times-to-failure, while <math>{{\sigma' }}\,\!</math> represents the standard deviation of these data point logarithms. Specifically, the returned <math>{{\sigma' }}\,\!</math> is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.

Richard House at 04:44, 8 August 2012

2012-08-08T04:44:04Z

← Older revision		Revision as of 04:44, 8 August 2012
Line 11:		Line 11:

	*For the same <math>{{\sigma'}}</math> , the <math>pdf</math> 's skewness increases as <math>{\mu }'</math> increases.		*For the same <math>{{\sigma'}}</math> , the <math>pdf</math> 's skewness increases as <math>{\mu }'</math> increases.
	*For <math>{{\sigma' }}</math> values significantly greater than 1, the <math>pdf</math> rises very sharply in the beginning, i.e. for very small values of <math>T</math> near zero, and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential <math>pdf</math> or a Weibull <math>pdf</math> with <math>0<\beta <1</math> .		*For <math>{{\sigma' }}</math> values significantly greater than 1, the <math>pdf</math> rises very sharply in the beginning, (i.e., for very small values of <math>T</math> near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential <math>pdf</math> or a Weibull <math>pdf</math> with <math>0<\beta <1</math> .
	*The parameter, <math>{\mu }'</math>, in terms of the logarithm of the <math>{T}'s</math> is also the scale parameter, and not the location parameter as in the case of the normal <math>pdf</math> .		*The parameter, <math>{\mu }'</math>, in terms of the logarithm of the <math>{T}'s</math> is also the scale parameter, and not the location parameter as in the case of the normal <math>pdf</math> .
	*The parameter <math>{{\sigma'}}</math>, or the standard deviation of the <math>{T}'s</math> in terms of their logarithm or of their <math>{T}'</math>, is also the shape parameter and not the scale parameter, as in the normal <math>pdf</math>, and assumes only positive values.		*The parameter <math>{{\sigma'}}</math>, or the standard deviation of the <math>{T}'s</math> in terms of their logarithm or of their <math>{T}'</math>, is also the shape parameter and not the scale parameter, as in the normal <math>pdf</math>, and assumes only positive values.


	'''Lognormal Distribution Parameters in ReliaSoft's Software'''		'''Lognormal Distribution Parameters in ReliaSoft's Software'''

	In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter <math>{\mu }'</math> represents the mean of the natural logarithms of the times-to-failure, while <math>{{\sigma' }}</math> represents the standard deviation of these data point logarithms. Specifically, the returned <math>{{\sigma' }}</math> is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.		In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter <math>{\mu }'</math> represents the mean of the natural logarithms of the times-to-failure, while <math>{{\sigma' }}</math> represents the standard deviation of these data point logarithms. Specifically, the returned <math>{{\sigma' }}</math> is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.

Lisa Hacker at 02:46, 7 August 2012

2012-08-07T02:46:26Z

← Older revision		Revision as of 02:46, 7 August 2012
Line 8:		Line 8:
	*The degree of skewness increases as <math>{{\sigma'}}</math> increases, for a given <math>\mu'</math>		*The degree of skewness increases as <math>{{\sigma'}}</math> increases, for a given <math>\mu'</math>

	~~==Effect of Mean==~~
	[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]		[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]

Lisa Hacker: /* Effect of Standard Deviation */

2012-08-07T02:46:16Z

Effect of Standard Deviation

← Older revision		Revision as of 02:46, 7 August 2012
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	<noinclude>		<noinclude>
	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	~~==Effect of Standard Deviation==~~
	[[Image:WB.10 effect of sigma.png\|center\|400px\| ]]		[[Image:WB.10 effect of sigma.png\|center\|400px\| ]]
	<math></math>		<math></math>

Lisa Hacker: /* Effect of Mean */

2012-08-07T02:45:49Z

Effect of Mean

← Older revision		Revision as of 02:45, 7 August 2012
Line 10:		Line 10:

	==Effect of Mean==		==Effect of Mean==
	[[Image:WB.10 lognormal pdf.png\|center\|~~250px~~\| ]]		[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]

	*For the same <math>{{\sigma'}}</math> , the <math>pdf</math> 's skewness increases as <math>{\mu }'</math> increases.		*For the same <math>{{\sigma'}}</math> , the <math>pdf</math> 's skewness increases as <math>{\mu }'</math> increases.

Lisa Hacker: /* Effect of Standard Deviation */

2012-08-07T02:45:37Z

Effect of Standard Deviation

← Older revision		Revision as of 02:45, 7 August 2012
Line 2:		Line 2:
	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	==Effect of Standard Deviation==		==Effect of Standard Deviation==
	[[Image:WB.10 effect of sigma.png\|center\|~~250px~~\| ]]		[[Image:WB.10 effect of sigma.png\|center\|400px\| ]]
	<math></math>		<math></math>

← Older revision		Revision as of 18:08, 24 September 2012
Line 2:		Line 2:
	''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>		''This article also appears in the [[The_Lognormal_Distribution\|Life Data Analysis Reference]] and [[Distributions_Used_in_Accelerated_Testing\|Accelerated Life Testing Data Analysis Reference]] books.'' </noinclude>
	[[Image:WB.10 effect of sigma.png\|center\|400px\| ]]		[[Image:WB.10 effect of sigma.png\|center\|400px\| ]]
	~~<math></math>~~

	*The lognormal distribution is a distribution skewed to the right.		*The lognormal distribution is a distribution skewed to the right.
	*The ''pdf'' starts at zero, increases to its mode, and decreases thereafter.		*The ''pdf'' starts at zero, increases to its mode, and decreases thereafter.
	*The degree of skewness increases as <math>{{\sigma'}}\,\!</math> increases, for a given <math>\mu'\,\!</math>		*The degree of skewness increases as <math>{{\sigma'}}\,\!</math> increases, for a given <math>\mu'\,\!</math>

	[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]		[[Image:WB.10 lognormal pdf.png\|center\|400px\| ]]

	*For the same <math>{{\sigma'}}\,\!</math> , the ''pdf'' 's skewness increases as <math>{\mu }'\,\!</math> increases.		*For the same <math>{{\sigma'}}\,\!</math>, the ''pdf'' 's skewness increases as <math>{\mu }'\,\!</math> increases.
	*For <math>{{\sigma' }}\,\!</math> values significantly greater than 1, the ''pdf'' rises very sharply in the beginning, (i.e., for very small values of <math>T\,\!</math> near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential ''pdf'' or a Weibull ''pdf'' with <math>0<\beta <1\,\!</math> .		*For <math>{{\sigma' }}\,\!</math> values significantly greater than 1, the ''pdf'' rises very sharply in the beginning, (i.e., for very small values of <math>T\,\!</math> near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential ''pdf'' or a Weibull ''pdf'' with <math>0<\beta <1\,\!</math>.
	*The parameter, <math>{\mu }'\,\!</math>, in terms of the logarithm of the <math>{T}'s\,\!</math> is also the scale parameter, and not the location parameter as in the case of the normal ''pdf''.		*The parameter, <math>{\mu }'\,\!</math>, in terms of the logarithm of the <math>{T}'s\,\!</math> is also the scale parameter, and not the location parameter as in the case of the normal ''pdf''.
	*The parameter <math>{{\sigma'}}\,\!</math>, or the standard deviation of the <math>{T}'s\,\!</math> in terms of their logarithm or of their <math>{T}'\,\!</math>, is also the shape parameter and not the scale parameter, as in the normal ''pdf'', and assumes only positive values.		*The parameter <math>{{\sigma'}}\,\!</math>, or the standard deviation of the <math>{T}'s\,\!</math> in terms of their logarithm or of their <math>{T}'\,\!</math>, is also the shape parameter and not the scale parameter, as in the normal ''pdf'', and assumes only positive values.

	'''Lognormal Distribution Parameters in ReliaSoft's Software'''		'''Lognormal Distribution Parameters in ReliaSoft's Software'''

	In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter <math>{\mu }'\,\!</math> represents the mean of the natural logarithms of the times-to-failure, while <math>{{\sigma' }}\,\!</math> represents the standard deviation of these data point logarithms. Specifically, the returned <math>{{\sigma' }}\,\!</math> is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.		In ReliaSoft's software, the parameters returned for the lognormal distribution are always logarithmic. That is: the parameter <math>{\mu }'\,\!</math> represents the mean of the natural logarithms of the times-to-failure, while <math>{{\sigma' }}\,\!</math> represents the standard deviation of these data point logarithms. Specifically, the returned <math>{{\sigma' }}\,\!</math> is the square root of the variance of the natural logarithms of the data points. Even though the application denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution, and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, as well as the standard deviation can be obtained through the QCP or the Function Wizard.