The Normal Distribution

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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/life_data_analysis

Chapter 9: The Normal Distribution


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Chapter 9  
The Normal Distribution  

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Available Software:
Weibull++

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More Resources:
Weibull++ Examples Collection



The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is the most widely-used general purpose distribution. It is for this reason that it is included among the lifetime distributions commonly used for reliability and life data analysis. There are some who argue that the normal distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small standard deviation, the issue of negative failure times should not present itself as a problem. Nevertheless, the normal distribution has been shown to be useful for modeling the lifetimes of consumable items, such as printer toner cartridges.

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/life_data_analysis

Chapter 9: The Normal Distribution


Weibullbox.png

Chapter 9  
The Normal Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection



The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is the most widely-used general purpose distribution. It is for this reason that it is included among the lifetime distributions commonly used for reliability and life data analysis. There are some who argue that the normal distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small standard deviation, the issue of negative failure times should not present itself as a problem. Nevertheless, the normal distribution has been shown to be useful for modeling the lifetimes of consumable items, such as printer toner cartridges.

Template loop detected: Template:Normal probability density function

Template loop detected: Template:Normal statistical properties

Template loop detected: Template:Characteristics of the normal distribution

Template loop detected: Template:Normal distribution estimation of the parameters

Template loop detected: Template:Normal distribution confidence bounds

Template loop detected: Template:Nd bayesian confidence bounds

General Examples

Example 8

Six units are tested to failure with the following hours-to-failure data obtained: 12125, 11260, 12080, 12825, 13550 and 14670 hours. Assuming the data are normally distributed, do the following: 8-1. Find the parameters for the data. (Use Rank Regression on X to duplicate the results shown in this example.) 8-2. Obtain the probability plot for the data with 90%, two-sided Type 1 confidence bounds. 8-3. Obtain the [math]\displaystyle{ pdf }[/math] plot for these data.

Solutions to Example 8

8-1. The next figure shows the data as entered in Weibull++, as well as the calculated parameters.

[math]\displaystyle{ }[/math]

11260folio.png


8-2. Obtain the probability plot as before. To plot confidence bounds, from the Plot Options menu choose Confidence Bounds and then Show Confidence Bounds. On the Type and Settings page of the Confidence Bounds window, select Two Sided Bounds, make sure Type 1 is selected, and then enter 90 in the Confidence level, % box, and click OK, as shown next.

[math]\displaystyle{ }[/math]

11260CBS.png


The following plot should appear on your screen:

[math]\displaystyle{ }[/math]

11260plot.png


8-3. From the Special Plot Type menu choose Pdf Plot. The following plot should appear on your screen.

11260pdfplot.png

[math]\displaystyle{ }[/math]

Example 9

Using the data and results from the previous example (and RRX), do the following: 9-1. Using the Quick Calculation Pad, determine the reliability for a mission of 11,000 hours, as well as the upper and lower two-sided 90% confidence limit on this reliability. 9-2. Using the Quick Calculation Pad, determine the MTTF, as well as the upper and lower two-sided 90% confidence limit on this MTTF.

Solutions to Example 9

Both of these results are easily obtained from the QCP. The QCP with results for both cases is shown in the next two figures.

QcpPOF.png
Qcpmeanlife.png

Example 10

Using the data from Example 8, and using the rank regression on X analysis method (RRX), obtain tabulated values for the failure rate for 10 different mission end times. The mission end times are 1,000 to 10,000 hours, using increments of 1,000 hours.

Solution to Example 10

This can be easily accomplished via the use of the Function Wizard, available in either Weibull++'s Reports or the General Spreadsheet. (For more information on these features, please refer to the Weibull++ User's Guide.) We will illustrate this using the General Spreadsheet.

First, click on Insert General Spread Sheet from the Folio menu.

Generalspreadsheeticon.png

Type Time in cell A1 and Failure Rate in cell B1. Then enter 1000 through 10000 in cells A2 to A11. Finally, place the cursor into cell B2, as shown next.

Plainspreadsheet.png

Open the Function Wizard by selecting Function Wizard from the Data menu or by clicking the Function Wizard icon.

Functionwizard.png

Select FAILURERATE from the list of functions. Enter A2 for Time; this indicates that the time input for the equation will be obtained from the specified cell in the worksheet. To specify the existing Weibull++ analysis that the function result will be based on, click Select... to open the Select Folio/Data Sheet window and then navigate to the desired sheet.

A2functionwizard.png
A2selectdatasheet.png

Click OK to close the window and return to the Function Wizard. Click Add to Equation to update the box at the bottom of the window with the function code that will be inserted into the spreadsheet.

FunctionwizardFR.png

Click Insert to close the window and insert the function code into the active cell in the General Spreadsheet. Copy the function into cells B3 through B11. One way to do this is to position the mouse over the bottom right corner of cell B2 and when the cursor turns into a plus symbol (+), click and drag the mouse to cell B11. By selecting one of the cells that you copied the function into, you can see that the cell reference was updated to match the current row, as shown next with cell B11 selected. The results are as follows:

Folio^E.png

Example 11

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
Data point index Last Inspected State End Time
1 30 32
2 32 35
3 35 37
4 37 40
5 42 42
6 45 45
7 50 50
8 55 55

Solution to Example 11

This is a sequence of interval times-to-failure. This data set can be entered into Weibull++ by creating a data sheet that can be used to analyze times-to-failure data with interval and left censored data.

[math]\displaystyle{ }[/math]

Lastinspected.png
Lastinspectedsheet.png

The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align} }[/math]

For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align} }[/math]

For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align} }[/math]


A plot of the MLE solution is shown next.

[math]\displaystyle{ }[/math]

Lastinspectedplot.png

Example 12

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.4 - Non-Grouped Data ,for Example 12
Data point index State F or S State End Time
1 F 2
2 F 5
3 F 11
4 F 23
5 F 29
6 F 37
7 F 43
8 F 59

Solution to Example 12

This data set can be entered into Weibull++ by creating a Data Sheet appropriate for the entry of non-grouped times-to-failure data. The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align} }[/math]

Example 13

Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Table 8.5 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)
Data point index Last Inspected State End Time
1 F 2
2 S 3
3 F 5
4 S 7
5 F 11
6 S 13
7 S 17
8 S 19
9 F 23
10 F 29
11 S 31
12 F 37
13 S 41
14 F 43
15 S 47
16 S 53
17 F 59
18 S 61
19 S 67

Solution to Example 13

This augments the previous example by adding eleven suspensions to the data set. This data set can be entered into Weibull++ by selecting the data sheet for Times to Failure and with Right Censored Data (Suspensions). The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align} }[/math]

Example 14

Suppose our data set includes left and right censored, interval censored and complete data as shown in the following table.

Solution to Example 14

This data set can be entered into Weibull++ by selecting the data type Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations.

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07. \end{align} }[/math]

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/life_data_analysis

Chapter 9: The Normal Distribution


Weibullbox.png

Chapter 9  
The Normal Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection



The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is the most widely-used general purpose distribution. It is for this reason that it is included among the lifetime distributions commonly used for reliability and life data analysis. There are some who argue that the normal distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small standard deviation, the issue of negative failure times should not present itself as a problem. Nevertheless, the normal distribution has been shown to be useful for modeling the lifetimes of consumable items, such as printer toner cartridges.

Template loop detected: Template:Normal probability density function

Template loop detected: Template:Normal statistical properties

Template loop detected: Template:Characteristics of the normal distribution

Template loop detected: Template:Normal distribution estimation of the parameters

Template loop detected: Template:Normal distribution confidence bounds

Template loop detected: Template:Nd bayesian confidence bounds

General Examples

Example 8

Six units are tested to failure with the following hours-to-failure data obtained: 12125, 11260, 12080, 12825, 13550 and 14670 hours. Assuming the data are normally distributed, do the following: 8-1. Find the parameters for the data. (Use Rank Regression on X to duplicate the results shown in this example.) 8-2. Obtain the probability plot for the data with 90%, two-sided Type 1 confidence bounds. 8-3. Obtain the [math]\displaystyle{ pdf }[/math] plot for these data.

Solutions to Example 8

8-1. The next figure shows the data as entered in Weibull++, as well as the calculated parameters.

[math]\displaystyle{ }[/math]

11260folio.png


8-2. Obtain the probability plot as before. To plot confidence bounds, from the Plot Options menu choose Confidence Bounds and then Show Confidence Bounds. On the Type and Settings page of the Confidence Bounds window, select Two Sided Bounds, make sure Type 1 is selected, and then enter 90 in the Confidence level, % box, and click OK, as shown next.

[math]\displaystyle{ }[/math]

11260CBS.png


The following plot should appear on your screen:

[math]\displaystyle{ }[/math]

11260plot.png


8-3. From the Special Plot Type menu choose Pdf Plot. The following plot should appear on your screen.

11260pdfplot.png

[math]\displaystyle{ }[/math]

Example 9

Using the data and results from the previous example (and RRX), do the following: 9-1. Using the Quick Calculation Pad, determine the reliability for a mission of 11,000 hours, as well as the upper and lower two-sided 90% confidence limit on this reliability. 9-2. Using the Quick Calculation Pad, determine the MTTF, as well as the upper and lower two-sided 90% confidence limit on this MTTF.

Solutions to Example 9

Both of these results are easily obtained from the QCP. The QCP with results for both cases is shown in the next two figures.

QcpPOF.png
Qcpmeanlife.png

Example 10

Using the data from Example 8, and using the rank regression on X analysis method (RRX), obtain tabulated values for the failure rate for 10 different mission end times. The mission end times are 1,000 to 10,000 hours, using increments of 1,000 hours.

Solution to Example 10

This can be easily accomplished via the use of the Function Wizard, available in either Weibull++'s Reports or the General Spreadsheet. (For more information on these features, please refer to the Weibull++ User's Guide.) We will illustrate this using the General Spreadsheet.

First, click on Insert General Spread Sheet from the Folio menu.

Generalspreadsheeticon.png

Type Time in cell A1 and Failure Rate in cell B1. Then enter 1000 through 10000 in cells A2 to A11. Finally, place the cursor into cell B2, as shown next.

Plainspreadsheet.png

Open the Function Wizard by selecting Function Wizard from the Data menu or by clicking the Function Wizard icon.

Functionwizard.png

Select FAILURERATE from the list of functions. Enter A2 for Time; this indicates that the time input for the equation will be obtained from the specified cell in the worksheet. To specify the existing Weibull++ analysis that the function result will be based on, click Select... to open the Select Folio/Data Sheet window and then navigate to the desired sheet.

A2functionwizard.png
A2selectdatasheet.png

Click OK to close the window and return to the Function Wizard. Click Add to Equation to update the box at the bottom of the window with the function code that will be inserted into the spreadsheet.

FunctionwizardFR.png

Click Insert to close the window and insert the function code into the active cell in the General Spreadsheet. Copy the function into cells B3 through B11. One way to do this is to position the mouse over the bottom right corner of cell B2 and when the cursor turns into a plus symbol (+), click and drag the mouse to cell B11. By selecting one of the cells that you copied the function into, you can see that the cell reference was updated to match the current row, as shown next with cell B11 selected. The results are as follows:

Folio^E.png

Example 11

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
Data point index Last Inspected State End Time
1 30 32
2 32 35
3 35 37
4 37 40
5 42 42
6 45 45
7 50 50
8 55 55

Solution to Example 11

This is a sequence of interval times-to-failure. This data set can be entered into Weibull++ by creating a data sheet that can be used to analyze times-to-failure data with interval and left censored data.

[math]\displaystyle{ }[/math]

Lastinspected.png
Lastinspectedsheet.png

The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align} }[/math]

For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align} }[/math]

For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align} }[/math]


A plot of the MLE solution is shown next.

[math]\displaystyle{ }[/math]

Lastinspectedplot.png

Example 12

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.4 - Non-Grouped Data ,for Example 12
Data point index State F or S State End Time
1 F 2
2 F 5
3 F 11
4 F 23
5 F 29
6 F 37
7 F 43
8 F 59

Solution to Example 12

This data set can be entered into Weibull++ by creating a Data Sheet appropriate for the entry of non-grouped times-to-failure data. The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align} }[/math]

Example 13

Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Table 8.5 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)
Data point index Last Inspected State End Time
1 F 2
2 S 3
3 F 5
4 S 7
5 F 11
6 S 13
7 S 17
8 S 19
9 F 23
10 F 29
11 S 31
12 F 37
13 S 41
14 F 43
15 S 47
16 S 53
17 F 59
18 S 61
19 S 67

Solution to Example 13

This augments the previous example by adding eleven suspensions to the data set. This data set can be entered into Weibull++ by selecting the data sheet for Times to Failure and with Right Censored Data (Suspensions). The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align} }[/math]

Example 14

Suppose our data set includes left and right censored, interval censored and complete data as shown in the following table.

Solution to Example 14

This data set can be entered into Weibull++ by selecting the data type Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations.

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07. \end{align} }[/math]

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/life_data_analysis

Chapter 9: The Normal Distribution


Weibullbox.png

Chapter 9  
The Normal Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection



The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is the most widely-used general purpose distribution. It is for this reason that it is included among the lifetime distributions commonly used for reliability and life data analysis. There are some who argue that the normal distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small standard deviation, the issue of negative failure times should not present itself as a problem. Nevertheless, the normal distribution has been shown to be useful for modeling the lifetimes of consumable items, such as printer toner cartridges.

Template loop detected: Template:Normal probability density function

Template loop detected: Template:Normal statistical properties

Template loop detected: Template:Characteristics of the normal distribution

Template loop detected: Template:Normal distribution estimation of the parameters

Template loop detected: Template:Normal distribution confidence bounds

Template loop detected: Template:Nd bayesian confidence bounds

General Examples

Example 8

Six units are tested to failure with the following hours-to-failure data obtained: 12125, 11260, 12080, 12825, 13550 and 14670 hours. Assuming the data are normally distributed, do the following: 8-1. Find the parameters for the data. (Use Rank Regression on X to duplicate the results shown in this example.) 8-2. Obtain the probability plot for the data with 90%, two-sided Type 1 confidence bounds. 8-3. Obtain the [math]\displaystyle{ pdf }[/math] plot for these data.

Solutions to Example 8

8-1. The next figure shows the data as entered in Weibull++, as well as the calculated parameters.

[math]\displaystyle{ }[/math]

11260folio.png


8-2. Obtain the probability plot as before. To plot confidence bounds, from the Plot Options menu choose Confidence Bounds and then Show Confidence Bounds. On the Type and Settings page of the Confidence Bounds window, select Two Sided Bounds, make sure Type 1 is selected, and then enter 90 in the Confidence level, % box, and click OK, as shown next.

[math]\displaystyle{ }[/math]

11260CBS.png


The following plot should appear on your screen:

[math]\displaystyle{ }[/math]

11260plot.png


8-3. From the Special Plot Type menu choose Pdf Plot. The following plot should appear on your screen.

11260pdfplot.png

[math]\displaystyle{ }[/math]

Example 9

Using the data and results from the previous example (and RRX), do the following: 9-1. Using the Quick Calculation Pad, determine the reliability for a mission of 11,000 hours, as well as the upper and lower two-sided 90% confidence limit on this reliability. 9-2. Using the Quick Calculation Pad, determine the MTTF, as well as the upper and lower two-sided 90% confidence limit on this MTTF.

Solutions to Example 9

Both of these results are easily obtained from the QCP. The QCP with results for both cases is shown in the next two figures.

QcpPOF.png
Qcpmeanlife.png

Example 10

Using the data from Example 8, and using the rank regression on X analysis method (RRX), obtain tabulated values for the failure rate for 10 different mission end times. The mission end times are 1,000 to 10,000 hours, using increments of 1,000 hours.

Solution to Example 10

This can be easily accomplished via the use of the Function Wizard, available in either Weibull++'s Reports or the General Spreadsheet. (For more information on these features, please refer to the Weibull++ User's Guide.) We will illustrate this using the General Spreadsheet.

First, click on Insert General Spread Sheet from the Folio menu.

Generalspreadsheeticon.png

Type Time in cell A1 and Failure Rate in cell B1. Then enter 1000 through 10000 in cells A2 to A11. Finally, place the cursor into cell B2, as shown next.

Plainspreadsheet.png

Open the Function Wizard by selecting Function Wizard from the Data menu or by clicking the Function Wizard icon.

Functionwizard.png

Select FAILURERATE from the list of functions. Enter A2 for Time; this indicates that the time input for the equation will be obtained from the specified cell in the worksheet. To specify the existing Weibull++ analysis that the function result will be based on, click Select... to open the Select Folio/Data Sheet window and then navigate to the desired sheet.

A2functionwizard.png
A2selectdatasheet.png

Click OK to close the window and return to the Function Wizard. Click Add to Equation to update the box at the bottom of the window with the function code that will be inserted into the spreadsheet.

FunctionwizardFR.png

Click Insert to close the window and insert the function code into the active cell in the General Spreadsheet. Copy the function into cells B3 through B11. One way to do this is to position the mouse over the bottom right corner of cell B2 and when the cursor turns into a plus symbol (+), click and drag the mouse to cell B11. By selecting one of the cells that you copied the function into, you can see that the cell reference was updated to match the current row, as shown next with cell B11 selected. The results are as follows:

Folio^E.png

Example 11

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
Data point index Last Inspected State End Time
1 30 32
2 32 35
3 35 37
4 37 40
5 42 42
6 45 45
7 50 50
8 55 55

Solution to Example 11

This is a sequence of interval times-to-failure. This data set can be entered into Weibull++ by creating a data sheet that can be used to analyze times-to-failure data with interval and left censored data.

[math]\displaystyle{ }[/math]

Lastinspected.png
Lastinspectedsheet.png

The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align} }[/math]

For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align} }[/math]

For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align} }[/math]


A plot of the MLE solution is shown next.

[math]\displaystyle{ }[/math]

Lastinspectedplot.png

Example 12

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.4 - Non-Grouped Data ,for Example 12
Data point index State F or S State End Time
1 F 2
2 F 5
3 F 11
4 F 23
5 F 29
6 F 37
7 F 43
8 F 59

Solution to Example 12

This data set can be entered into Weibull++ by creating a Data Sheet appropriate for the entry of non-grouped times-to-failure data. The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align} }[/math]

Example 13

Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Table 8.5 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)
Data point index Last Inspected State End Time
1 F 2
2 S 3
3 F 5
4 S 7
5 F 11
6 S 13
7 S 17
8 S 19
9 F 23
10 F 29
11 S 31
12 F 37
13 S 41
14 F 43
15 S 47
16 S 53
17 F 59
18 S 61
19 S 67

Solution to Example 13

This augments the previous example by adding eleven suspensions to the data set. This data set can be entered into Weibull++ by selecting the data sheet for Times to Failure and with Right Censored Data (Suspensions). The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align} }[/math]

Example 14

Suppose our data set includes left and right censored, interval censored and complete data as shown in the following table.

Solution to Example 14

This data set can be entered into Weibull++ by selecting the data type Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations.

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07. \end{align} }[/math]

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/life_data_analysis

Chapter 9: The Normal Distribution


Weibullbox.png

Chapter 9  
The Normal Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection



The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is the most widely-used general purpose distribution. It is for this reason that it is included among the lifetime distributions commonly used for reliability and life data analysis. There are some who argue that the normal distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small standard deviation, the issue of negative failure times should not present itself as a problem. Nevertheless, the normal distribution has been shown to be useful for modeling the lifetimes of consumable items, such as printer toner cartridges.

Template loop detected: Template:Normal probability density function

Template loop detected: Template:Normal statistical properties

Template loop detected: Template:Characteristics of the normal distribution

Template loop detected: Template:Normal distribution estimation of the parameters

Template loop detected: Template:Normal distribution confidence bounds

Template loop detected: Template:Nd bayesian confidence bounds

General Examples

Example 8

Six units are tested to failure with the following hours-to-failure data obtained: 12125, 11260, 12080, 12825, 13550 and 14670 hours. Assuming the data are normally distributed, do the following: 8-1. Find the parameters for the data. (Use Rank Regression on X to duplicate the results shown in this example.) 8-2. Obtain the probability plot for the data with 90%, two-sided Type 1 confidence bounds. 8-3. Obtain the [math]\displaystyle{ pdf }[/math] plot for these data.

Solutions to Example 8

8-1. The next figure shows the data as entered in Weibull++, as well as the calculated parameters.

[math]\displaystyle{ }[/math]

11260folio.png


8-2. Obtain the probability plot as before. To plot confidence bounds, from the Plot Options menu choose Confidence Bounds and then Show Confidence Bounds. On the Type and Settings page of the Confidence Bounds window, select Two Sided Bounds, make sure Type 1 is selected, and then enter 90 in the Confidence level, % box, and click OK, as shown next.

[math]\displaystyle{ }[/math]

11260CBS.png


The following plot should appear on your screen:

[math]\displaystyle{ }[/math]

11260plot.png


8-3. From the Special Plot Type menu choose Pdf Plot. The following plot should appear on your screen.

11260pdfplot.png

[math]\displaystyle{ }[/math]

Example 9

Using the data and results from the previous example (and RRX), do the following: 9-1. Using the Quick Calculation Pad, determine the reliability for a mission of 11,000 hours, as well as the upper and lower two-sided 90% confidence limit on this reliability. 9-2. Using the Quick Calculation Pad, determine the MTTF, as well as the upper and lower two-sided 90% confidence limit on this MTTF.

Solutions to Example 9

Both of these results are easily obtained from the QCP. The QCP with results for both cases is shown in the next two figures.

QcpPOF.png
Qcpmeanlife.png

Example 10

Using the data from Example 8, and using the rank regression on X analysis method (RRX), obtain tabulated values for the failure rate for 10 different mission end times. The mission end times are 1,000 to 10,000 hours, using increments of 1,000 hours.

Solution to Example 10

This can be easily accomplished via the use of the Function Wizard, available in either Weibull++'s Reports or the General Spreadsheet. (For more information on these features, please refer to the Weibull++ User's Guide.) We will illustrate this using the General Spreadsheet.

First, click on Insert General Spread Sheet from the Folio menu.

Generalspreadsheeticon.png

Type Time in cell A1 and Failure Rate in cell B1. Then enter 1000 through 10000 in cells A2 to A11. Finally, place the cursor into cell B2, as shown next.

Plainspreadsheet.png

Open the Function Wizard by selecting Function Wizard from the Data menu or by clicking the Function Wizard icon.

Functionwizard.png

Select FAILURERATE from the list of functions. Enter A2 for Time; this indicates that the time input for the equation will be obtained from the specified cell in the worksheet. To specify the existing Weibull++ analysis that the function result will be based on, click Select... to open the Select Folio/Data Sheet window and then navigate to the desired sheet.

A2functionwizard.png
A2selectdatasheet.png

Click OK to close the window and return to the Function Wizard. Click Add to Equation to update the box at the bottom of the window with the function code that will be inserted into the spreadsheet.

FunctionwizardFR.png

Click Insert to close the window and insert the function code into the active cell in the General Spreadsheet. Copy the function into cells B3 through B11. One way to do this is to position the mouse over the bottom right corner of cell B2 and when the cursor turns into a plus symbol (+), click and drag the mouse to cell B11. By selecting one of the cells that you copied the function into, you can see that the cell reference was updated to match the current row, as shown next with cell B11 selected. The results are as follows:

Folio^E.png

Example 11

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
Data point index Last Inspected State End Time
1 30 32
2 32 35
3 35 37
4 37 40
5 42 42
6 45 45
7 50 50
8 55 55

Solution to Example 11

This is a sequence of interval times-to-failure. This data set can be entered into Weibull++ by creating a data sheet that can be used to analyze times-to-failure data with interval and left censored data.

[math]\displaystyle{ }[/math]

Lastinspected.png
Lastinspectedsheet.png

The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align} }[/math]

For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align} }[/math]

For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align} }[/math]


A plot of the MLE solution is shown next.

[math]\displaystyle{ }[/math]

Lastinspectedplot.png

Example 12

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.4 - Non-Grouped Data ,for Example 12
Data point index State F or S State End Time
1 F 2
2 F 5
3 F 11
4 F 23
5 F 29
6 F 37
7 F 43
8 F 59

Solution to Example 12

This data set can be entered into Weibull++ by creating a Data Sheet appropriate for the entry of non-grouped times-to-failure data. The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align} }[/math]

Example 13

Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Table 8.5 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)
Data point index Last Inspected State End Time
1 F 2
2 S 3
3 F 5
4 S 7
5 F 11
6 S 13
7 S 17
8 S 19
9 F 23
10 F 29
11 S 31
12 F 37
13 S 41
14 F 43
15 S 47
16 S 53
17 F 59
18 S 61
19 S 67

Solution to Example 13

This augments the previous example by adding eleven suspensions to the data set. This data set can be entered into Weibull++ by selecting the data sheet for Times to Failure and with Right Censored Data (Suspensions). The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align} }[/math]

Example 14

Suppose our data set includes left and right censored, interval censored and complete data as shown in the following table.

Solution to Example 14

This data set can be entered into Weibull++ by selecting the data type Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations.

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07. \end{align} }[/math]

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/life_data_analysis

Chapter 9: The Normal Distribution


Weibullbox.png

Chapter 9  
The Normal Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection



The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is the most widely-used general purpose distribution. It is for this reason that it is included among the lifetime distributions commonly used for reliability and life data analysis. There are some who argue that the normal distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small standard deviation, the issue of negative failure times should not present itself as a problem. Nevertheless, the normal distribution has been shown to be useful for modeling the lifetimes of consumable items, such as printer toner cartridges.

Template loop detected: Template:Normal probability density function

Template loop detected: Template:Normal statistical properties

Template loop detected: Template:Characteristics of the normal distribution

Template loop detected: Template:Normal distribution estimation of the parameters

Template loop detected: Template:Normal distribution confidence bounds

Template loop detected: Template:Nd bayesian confidence bounds

General Examples

Example 8

Six units are tested to failure with the following hours-to-failure data obtained: 12125, 11260, 12080, 12825, 13550 and 14670 hours. Assuming the data are normally distributed, do the following: 8-1. Find the parameters for the data. (Use Rank Regression on X to duplicate the results shown in this example.) 8-2. Obtain the probability plot for the data with 90%, two-sided Type 1 confidence bounds. 8-3. Obtain the [math]\displaystyle{ pdf }[/math] plot for these data.

Solutions to Example 8

8-1. The next figure shows the data as entered in Weibull++, as well as the calculated parameters.

[math]\displaystyle{ }[/math]

11260folio.png


8-2. Obtain the probability plot as before. To plot confidence bounds, from the Plot Options menu choose Confidence Bounds and then Show Confidence Bounds. On the Type and Settings page of the Confidence Bounds window, select Two Sided Bounds, make sure Type 1 is selected, and then enter 90 in the Confidence level, % box, and click OK, as shown next.

[math]\displaystyle{ }[/math]

11260CBS.png


The following plot should appear on your screen:

[math]\displaystyle{ }[/math]

11260plot.png


8-3. From the Special Plot Type menu choose Pdf Plot. The following plot should appear on your screen.

11260pdfplot.png

[math]\displaystyle{ }[/math]

Example 9

Using the data and results from the previous example (and RRX), do the following: 9-1. Using the Quick Calculation Pad, determine the reliability for a mission of 11,000 hours, as well as the upper and lower two-sided 90% confidence limit on this reliability. 9-2. Using the Quick Calculation Pad, determine the MTTF, as well as the upper and lower two-sided 90% confidence limit on this MTTF.

Solutions to Example 9

Both of these results are easily obtained from the QCP. The QCP with results for both cases is shown in the next two figures.

QcpPOF.png
Qcpmeanlife.png

Example 10

Using the data from Example 8, and using the rank regression on X analysis method (RRX), obtain tabulated values for the failure rate for 10 different mission end times. The mission end times are 1,000 to 10,000 hours, using increments of 1,000 hours.

Solution to Example 10

This can be easily accomplished via the use of the Function Wizard, available in either Weibull++'s Reports or the General Spreadsheet. (For more information on these features, please refer to the Weibull++ User's Guide.) We will illustrate this using the General Spreadsheet.

First, click on Insert General Spread Sheet from the Folio menu.

Generalspreadsheeticon.png

Type Time in cell A1 and Failure Rate in cell B1. Then enter 1000 through 10000 in cells A2 to A11. Finally, place the cursor into cell B2, as shown next.

Plainspreadsheet.png

Open the Function Wizard by selecting Function Wizard from the Data menu or by clicking the Function Wizard icon.

Functionwizard.png

Select FAILURERATE from the list of functions. Enter A2 for Time; this indicates that the time input for the equation will be obtained from the specified cell in the worksheet. To specify the existing Weibull++ analysis that the function result will be based on, click Select... to open the Select Folio/Data Sheet window and then navigate to the desired sheet.

A2functionwizard.png
A2selectdatasheet.png

Click OK to close the window and return to the Function Wizard. Click Add to Equation to update the box at the bottom of the window with the function code that will be inserted into the spreadsheet.

FunctionwizardFR.png

Click Insert to close the window and insert the function code into the active cell in the General Spreadsheet. Copy the function into cells B3 through B11. One way to do this is to position the mouse over the bottom right corner of cell B2 and when the cursor turns into a plus symbol (+), click and drag the mouse to cell B11. By selecting one of the cells that you copied the function into, you can see that the cell reference was updated to match the current row, as shown next with cell B11 selected. The results are as follows:

Folio^E.png

Example 11

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
Data point index Last Inspected State End Time
1 30 32
2 32 35
3 35 37
4 37 40
5 42 42
6 45 45
7 50 50
8 55 55

Solution to Example 11

This is a sequence of interval times-to-failure. This data set can be entered into Weibull++ by creating a data sheet that can be used to analyze times-to-failure data with interval and left censored data.

[math]\displaystyle{ }[/math]

Lastinspected.png
Lastinspectedsheet.png

The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align} }[/math]

For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align} }[/math]

For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align} }[/math]


A plot of the MLE solution is shown next.

[math]\displaystyle{ }[/math]

Lastinspectedplot.png

Example 12

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.4 - Non-Grouped Data ,for Example 12
Data point index State F or S State End Time
1 F 2
2 F 5
3 F 11
4 F 23
5 F 29
6 F 37
7 F 43
8 F 59

Solution to Example 12

This data set can be entered into Weibull++ by creating a Data Sheet appropriate for the entry of non-grouped times-to-failure data. The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align} }[/math]

Example 13

Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Table 8.5 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)
Data point index Last Inspected State End Time
1 F 2
2 S 3
3 F 5
4 S 7
5 F 11
6 S 13
7 S 17
8 S 19
9 F 23
10 F 29
11 S 31
12 F 37
13 S 41
14 F 43
15 S 47
16 S 53
17 F 59
18 S 61
19 S 67

Solution to Example 13

This augments the previous example by adding eleven suspensions to the data set. This data set can be entered into Weibull++ by selecting the data sheet for Times to Failure and with Right Censored Data (Suspensions). The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align} }[/math]

Example 14

Suppose our data set includes left and right censored, interval censored and complete data as shown in the following table.

Solution to Example 14

This data set can be entered into Weibull++ by selecting the data type Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations.

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07. \end{align} }[/math]

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/life_data_analysis

Chapter 9: The Normal Distribution


Weibullbox.png

Chapter 9  
The Normal Distribution  

Synthesis-icon.png

Available Software:
Weibull++

Examples icon.png

More Resources:
Weibull++ Examples Collection



The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is the most widely-used general purpose distribution. It is for this reason that it is included among the lifetime distributions commonly used for reliability and life data analysis. There are some who argue that the normal distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small standard deviation, the issue of negative failure times should not present itself as a problem. Nevertheless, the normal distribution has been shown to be useful for modeling the lifetimes of consumable items, such as printer toner cartridges.

Template loop detected: Template:Normal probability density function

Template loop detected: Template:Normal statistical properties

Template loop detected: Template:Characteristics of the normal distribution

Template loop detected: Template:Normal distribution estimation of the parameters

Template loop detected: Template:Normal distribution confidence bounds

Template loop detected: Template:Nd bayesian confidence bounds

General Examples

Example 8

Six units are tested to failure with the following hours-to-failure data obtained: 12125, 11260, 12080, 12825, 13550 and 14670 hours. Assuming the data are normally distributed, do the following: 8-1. Find the parameters for the data. (Use Rank Regression on X to duplicate the results shown in this example.) 8-2. Obtain the probability plot for the data with 90%, two-sided Type 1 confidence bounds. 8-3. Obtain the [math]\displaystyle{ pdf }[/math] plot for these data.

Solutions to Example 8

8-1. The next figure shows the data as entered in Weibull++, as well as the calculated parameters.

[math]\displaystyle{ }[/math]

11260folio.png


8-2. Obtain the probability plot as before. To plot confidence bounds, from the Plot Options menu choose Confidence Bounds and then Show Confidence Bounds. On the Type and Settings page of the Confidence Bounds window, select Two Sided Bounds, make sure Type 1 is selected, and then enter 90 in the Confidence level, % box, and click OK, as shown next.

[math]\displaystyle{ }[/math]

11260CBS.png


The following plot should appear on your screen:

[math]\displaystyle{ }[/math]

11260plot.png


8-3. From the Special Plot Type menu choose Pdf Plot. The following plot should appear on your screen.

11260pdfplot.png

[math]\displaystyle{ }[/math]

Example 9

Using the data and results from the previous example (and RRX), do the following: 9-1. Using the Quick Calculation Pad, determine the reliability for a mission of 11,000 hours, as well as the upper and lower two-sided 90% confidence limit on this reliability. 9-2. Using the Quick Calculation Pad, determine the MTTF, as well as the upper and lower two-sided 90% confidence limit on this MTTF.

Solutions to Example 9

Both of these results are easily obtained from the QCP. The QCP with results for both cases is shown in the next two figures.

QcpPOF.png
Qcpmeanlife.png

Example 10

Using the data from Example 8, and using the rank regression on X analysis method (RRX), obtain tabulated values for the failure rate for 10 different mission end times. The mission end times are 1,000 to 10,000 hours, using increments of 1,000 hours.

Solution to Example 10

This can be easily accomplished via the use of the Function Wizard, available in either Weibull++'s Reports or the General Spreadsheet. (For more information on these features, please refer to the Weibull++ User's Guide.) We will illustrate this using the General Spreadsheet.

First, click on Insert General Spread Sheet from the Folio menu.

Generalspreadsheeticon.png

Type Time in cell A1 and Failure Rate in cell B1. Then enter 1000 through 10000 in cells A2 to A11. Finally, place the cursor into cell B2, as shown next.

Plainspreadsheet.png

Open the Function Wizard by selecting Function Wizard from the Data menu or by clicking the Function Wizard icon.

Functionwizard.png

Select FAILURERATE from the list of functions. Enter A2 for Time; this indicates that the time input for the equation will be obtained from the specified cell in the worksheet. To specify the existing Weibull++ analysis that the function result will be based on, click Select... to open the Select Folio/Data Sheet window and then navigate to the desired sheet.

A2functionwizard.png
A2selectdatasheet.png

Click OK to close the window and return to the Function Wizard. Click Add to Equation to update the box at the bottom of the window with the function code that will be inserted into the spreadsheet.

FunctionwizardFR.png

Click Insert to close the window and insert the function code into the active cell in the General Spreadsheet. Copy the function into cells B3 through B11. One way to do this is to position the mouse over the bottom right corner of cell B2 and when the cursor turns into a plus symbol (+), click and drag the mouse to cell B11. By selecting one of the cells that you copied the function into, you can see that the cell reference was updated to match the current row, as shown next with cell B11 selected. The results are as follows:

Folio^E.png

Example 11

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
Data point index Last Inspected State End Time
1 30 32
2 32 35
3 35 37
4 37 40
5 42 42
6 45 45
7 50 50
8 55 55

Solution to Example 11

This is a sequence of interval times-to-failure. This data set can be entered into Weibull++ by creating a data sheet that can be used to analyze times-to-failure data with interval and left censored data.

[math]\displaystyle{ }[/math]

Lastinspected.png
Lastinspectedsheet.png

The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align} }[/math]

For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align} }[/math]

For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align} }[/math]


A plot of the MLE solution is shown next.

[math]\displaystyle{ }[/math]

Lastinspectedplot.png

Example 12

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.4 - Non-Grouped Data ,for Example 12
Data point index State F or S State End Time
1 F 2
2 F 5
3 F 11
4 F 23
5 F 29
6 F 37
7 F 43
8 F 59

Solution to Example 12

This data set can be entered into Weibull++ by creating a Data Sheet appropriate for the entry of non-grouped times-to-failure data. The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align} }[/math]

Example 13

Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Table 8.5 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)
Data point index Last Inspected State End Time
1 F 2
2 S 3
3 F 5
4 S 7
5 F 11
6 S 13
7 S 17
8 S 19
9 F 23
10 F 29
11 S 31
12 F 37
13 S 41
14 F 43
15 S 47
16 S 53
17 F 59
18 S 61
19 S 67

Solution to Example 13

This augments the previous example by adding eleven suspensions to the data set. This data set can be entered into Weibull++ by selecting the data sheet for Times to Failure and with Right Censored Data (Suspensions). The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align} }[/math]

Example 14

Suppose our data set includes left and right censored, interval censored and complete data as shown in the following table.

Solution to Example 14

This data set can be entered into Weibull++ by selecting the data type Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations.

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07. \end{align} }[/math]

General Examples

Example 8

Six units are tested to failure with the following hours-to-failure data obtained: 12125, 11260, 12080, 12825, 13550 and 14670 hours. Assuming the data are normally distributed, do the following: 8-1. Find the parameters for the data. (Use Rank Regression on X to duplicate the results shown in this example.) 8-2. Obtain the probability plot for the data with 90%, two-sided Type 1 confidence bounds. 8-3. Obtain the [math]\displaystyle{ pdf }[/math] plot for these data.

Solutions to Example 8

8-1. The next figure shows the data as entered in Weibull++, as well as the calculated parameters.

[math]\displaystyle{ }[/math]

11260folio.png


8-2. Obtain the probability plot as before. To plot confidence bounds, from the Plot Options menu choose Confidence Bounds and then Show Confidence Bounds. On the Type and Settings page of the Confidence Bounds window, select Two Sided Bounds, make sure Type 1 is selected, and then enter 90 in the Confidence level, % box, and click OK, as shown next.

[math]\displaystyle{ }[/math]

11260CBS.png


The following plot should appear on your screen:

[math]\displaystyle{ }[/math]

11260plot.png


8-3. From the Special Plot Type menu choose Pdf Plot. The following plot should appear on your screen.

11260pdfplot.png

[math]\displaystyle{ }[/math]

Example 9

Using the data and results from the previous example (and RRX), do the following: 9-1. Using the Quick Calculation Pad, determine the reliability for a mission of 11,000 hours, as well as the upper and lower two-sided 90% confidence limit on this reliability. 9-2. Using the Quick Calculation Pad, determine the MTTF, as well as the upper and lower two-sided 90% confidence limit on this MTTF.

Solutions to Example 9

Both of these results are easily obtained from the QCP. The QCP with results for both cases is shown in the next two figures.

QcpPOF.png
Qcpmeanlife.png

Example 10

Using the data from Example 8, and using the rank regression on X analysis method (RRX), obtain tabulated values for the failure rate for 10 different mission end times. The mission end times are 1,000 to 10,000 hours, using increments of 1,000 hours.

Solution to Example 10

This can be easily accomplished via the use of the Function Wizard, available in either Weibull++'s Reports or the General Spreadsheet. (For more information on these features, please refer to the Weibull++ User's Guide.) We will illustrate this using the General Spreadsheet.

First, click on Insert General Spread Sheet from the Folio menu.

Generalspreadsheeticon.png

Type Time in cell A1 and Failure Rate in cell B1. Then enter 1000 through 10000 in cells A2 to A11. Finally, place the cursor into cell B2, as shown next.

Plainspreadsheet.png

Open the Function Wizard by selecting Function Wizard from the Data menu or by clicking the Function Wizard icon.

Functionwizard.png

Select FAILURERATE from the list of functions. Enter A2 for Time; this indicates that the time input for the equation will be obtained from the specified cell in the worksheet. To specify the existing Weibull++ analysis that the function result will be based on, click Select... to open the Select Folio/Data Sheet window and then navigate to the desired sheet.

A2functionwizard.png
A2selectdatasheet.png

Click OK to close the window and return to the Function Wizard. Click Add to Equation to update the box at the bottom of the window with the function code that will be inserted into the spreadsheet.

FunctionwizardFR.png

Click Insert to close the window and insert the function code into the active cell in the General Spreadsheet. Copy the function into cells B3 through B11. One way to do this is to position the mouse over the bottom right corner of cell B2 and when the cursor turns into a plus symbol (+), click and drag the mouse to cell B11. By selecting one of the cells that you copied the function into, you can see that the cell reference was updated to match the current row, as shown next with cell B11 selected. The results are as follows:

Folio^E.png

Example 11

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
Data point index Last Inspected State End Time
1 30 32
2 32 35
3 35 37
4 37 40
5 42 42
6 45 45
7 50 50
8 55 55

Solution to Example 11

This is a sequence of interval times-to-failure. This data set can be entered into Weibull++ by creating a data sheet that can be used to analyze times-to-failure data with interval and left censored data.

[math]\displaystyle{ }[/math]

Lastinspected.png
Lastinspectedsheet.png

The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align} }[/math]

For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align} }[/math]

For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align} }[/math]


A plot of the MLE solution is shown next.

[math]\displaystyle{ }[/math]

Lastinspectedplot.png

Example 12

Eight units are being reliability tested and the following is a table of their times-to-failure:


Table 8.4 - Non-Grouped Data ,for Example 12
Data point index State F or S State End Time
1 F 2
2 F 5
3 F 11
4 F 23
5 F 29
6 F 37
7 F 43
8 F 59

Solution to Example 12

This data set can be entered into Weibull++ by creating a Data Sheet appropriate for the entry of non-grouped times-to-failure data. The computed parameters for maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align} }[/math]

Example 13

Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Table 8.5 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)
Data point index Last Inspected State End Time
1 F 2
2 S 3
3 F 5
4 S 7
5 F 11
6 S 13
7 S 17
8 S 19
9 F 23
10 F 29
11 S 31
12 F 37
13 S 41
14 F 43
15 S 47
16 S 53
17 F 59
18 S 61
19 S 67

Solution to Example 13

This augments the previous example by adding eleven suspensions to the data set. This data set can be entered into Weibull++ by selecting the data sheet for Times to Failure and with Right Censored Data (Suspensions). The parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align} }[/math]

Example 14

Suppose our data set includes left and right censored, interval censored and complete data as shown in the following table.

Solution to Example 14

This data set can be entered into Weibull++ by selecting the data type Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations.

The computed parameters using maximum likelihood are:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42. \end{align} }[/math]


For rank regression on x:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17. \end{align} }[/math]


For rank regression on y:

[math]\displaystyle{ \begin{align} & \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07. \end{align} }[/math]