Normal Distribution - Probability Plotting

	Normal Distribution - Probability Plotting

The following section describes probability plotting for the normal distribution using the Reliability & Maintenance Analyst. The manual method is located here.

Probability plotting is an excellent method for determining goodness-of-fit. To determine the goodness-of-fit, select the "Transformed" option in the Plot Type frame, and click the "Plot" button. If the plotted points form a straight line, the distribution provides a good time to fail model for the data. The "R-Squared" value is a measure of how well the data forms a straight line. An R-Squared value of 1.0 indicates a perfectly straight line. R-Squared is also known as the coefficient of determination. To estimate the parameters of the normal distribution using probability plotting, follow these steps:

Enter the data using one of the data entry grids, or connect to a database.
Select the "Parameter Estimation"
Select "Normal"
Select "Probability Plot"

The figure below shows the normal probability plotting screen using the data in the file "Demo2.dat".

Clicking the "Plot" button gives a probability plot with 2 options - transformed data and untransformed data. The transformed data produced a plot of the data that has been transformed to produce a straight line with a 95% confidence interval surrounding it. If the plotted points do not follow a straight line, the normal distribution with the estimated parameters does not provide an adequate time to fail model. The untransformed option plots the cumulative distribution function against time. This is useful for determining the probability of failure at a given time. The title of the graphs can be changed by editing the text in the Graph Title frame. To check the spelling of the title, click the "Spell Check" button.

To predict reliability or time-to-fail using the estimated parameters use the Predicting Module.

Manual Probability Plotting

By rearranging the normal cumulative distribution function, a linear expression can be obtained.

where F(x) is the normal cumulative distribution function, and

is the inverse of the standard normal cumulative distribution function.

It can be seen that by plotting x versus the resulting y-intercept equals m and the resulting slope equals s . The cumulative distribution function, F(x), is usually estimated from the median rank, but other estimates such as the mean rank and the Kaplan-Meier product limit estimator are also used.

An alternative to plotting x versus on conventional graph paper is to plot x versus F(x) on specialized probability paper. The advantage of probability paper is that the values of do not have to be computed. Computers have made this technique obsolete.

Example

Use probability plotting to determine the parameters of the normal distribution given the following data set. A "c" following an entry indicates the censoring.

Time to Fail

150 c

183 c

235

157 c

209

235

167 c

216 c

248 c

179

217 c

257

Solution
The table below is constructed to obtain the data for plotting. By plotting the last column of the data below on the x-axis, and the time to fail on the y-axis, the slope of the resulting line is 34.8, which is the estimated value of s . The y-intercept is 235.3, which is the estimated value of m.

Time to
Fail

Median
Rank, F(x)

150 c

157 c

167 c

179

1.3000

1.3000

0.0806

-1.4008

183 c

209

1.4625

2.7625

0.1986

-0.8467

216 c

217 c

235

2.0475

4.8100

0.3637

-0.3486

235

2.0475

6.8575

0.5288

0.0723

248 c

257

3.0713

9.9288

0.7765

0.7605

Nonparametric confidence limits for reliability can be added to a normal probability plot using 5% and 95% ranks. The plotting position for the standard normal inverse of the 5% and 95% ranks is

Click here to download this example in Microsoft Excel.