Probability and Statistics for Reliability

The 2 Parameter Lognormal Distribution 7 Formulas

This is part of a short series on the common life data distributions.

The Lognormal distribution is a versatile and continuous distribution. It is similar to the Weibull in flexibility with just slightly fatter tails in most circumstances. It is commonly used to describe time to repair behavior. This short article focuses on 7 formulas of the Lognormal Distribution.

If you want to know more about fitting a set of data to a distribution, well that is in another article.

It has the essential formulas that you may find useful when answering specific questions. Knowing a distribution’s set of parameters does provide, along with the right formulas, a quick means to answer a wide range of reliability related questions. [Read more…]

by Fred Schenkelberg Leave a Comment

The 1 Parameter Exponential Distribution 7 Formulas

This is part of a short series on the common life data distributions.

The Exponential distribution is popular and useful in isolated situations. It has some nice features and flexibility that support it’s popularity. This short article focuses on 7 formulas of the Exponential Distribution.

If you want to know more about fitting a set of data to a distribution, well that is in another article.

by Fred Schenkelberg Leave a Comment

The Gamma Distribution 7 Formulas

This is part of a short series on the common life data distributions.

The Gamma distribution is routinely used to describe systems undergoing sequences of events or shocks which lead to eventual failure. Also used to describe renewal processes. This short article focuses on 7 formulas of the Gamma Distribution.

If you want to know more about fitting a set of data to a distribution, well that is in another article.

by Fred Schenkelberg 5 Comments

The 2 Parameter Weibull Distribution 7 Formulas

This is part of a short series on the common life data distributions.

The Weibull distribution is both popular and useful. It has some nice features and flexibility that support its popularity. This short article focuses on 7 formulas of the Weibull Distribution.

If you want to know more about fitting a set of data to a distribution, well that is in another article.

by Fred Schenkelberg Leave a Comment

How to Calculate Reliability Given 3 Different Distributions

On occasion, we want to estimate the reliability of an item at a specific time.

Maybe we are considering extending the warranty period, for example, and want to know the probability of no failures over one year instead of over the current 3 months.

Or, let’s say you talked to a bearing vendor and have the Weibull parameters and wish to know the reliability value over 2 years.

Whatever specific situation, you have the life distributions parameters. You just need to calculate reliability at a specific time. We can do that and let’s try it with three distributions using their respective reliability functions: exponential, Weibull, and lognormal. [Read more…]

by Fred Schenkelberg 2 Comments

The Many Ways We Use Variance

The term variance is a statistical concept related to the spread or dispersion of a set of data. Second to the mean, it a common value we may calculate.

We find standard deviation easier to understand and use (it uses the same units as the data) whereas variance uses the units squared.

We use variance in quite a few different ways. Let’s review just a few. [Read more…]

by Fred Schenkelberg 2 Comments

Siegel-Tukey Test for Differences in Scale

There are a few different reasons we explore differences in scale.

Keep in mind that the scale of a dataset is basically the spread of the data. For most datasets, we’re examining the variance.

Hypothesis tests comparing means vary depending on the assumption of equal variances. Thus testing that assumption requires methods to adequately test the homogeneity of variances. The F-test should come to mind as it is a common approach.

Some datasets do not lend themselves to using the F-test, which is applicable using real numbers. Some datasets gather information that is ordinal or interval data, thus we need another approach to test for differences in scale. [Read more…]

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Hartley’s Test for Variance Homogeneity

The Hartley test is an extension of the F distribution-based hypothesis test checking if two samples have different variances.

The F test works with two samples allowing us to compare two population variances based on the two samples. This test does not work for three or more populations. We could conduct multiple pairwise comparisons, yet the probability of an erroneous result is significant.

Bartlett’s Test and Levene’s Test are non-parametric checks for homogeneity of variances. Bartlett’s Test pretty much expects the underlying data to be normally distributed.

Levene’s Test is a better choice when you’re not sure the data is normal. Both are conservative and time-consuming to calculate.

We need another way to check for equal variances. [Read more…]

by Fred Schenkelberg 2 Comments

The Sum of Squares Concept

The data analysis course professor tended to focus on the practical application of statistics.

Avoiding statistical theory was fine with me. Learning statistics for me was on how to solve problems, optimize designs, and understanding data.

Then one lecture started with the question, “Why do we sum squares regression analysis, ANOVA calculations, and with other statistical methods?” He paused waiting for one us to answer.

We didn’t. I feared the upcoming lecture would include arcane derivations and burdensome theoretical annotations. It didn’t. [Read more…]

by Fred Schenkelberg 9 Comments

Bartlett’s Test for Homogeneity of Variances

A common assumption when comparing three or more normal population means is they have similar (the same) population variances.

ANOVA and some DOE analysis results rely on the underlying data having similar variances. If this assumption is not true, the conclusions suggested by the ANOVA or DOE may be misleading.

It doesn’t take long to check. [Read more…]

by Fred Schenkelberg Leave a Comment

Confidence Interval Interpretations and Misunderstandings

Confidence intervals (CIs)provide a means to judge point estimates based on a sample from the population.

If that statement excites you, you may well have the makings of a fine statistician.

CIs are a form of internal estimate and specify a range within which a parameter may reside. CIs helps us evaluate the trustworthiness of point estimates. [Read more…]

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Plotting Repairable System Failure Data

A good plot reveals the data’s story.

Repairable system data is what is called by statisticians a renewal process.

The repair activity may restore the system to as good as new. Sometimes, the repair pretty much leaves the system in a state similar to just before the repair.

What happens most often, though, is the chance of system failure changes after each repair activity.

A simple plot can help us see what is happening. [Read more…]

by Fred Schenkelberg Leave a Comment

Two Proportions Hypothesis Testing

In the article, Hypothesis Tests for Proportion, the comparison is between a given value and the sample. In this case, let’s compare two populations. We take a sample which provides a proportion representing each population and determines if the populations are different from each other based on the two samples.

The exact solution uses the Binomial distribution, yet when np and 1 – np are greater than 5, then we can use a normal approximation for the test statistic and critical value. [Read more…]

by Fred Schenkelberg 6 Comments

Success Testing Formula Derivation

The planning of environmental or reliability testing becomes a question of sample size at some point.

It’s probably the most common question I hear as a reliability engineer – how many samples do we need. [Read more…]

by Fred Schenkelberg 4 Comments

How to Read an OC curve

The operating characteristic curve, OC curve, visualizes a sampling plan.

At times, we select a sample from a group of items and evaluate them. Does this lot of widgets meet the specifications? Does this batch measure up? [Read more…]