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In hypothesis testing, we set a null and alternative hypothesis. We are seeking evidence that the alternative hypothesis is true given the sample data. By using a sample from a population and not measuring every item in the population, we need to consider a couple of unwanted outcomes. Statisticians have named these unwanted results Type I and Type II Errors. [Read more…]
In the situation where you have a sample and would like to know if the population represented by the sample has a mean different than some specification, then this is the test for you. Oh, you also know, which is actually rather rare in practice, the actual variance of the population you drew the sample. [Read more…]
In system modeling and fault tree analysis (FTA) we use a set of similar calculations based on Boolean logic, the AND and OR gate probability calculations. Within FTA, the AND and OR gates are just two of many possible ways to model a system. Within system modeling, often reliability block diagrams (RBD) we model parallel and series elements of a system.
In order to do these basic calculations, we need to consider a few assumptions then proceed to the math.
When confronted with a stack of data, do you think about creating a histogram, too? Just tallied the 50th measurement of a new process – just means it’s time to craft a histogram, right?
There isn’t another data analysis tool as versatile. A histogram (bar chart) can deal with count, categorical, and continuous data (technically, the first two graphs would be bar charts). It like a lot of data yet reveals secretes of even smaller sets. A histogram should be on your shortlist of most often graphing tools. [Read more…]
Sometimes we just need a simple plot of a few data points. When there is scant data a histogram or box plot just is not informative. This is a great use for a one dimensional scatter plot, dot plot, or a what is called a strip chart in R.
The basic idea is to see where the data lines along a line. For example, let say we have 20 times to first failure. A table of numbers is not all that helpful. We could explore using a cumulative distribution plot (Weibull analysis), yet it would be difficult to fit a distribution with so little data.
Let’s turn to a strip chart to get a look at the data. [Read more…]
In a meeting the other day, the presenter was talking about a range of different failures for the product in question. She talked about each issue, a bit about the failure analysis, yet didn’t reveal which failures occurred more or less often.
She did provide a handout with a listing of the problems in order of the product field age and listing of the failure name (component or system involved). So, I grabbed a piece of paper to create a frequency table so I could quickly determine which problems occurred more often than others. [Read more…]
The clever Dr. John Snow mapped cholera cases during the epidemic of 1854 on a street map of the area. This type of mapping now called a measles chart, or defect location check sheet, or defect map, is useful when exploring the effect of location data.
The name measles chart may have come from the habit of using an image of drawing of a product and adding small red dots to signify defect locations.
Received a question from a reader this morning that will make a nice tutorial.
A box contains 27 black and 3 red balls. A random sample of 5 balls is drawn without replacement. What is the probability that the sample contains one red ball?
So here’s my thinking and two ways to solve this problem. Instead of red and black balls in an urn type problem, which is pretty abstract, let’s say we know 3 bad parts are in a bin of 30 total parts.
Most of us rely on accurate measurements. If these measurements are unreliable, then our decisions could be based on false information. How can we have confidence in our measurements?
The purpose of a measurement system analysis is to determine if a gauge is fit for use. This means that we can rely upon the measurements to give us a true indication of the parameter being measured. Our decisions will not be affected by erroneous data. So how can we know the quality of our measurements?
[Read more…]
The Friedman test is a non-parametric test used to test for differences between groups when the dependent variable is at least ordinal (could be continuous). The Friedman test is the non-parametric alternative to the one-way ANOVA with repeated measures (or the complete block design and a special case of the Durbin test). If the data is significantly different than normally distributed this becomes the preferred test over using an ANOVA.
The test procedure ranks each row (block) together, then considers the values of ranks by columns. The data is organized in to a matrix with B rows (blocks) and T columns (treatments) with a single operation in each cell of the matrix. [Read more…]
Let’s say you have shipped 1,000 products to your customer on January 1st. All are immediately placed into service. And each month since you have received a few product returns, what we are going to call failures. We also have fitted the data to a Weibull distribution. Then in May, your boss asks you to estimate how many failures to expect in June.
This is a simple example as we’re not shipping units every month, nor changing the product design or assembly process. We also have worked out the fitted Weibull parameters already. That leaves the calculation of how many failures we should expect over the next month. [Read more…]
When looking at a pile of data, sometimes there is a data point that is not like the others. It attracts attention as it is different than the rest of the data.
When I spot something odd in a dataset, I wonder if there is something to learn here. Is this an opportunity to make a discovery or improve a process?
All too often it is tempting to remove the outlier as a mistake. Or to drop the outlier as it doesn’t make any sense and ‘messes up’ the analysis. [Read more…]
The McNemar test is a nonparametric statistical test to compare dichotomous (unique) results of paired data.
If you are comparing survey results (favorable/unfavorable) for a group of potential customers given two ad campaigns, or evaluating the performance of two vendors in a set of prototype units, or determining if a maintenance procedure is effective for a set of equipment, this test permits the detection of changes.
The McNemar test is similar to the χ2 test. The McNemar only works with a two by two table, where the χ2 test works with larger tables. The χ2 test is checking for independence, while the McNemar test is looking for consistency in results.
Let’s examine an example where a group of people are surveyed about a prototype design, before and after a presentation. [Read more…]
This is part of a short series on the common distributions.
The Triangle distribution is univariate continuous distribution. This short article focuses on 4 formulas of the triangle distribution.
The distribution becomes a standard triangle distribution when a = 0, b = 1, thus it has a mean at the $- \sqrt{{c}/{2}\;} -$ and the median is at $- 1-\sqrt{{\left( 1-c \right)}/{2}\;}-$. The distribution becomes a symmetrical triangle distribution when $- c={\left( b-a \right)}/{2}\;-$.
The triangle distribution is used to approximate distributions when the actual distribution is unknown and bounded, often useful for Monte Carlo simulations. Other applications include subjective representation when there is evidence of bounds and a mode, or as a substitution to the beta distribution since it is bounded. [Read more…]
This is part of a short series on the common distributions.
The Uniform distribution is a univariate continuous distribution. This short article focuses on 7 formulas of the Uniform Distribution. A common application is as a non-informative prior. Another application is to model a bounded parameter. The uniform distribution also finds application in random number generation. [Read more…]