Sample Sizes – Surveys

Introduction

How many responses are needed for a survey? This question requires specifying the desired confidence and the accuracy of the survey results.

The Bernoulli Trial

A Bernoulli trial is an event that has two possible outcomes. Consider the case where the only possible outcomes are success or failure. Let the probability of a success is p and the probability of failure equals q.  The probabilities of all possible events must equal 1, so q = 1-p. These relationships are expressed mathematically as 

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Process Capability VII – Confidence Limits

Introduction

In prior articles on process capability, sample statistics and SPC statistics were assumed to be population parameters and ignored sampling variability. This article reviews the analytic methods that can be used to develop confidence bounds on the process capability indices.

$-P_p-$ Index

The P_p index calculation requires an estimate of the parameter σ. The index is calculated as:

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Process Capability – Non-Normal Variables

The Situation

You have a process that is not capable because sample measurements or SPC data indicate that some characteristics have too much variability. The calculated Cpk’s are too small. What do you do?

Assuming the data is correct, a course of action is to review the assumption is that the measurements are normally distributed. For most situations, this is a reasonable assumption, but other statistical distributions may provide a better description of the data variation. 

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Process Capability V – Variation Reduction

Introduction

In the prior article, Process Capability IV, vehicle wheel toe alignment showed excessive variation. Because the vehicle assembly process is very long and involves many steps, the toe alignment problem required brainstorm which serial production steps, factors, and levels could be responsible for the variation. The most likely process steps were examined and the problem area identified as a wheel alignment machine that needed maintenance and calibration.

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Telematics Analytics

This post provides a list of my telematics analytics articles, a short description and a link. The goal is to make it easy for the reader to locate any of the articles. It is recommended to read these articles in the order presented.  This post will be updated when additional posts are written on this topic.

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Process Capability Articles

This post provides a list of my process capability articles, a short description and a link. The goal is to make it easy for the reader to locate any of the articles. It is recommended to read these articles in the order presented.  This post will be updated when additional posts are written on the Process Capability topic.

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Process Capability IV – Improve the Process

In prior articles of this series, the process was not capable. The Cp ~0.46 so even if centered, a high percentage of parts would be non-conforming! In industry, it is common to see a Cp of about 1. The paradigm is that if the process range is just contained within the tolerance range, it is satisfactory.  Such processes are marginally capable. A process mean shift away from the target or an increase in variation significantly increases the number of non-conforming parts.  Process Capability needs to be increased. At a minimum, Cp should be greater than 1.33, but larger is better.

There are two approaches to improving process capability. One is to center the process and the other is to reduce variation. Both need to be accomplished.

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Introduction

In the article “PCA I”, the main process performance indices were defined. In the article “PCA II”, ways to calculate the percent defective were introduced. In this article, the mathematical connection between the Cp index and percent defectives is made.

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Introduction

In my prior article, PCA was defined for prototype samples and serial production. It’s application to one and two sided distributions was discussed. The P_p, P_pk, C_p, and C_pk indices were defined. This article provides methods to estimate the percent defective.

The Data

The engineering tolerance for a critical characteristic is 10±0.2. So the lower specification limit (LSL) is 9.8 and the upper specification limit (USL) is 10.2. A sample of 30 parts provided measurements of the critical characteristic. The sample mean ($-\bar{x}-$) was 9.951 and the sample standard deviation (s) was 0.1825.

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Introduction

The analysis of telematics data with two or more parameters is a complex process. The analysis of multiple parameters using contour plots is a powerful tool since a lot of information is captured in the graphics.

The best results come from a team effort. For engineering data, the team may consist of the design/development engineers, a programmer, and a reliability engineer or a statistician. The following is an analysis of engine speed, engine torque, and the transmission gear state to describe the process.

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Introduction

Vehicle operation is highly variable causing parameter and states data to change unpredictably. A way to deal with this data is Markov Analysis.

Markov Processes

A Markov Process is defined a having a finite number of states and the probability of transitioning from the current state to another state is between 0 and 1. A Markov transition matrix is used to count these transitions, table 1. [Read more…]

Introduction

Telematics data presents the opportunity to characterize the vehicle lifetime usage. This information is used to validate development and testing targets. Because there can be bad or missing data, it needs to be reviewed prior to analysis to have confidence in the analysis. [Read more…]

Introduction

In my prior article, “A Primer on Probability Distributions”, the usage of different types of histograms to display data was discussed. The histogram would acquire a bell shape if the data were normally distributed. The main limitation of the histogram approach is that the shape of the histogram can change radically with the selection of the bins used. Instead, a probability plot could be used.

This article discussed probability plots. They are extremely useful for the analysis of big and small data sets.

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Introduction

Frequently, data collection is the most time consuming and expensive part of a project. Consequently, people work with small sample data. There is too little data to plot a histogram, so the analysis assumes the underlying population is normally distributed.

A frequent error is to assume the sample average and standard deviation are the population normal mean and standard deviation. When small sample sizes are being analyzed, these assumptions lead to estimation errors.

Methods to make better estimates are discussed in this article. [Read more…]