Statistics Article

Multiple Authors
By: Donald J. Wheeler, Al Pfadt

In memory of Al Phadt, Ph.D.

This article is a reprint of a paper Al and I presented several years ago. It illustrates how the interpretation and visual display of data in their context can facilitate discovery. Al’s integrated approach is a classic example not only for clinical practitioners but also for everyone who needs to turn data into knowledge.

This is an example of how process behavior charts were used to (1) evaluate outcomes; and (2) assist in making clinical decisions in the treatment of severe, potentially life-threatening, self-injurious behavior (viz., self-inflicted wounds to the body caused by head-banging and biting the wrists and fingers). The treatment of Peter, a 25-year-old man with autistic disorder who functions in the severe range of intellectual disability and has been blind since birth, is described from two points of view. First, from the perspective of Dr. Al Pfadt, the behavioral psychologist who constructed and analyzed the charts shown here; then from the perspective of Peter’s parents. The following material was provided by Pfadt and his colleagues at the New York State Institute for Basic Research in Developmental Disabilities and is used with the permission of Peter’s parents.

Donald J. Wheeler’s picture

By: Donald J. Wheeler

The shape parameters for a probability model are called skewness and kurtosis. While skewness at least sounds like something we might understand, kurtosis simply sounds like jargon. Here we’ll use some examples to visualize just what happens to a probability model as kurtosis increases. Then we’ll combine the visible effects of both skewness and kurtosis to see how they combine to “shape” probability models

Last month, we found that while an increase in positive skewness measures a nearly invisible increase in the area of the upper tail, it has two visible manifestations. Specifically, these are a substantially shorter lower tail and an appreciable shift of the mode to the left. Figure 1 summarizes these results for the three comparisons made.

Figure 1: Effects of increasing positive skewness while holding kurtosis constant

Figure 1 shows what happens when we hold the kurtosis constant and increase the skewness. In the comparisons that follow, we’ll look at what happens when we increase the kurtosis while holding the skewness constant.

Alan Metzel’s picture

By: Alan Metzel

Almost seven years ago, Quality Digest presented a short article by Matthew Barsalou titled “A Worksheet for Ishikawa Diagrams.” At the time, I commented concerning enhancements that provide greater granularity. Indicating that he would probably have little time to devote to such a project, Barsalou graciously invited me to expand upon his work. As one of the few positive outcomes of the recent storms that have raged across the United States, I’ve finally completed that task. In thanks to Barsalou—and with tribute to his mentor—I refer to this effort as the “Enhanced Perkin Tracker.”

Story update 1/23/2023: A previous version of this story linked to the wrong Excel file. That Excel link is now correct.

For those who might be unfamiliar with the Ishikawa diagram, it’s a graphic problem-solving tool used to relate multiple potential causes to a single effect in a rational manner. Based on its shape, it’s easy to understand why it’s often referred to as a “fishbone” or “herringbone” diagram.

Donald J. Wheeler’s picture

By: Donald J. Wheeler

The computation for skewness does not fully describe everything that happens as a distribution becomes more skewed. Here we shall use some examples to visualize just what skewness does—and does not—involve.

The mean for a probability model describes the balance point. The standard deviation describes the dispersion about the mean. Yet a simple description of skewness is elusive. Depending on which book you read, skewness may be described as having something to do with the relative size of the two tails, or with the weight of the heavier tail of a probability model.

By far the easiest way to understand what increasing skewness does to a probability model is to compare models with different amounts of skewness. But before we can do this, we have to first standardize those models. This is because skewness is defined in terms of standardized variables; skewness is what happens after we have taken into account differences in location and dispersion. (If we compare two distributions that have not been standardized, differences in location and dispersion may obscure differences in skewness.) So here we will be working with standardized distributions where the mean is always zero and the standard deviation is always equal to one.

Tony Boobier’s picture

By: Tony Boobier

Does your use of probabilities confuse your audience? Sometimes even using numbers can be misleading. The notion of a 1-in-a-100-year flood doesn’t prevent the possibility of flooding occurring in consecutive years. This description is no more than a statistical device for explaining the likelihood of flooding occurring.

Similarly, when we check the news for weather conditions and are told that there is a 90-percent chance of rain, this only means that on days with similar metrological conditions, it rained on 90 percent of them. As with the flooding comparison, it’s simply a mathematical method that expresses the likelihood of an incident occurring.

Donald J. Wheeler’s picture

By: Donald J. Wheeler

The cumulative sum (or Cusum) technique is occasionally offered as an alternative to process behavior charts, even though they have completely different objectives. Process behavior charts characterize whether a process has been operated predictably. Cusums assume that the process is already being operated predictably and look for deviations from the target value. Thus, by replacing process characterization with parameter estimation, Cusums beg the very question process behavior charts were created to address.

To illustrate the Cusum approach and compare it with an average chart, I’ll use the example from page 20 of Shewhart’s first book, Economic Control of Quality of Manufactured Product (Martino Fine books, 2015 reprint).These data consist of 204 measurements of electrical resistivity for an insulator. Shewhart organized them into 51 subgroups of size four, based upon the time order in which the measurements were obtained. Figure 1 gives the averages and ranges for these 51 subgroups.

Donald J. Wheeler’s picture

By: Donald J. Wheeler

Many people have been taught that capability indexes only apply to “normally distributed data.” This article will consider the various components of this idea to shed some light on what has, all too often, been based on superstition.

Capability indexes are statistics

Capability and performance indexes are arithmetic functions of the data. They are no different from an average or a range, just slightly more complex. The four basic indexes are the following:

The capability ratio, Cp, is an index number that compares the [space available within the specifications] with the [generic space required for any predictable process].

The performance ratio, Pp, is another index number that compares the [space available within specifications] with the [estimated space used by process in the past].

Danielle Underferth’s picture

By: Danielle Underferth

As municipalities clamor for a slice of President Biden’s $1.2 trillion infrastructure spending bill, one Johns Hopkins scientist is re-examining one of the basic elements of road-building: Determining the width of road lanes. But determining the width that provides the highest level of safety, access, and comfort for every road user—drivers, cyclists, and pedestrians—is complex, says Shima Hamidi, an assistant professor in Johns Hopkins’ Department of Environmental Health and Engineering, which is shared by the Whiting School of Engineering and the Bloomberg School of Public Health.

It’s a data problem, she says, and she wants to help cities solve it.

Hamidi is undertaking a massive collection of data on urban streets across the United States to answer one question: How low can cities go on street width to make room for bike lanes and wider sidewalks?

Tristan Mobbs’s picture

By: Tristan Mobbs

All too often the topic of fixing dirty data is neglected in the plethora of online media covering artificial intelligence (AI), data science, and analytics. This is wrong for many reasons.

To highlight just one, confidence in the quality of data is the vital foundation of all analysis. This topic remains relevant for all levels of complexity, from spreadsheets to complex machine-learning models.

So, I was delighted to review Susan Walsh’s book, Between the Spreadsheets: Classifying and Fixing Dirty Data (Facet Publishing, 2021). Here are some highlights from her book, and my own advice on who should read it.

Atul Minocha’s picture

By: Atul Minocha

Do you ever feel like you’re spending money like crazy on marketing and getting little or nothing in return? If so, you might be tempted to pull the plug on marketing altogether. That would be a big mistake.

An effective marketing strategy can mean the difference between your organization’s success and failure. To maximize your strategy, there are eight common marketing mistakes you should avoid at all costs.

#1 Focusing solely on data

Most marketers firmly believe the old saying, “What doesn’t get measured doesn’t get improved.” They track various metrics, hoping the data will show them how to improve customer engagement.

The problem is some of the most important elements of customer engagement—like emotional response—can’t be tracked easily. How do you measure whether or not you’re tugging at their heartstrings?

The real power of marketing comes from synergy of both the left brain (data) and the right brain (emotion). Focusing solely on the data will never lead to optimal results.

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