Q: What’s an effective approach for establishing, maintaining and optimizing an effective quality management system (QMS)?
Fort Worth, TX
A: The most effective approach is based on demonstrated need for a QMS and supported by the senior management of an organization, meaning the CEO and his or her direct reports. When everyone in an organization understands the need to address quality, there will be buy-in from everyone for a QMS. Consider two examples:
1. Many years ago, there was a prevailing perception in the customer base of a large retailer that the retailer sold seconds merchandise (products in which a mistake was made in making the item), even though that was not the case. To address and change this, the CEO of this organization created a quality assurance (QA) division with the director of QA reporting directly to the CEO. The fact that the director of QA attended weekly staff meetings with the CEO, along with other direct reports to the CEO, sent a powerful message throughout the retailer as well as the supplier community that quality of merchandise was extremely important to top management.
The organization also effectively used product inspection based on statistical sampling at suppliers’ facilities and retailers’ warehouses; it conducted pre-purchase and post-award product testing to ensure the products met requirements; it passed on information from customer complaints to the suppliers for their consideration and actions; and it helped buyers (purchasing officers/agents) establish performance specifications. Using these strategies, the retailer became known for first-class quality across all merchandise categories within two to three years, and its quality system came to be known as one of the best in the retail industry.
2. In another example, a children’s wear manufacturer had an established QMS, but the director of QA was not getting the necessary resources. Over a few months, he had his staff collect data on scrap, rejects and rework, and put cost figures to all these data. He managed to get 10 minutes of time in the executive vice president’s (EVP) staff meeting and presented on how much money the organization could save with a bit more resource devoted to quality management. This opened the EVP’s eyes to potential savings, and the director of QA got what he needed. From then on, every staff meeting with the EVP opened with a presentation on quality.
In both of these examples, quality management was strongly supported by senior management. In the first case, quality management was driven by customers. In the second case, it was driven by the bottom line.
Mehta Consulting LLC
Q: How robust is "median" as a statistic?
San Jose, CA
A: To be clear, the median is the center value of a set of observations taken from a broader population. As such, it is the 50th percentile. If a set of observations in rank order contains an uneven number of them, it is the observation in the middle; if the set contains an even number of observations, it is the mean of the two numbers closest to the middle.
How robust is it?
A statistic is robust if it is resistant to change even when some observations change. The mean of a data set, for example, will change if any observation in the data set changes. This is not so for the median. Individual observations may wander within limits and the median won’t budge. That property is partially responsible for the median’s appeal. But just how robust is it?You would have to define a specific quantitative measure of robustness to say.
When is the median appropriate?
In theory, if your data set fails the test of normality, and you can’t find a transformation to normality or another suitable distribution that fits, you can always default to a distribution-free test for comparing two or more treatments. Distribution-free tests often rely on the median as a measure of central tendency. Developers of distribution-free tests are careful to examine relative efficiency, measured as the ratio of the variance of the distribution-free statistic to the variance of the appropriate statistic under normality (or some other assumed distribution).
The relative efficiency of a distribution-free test statistic computed on data that are actually normally distributed is often low. This is one factor that prevents you from abandoning all normal theory statistics in favor of distribution-free tests. If the data are normal, using median instead of the mean is often inefficient.
It is important to understand that often, large data sets, even some from an underlying normal distribution, will fail a formal test of normality simply because the test criteria are very strict. In most practical applications, a straight line on a normal probability plot is sufficient evidence of normality or near-normality. Opinions will differ, but many applied statisticians will stick to normality unless there is clear evidence against it.
In many situations, there may be value in seeking the cause of non-normality. Data may actually come from multiple sources with different means, causing the appearance of non-normality in the aggregate data set. A distribution-free test in this environment may be technically correct, but it may miss the point of getting to the root causes of variation.
Still, when the data set is decidedly not normal and all other avenues have been traveled, the distribution-free test based on medians is appropriate.
How can you prevent the misuse of medians to sway perception?
You can’t. If someone’s paycheck depends on their ability to produce a summary that best supports their argument, little can be done to dissuade them from the practice. However, given the opportunity, you may be able to educate an errant user of medians by showing plots of the data, including a normal probability plot, to point out that the data can be summarized more efficiently. Bear in mind, however, that if the distribution is normal, the median and mean will be very close, so there may be little harm done by showing the median in place of the mean. When data analysts use medians, they should provide some justification explaining why that choice was made.
Are there measures of dispersion for the median that should be published along with them (for example, interquartile range)?
An interquartile range may be appropriate, but a statistic showing something closer to the full range of the data might do a better job of persuading the reader of vast uncertainty, assuming it exists. A graphical display, such as a box plot, is useful for showing the variability of the data that are summarized by the median.
There is a formula for the variance, ó2, of the median from any population, but it assumes a known distribution:
in which n is the number of observations, f is the density function of the population, and m is its median. In situations in which the distribution function may be assumed—for example, microbiological counts are often approximately lognormal—there may be value in showing the variance or the standard deviation of the median, simply to point out the uncertainty associated with the median.