In Search of the Median Ground

Polly Yurrathain’s uphill battle to calculate averages

Quality Manager Polly Yurrathain is in deep trouble. Her boss, Hammond Eggs, has been ranting for days about the diminishing quality of the company’s products. Their firm, which manufactures a variety of toy products, has undergone scrutiny by a major consumer journal; in addition, stores are complaining about high rates of return for the toys that Play With Us Inc. manufactures.

Eggs orders Yurrathain to hire “as many inspectors as needed” to keep defective products from leaving the assembly line and landing in stores. Yurrathain, who rarely responds to her boss with anything but a meek, “Yes, sir,” summons the courage to point out that a system of prevention is always preferable to one of detection. “I hired you to prevent mistakes,” her boss shouts. “And that hasn’t worked, so we’ll inspect every blasted toy that we make.” He orders Yurrathain to hire 10 inspectors immediately, at minimum wage.

Under pressure to get results, Yurrathain goes to the local employment office and pleads with people in line, finally attracting 10 who say they’ll show up Monday.

Although only eight workers actually appear on schedule, Yurrathain is undaunted and decides to teach them about X-bar and R charts on the first day. The training room is set with calculators at each station, and she begins to lecture. These new students do well with the concept of range:

However, they have a hard time calculating averages despite Yurrathain’s use of batting averages as a model. They seem to be totally puzzled by the statistical symbols:

She suddenly recalls the use of median charts, popularized by Paul Clifford after World War II, where only the median is recorded rather than the mean. Common practice called for sample sizes of three or five, so no math was necessary. For example:

Data: 14, 12, 17, 19, 15

To calculate the median, the data must be ordered from the smallest to the largest number; the middle number is the median. In this example, it would be 15.

Excited by this approach, Yurrathain teaches her motley crew how to do median charts, beginning with the following formula:

Unfortunately for her, one of the new recruits had been employed previously as an inspector. Just as Yurrathain finishes her brilliant lesson, this inspector asks why the A2 factor is different for median charts than for X-bar and R charts. He used his calculator to figure that the A2 factor for median charts, called A2 tilde (~), is about 25 percent larger than the A2 for X-bar and R charts.

Not wanting to embarrass herself by admitting that she doesn’t know, Yurrathain points out that the square root of 0.0625 is equal to 0.25. The former inspector accepts this answer because he isn’t adept at math. Is Yurrathain’s response appropriate?

The answer is no. Although she didn’t get caught by the unsuspecting inspector, Yurrathain was wrong. Median charts offer a good alternative to X-bar and R charts for two reasons:

They can be done by hand, so for those with limited math skills, creating the charts isn’t an overwhelming task.

By convention, a median chart shows not only the median value but also the values of the observations.

The real reason that control limits are about 25 percent wider than for X-bar and R charts deals with the difference in the way medians and averages are calculated.

The mean uses all the data in a sample to estimate the central location of the population.

The median orders the samples from smallest to largest and picks the middle number (assuming an odd sample size) as an estimate of the central location of the population. With a sample size of five, all data values are used in calculating the mean, but only one number is used to determine the median’s value. It’s as if the calculation of the median throws away the information in four of the five pieces of data. One would expect that the mean would be a more efficient estimator of central location of a population than would a median. (Note: An “efficient” estimator is one that is more precise in its ability to estimate a population parameter.)

Statistician Walter Shewhart was aware of the phenomenon and noted that the sampling distribution of sample medians will be about 25 percent more variable than the distribution of sample means. (Source: Shewhart, W.A. Economic Control of Quality of Manufactured Product [D. Van Nostrand, 1931])

Poor Yurrathain. She started out well, but because she was unwilling to take time to investigate, she was wrong.

About the author

Michael J. Cleary, Ph.D., founder and president of PQ Systems Inc., is a noted authority in the field of quality management and a professor emeritus of management science at Wright State University in Dayton, Ohio.

A 29-year professorship in management science has enabled Cleary to conduct extensive research and garner valuable experience in expanding quality management methods. He has published articles on quality management and statistical process control in a variety of academic and professional journals.