Mean vs. Median: A Beginner’s Guide

A common way to summarize a group of numbers — one most of us learned in grade school — is to find its mean, commonly called the average. But it’s not always the best measure.

Let’s say six kids go on a field trip, ages 10, 11, 10, 9, 13 and 12. It’s easy to add the ages and divide by six to get the group’s average age:

(10 + 11 + 10 + 9 + 13 + 12) / 6 = 10.8

Because all the ages are close, the average of 10.8 gives us a good picture of the group as a whole. But averages are less helpful when the values are skewed toward one end or if they include outliers.

For example, what if we add a much older chaperone to our field trip? With ages of 10, 11, 10, 9, 13, 12 and 46, the average age of the group rises considerably:

(10 + 11 + 10 + 9 + 13 + 12 + 46) / 7 = 15.9

Now the mean is not an accurate representation. The outlier skews the average, and no journalist should feel comfortable reporting it.

This is where calculating a median is handy. The median is the midpoint in an ordered list of values — the point at which half the values are higher and half lower. If the median household income in East Middletownburg is $50,000, then half the households earn more and half less.


26,500 school cafeterias uninspected

Thousands of school cafeterias went uninspected in the 2007-08 school year, we report today in the fourth major installment of our “Trouble on the Tray” investigation into school lunch safety.

In today’s story, reporters Blake Morrison and Peter Eisler worked with me to examine data on the number of schools in each state that met a federal requirement to have two cafeteria inspections annually. We found that in eight states, more than half of schools reporting failed to meet that standard in 2006-07 and 2007-08 school years.

Meanwhile, the series continues to draw attention on Capitol Hill. This week, Sen. Kirsten Gillibrand, D-N.Y., called on the federal government to increase its standards for meat used in school lunches and to cut contracts with companies that repeatedly did not meet standards.

When chickens stop laying eggs …

In part three of USA TODAY’s investigation into the quality of government-bought food for school lunches, we examine how its standards for microbial testing of school lunch beef are less stringent than those employed by fast food chains such as McDonald’s, Jack In the Box and KFC.

We also write about “spent hens” — birds that are past their egg-laying prime. Tough and stringy, these old birds typically are turned into pet food or compost. But egg producers struggling to find a market for all of them have had help from the federal government:

From 2001 though the first half of 2009, USA TODAY found, the government spent more than $145 million on spent-hen meat for schools — a total of more than 77 million pounds served in chicken patties and salads. Since 2007, 13.6 million pounds were purchased.

Both stories were heavily informed by analysis of data sets obtained from the USDA under the Freedom of Information Act. They included hundreds of thousands of orders from a federal inventory system and about 150,000 results of microbial tests of beef destined for school lunches.

Prompted in part by our series but also by last week’s recall of beef by a company we identified in Part Two of our series, one lawmaker has called for the government to investigate a supplier to the school lunch program. From another story of ours today:

Rep. Rosa DeLauro, D-Conn., called on the U.S. Department of Agriculture to “undertake a comprehensive examination” of the facility, Beef Packers, to “identify and correct any major problems” before it produces more beef.

To see all the stories in our series thus far, click here.

Project: School lunch safety

Today, my colleagues Blake Morrison, Peter Eisler and I published the second part of our investigation into the safety of food used in the National School Lunch Program. Today’s installment focuses on a California firm that kept receiving government contracts even after  it had been suspended from the program several times — twice because of failure to produce ground beef that was free of salmonella.

When the firm, Beef Packers, recalled beef last summer because of a salmonella outbreak in 11 states, the government decided not to recall beef made for school lunches that the company made around the same time:

The recall, announced by the government Aug. 6, covered only ground beef sent to certain retailers. In the days after it was announced, government and company spokesmen said meat sent to schools was not included. Documents obtained by USA TODAY through the Freedom of Information Act reveal a more complicated story — one that raises questions about whether the government took adequate steps to ensure that meat it bought for schoolchildren during the same period was safe.

To get at the story, we filed FOIA requests for several government data sets. They included the results of hundreds of thousands of microbial tests conducted by the USDA as well as a dump from an inventory system the government uses to track orders for the school lunch program.

Update, 12/7/2009: Morrison and Eisler report that Beef Packers issued its second recall this year for beef tainted with salmonella.

Bookshelf: Numbers in the Newsroom

Never underestimate the value of a compact guide to math, especially if you’re one of those journalists who thought  you could avoid numbers by becoming a writer. You shouldn’t — understanding numbers will help you get stories  others miss because of innumeracy.

One of the handiest resources I’ve found — and recommended just this week to a roomful of colleagues — is Sarah Cohen’s “Numbers in the Newsroom.” It’s a 108-page guide that covers the basics on percent change, rates, graphics, probability and much more. Cohen is a Pulitzer-winning former Washington Post staffer and one-time training director for Investigative Reporters and Editors. She’s now at Duke University, where she is the Knight Professor of the Practice of Journalism and Public Policy.

The book is a few years old, but its lessons are timeless. You can pick it up through IRE’s online store.

Have your own math book recommendations? List them below …