Tag Archives: Average

TWO CRITICALLY IMPORTANT WORDS: “ON AVERAGE”

Newspaper Reports (Misleading)

Recently, a team of 3 researchers conducted an investigation to see if children’s post-surgical pain could be reduced by “audio therapy.”

In the study, the participating children were randomly assigned to two main groups. Those in the treatment group listened—after surgery—to a self-selected audio book or musical playlist. Those in the control group did not listen to anything. Each child in each group was asked to rate his/her post-surgical pain level on two occasions: prior to and immediately after the time the treatment group received the audio therapy.

A newspaper summary of the results stated:

Patients listening to music and audiobooks reported feeling less pain after receiving the therapy, according to the study.”

This sentence makes it seem that audio therapy worked for everyone who got it. Not so.

Data in the published technical research report show that most children in the treatment group said their pain was lower after receiving the audio therapy. However, some of the children in that group had more pain after audio therapy than before. Also, some children in the control group had more pain reduction than some of the children who received audio therapy.

The quoted newspaper sentence should have begun with 2 important words: “On average.” By adding those two words, the misleading sentence from the newspaper report becomes a true and factual statement about the study’s results.

(The published research report carried this title: “The effect of audio therapy to treat postoperative pain in children undergoing major surgery: a randomized controlled trial.” It was published in a peer-reviewed journal called Pediatric Surgery.)

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A DRINKING HOLE

X-Bar

BACKGROUND & QUESTION:

Year after year, the annual statistical convention is held in the same city. This metropolis has 26 pubs, each named by an alphabet letter: A, B, C, … , Y, Z. The statisticians who are nice, kind, & considerate people go to a wide variety of these “drinking holes.” The mean statisticians, however, patronize just one of them. Which one?

ANSWER: The X-Bar

(This little effort at statistical humor comes from S. Huck)

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Beyond the Joke:

In statistics, the concept or numerical value of the arithmetic mean can be symbolized in various ways.

Many people use the letter M (often capitalized and italicized) to represent the arithmetic mean. A second way to do this is with the lower-case Greek letter, mu. (In several textbooks, M is used to represent the sample mean whereas mu designates the population mean.)

A third way to symbolize the arithmetic mean is with the letter X accompanied by a short, horizontal line positioned directly above the X. This line is referred to as a “bar,” and the entire symbol is read as “X-bar.”

Symbols for mean

In written statistical discussions, a bar can be positioned above letters (or symbols) other than the letter X. When this occurs, the bar indicates that the arithmetic mean has been (or should be) computed for the various numerical values of the variable represented by whatever letter or symbol has the bar above it. For example, if you see the lower-case letter r with a bar above it, you should refer to it as “r-bar” and guess that it represents the mean of a group of correlation coefficients.  

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STATISTICIANS ARE UNUSUAL YET QUITE AVERAGE

On the one hand, statisticians are a bit weird. On the other hand, they are altogether average. Here’s the proof:

  1. They often break the law and drive their cars on the MEDIAN.
  2. At dinner, they invariably want more desserts than anyone else, and they always want them ala-MODE.
  3. If you sum up their deviations, they lose their cool and get incredibly MEAN.

(This little effort at statistical humor comes from S. Huck)

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