User talk:Cwelling

AN EXAMPLE MATHEMATICAL LITERACY PROBLEM IN THE MEDIA Consider the following segment of a Yahoo News clip. The author, Francesca Levy, makes a mistake that the editor seems to have overlooked in the original post on http://realestate.yahoo.com/promo/americas-most-expensive-zip-codes-2010.html. The post has since been edited (Please see http://www.forbes.com/2010/09/27/most-expensive-zip-codes-2010-lifestyle-real-estate-zip-codes-10-intro.html.). Below is the relevant segment of the original article. [Excerpt begins below.] America's Most Expensive ZIP Codes 2010 …a

By Francesca Levy, Forbes.com Sep 27, 2010:

"Los Angeles has always been home to some of the world's most expensive real estate. But forget Beverly Hills, 90210: The new hot spot for multimillion-dollar mansions is Duarte, 91008…

"A scant 1,391 people live in 91008 ZIP code, and only 12 homes are currently on the market. So a single high-priced listing (like the mammoth nine-bedroom, built this year, that's selling for $19.8 million) is enough to skew the median price skyward" (http://realestate.yahoo.com/promo/americas-most-expensive-zip-codes-2010.html). (The bold emphasis is mine.)

We cannot help but notice the phrase “a single high-price listing…is enough to skew the median price skyward” (emphasis, mine). Of course, it is immediately obvious that the phrase should have been “a single high-price listing…is enough to skew the mean price skyward”. The very concern the author has is the very issue that using the median would help us avoid—an overstating of the average home-price. Cleary, the median’s reasonable resistance to the outlier ($19.8 million) is what makes the median a better measure of average than the mean, in this scenario. Therefore, it is not very wise to speak of skewing “the median price skyward”.

Below is a basic statistical analysis of the most likely listed 12 home prices that Ms. Levy references. http://realestate.yahoo.com/search/California/Duarte/homes-for-sale confirms the home prices that I used below.

Data: Home Prices in Bradbury, California 91008:

$19,800,000; $1,250,000; $5,950,000; $4,680,000; $630,000; $5,880,000; $2,980,000; $1,299,000; $5,888,000; $5,850,000; $1,875,000; $5,980,000.

Mean and Five Number Summary Data (with outlier, i.e. $19,800,000) for n = 12:

Mean = $5,171,833;

Minimum = $630,000;
Q1 = $1,587,000;
Q2 (Median) = $5,265,000;
Q3 = $5,919,000;
Maximum = $19,800,000.

Mean and Five Number Summary Data (without outlier) for n* = 11:

(Remove the $19,800,000 home price.)

Mean* = $3,842,000;

Minimum* = $630,000;
Q1* =  $1,299,000;
Q2*(Median*) = = $4,680,000;
Q3*  = $5,888,000;
Maximum* = $5,980,000.

(Note: http://realestate.yahoo.com/search/California/Duarte/homes-for-sale confirms these prices; but I lost the original source/site of the listing which listed exactly 12 home prices for the range $630,000 to $19,800,000 of my data set.)

In order to challenge Ms. Levy’s statement, let us compare the “n = 12” data with outlier summary with the “n = 11” data without outlier summary. We see that it the mean that received the greatest skewing. The median was, as expected, more resistant to skewing. This is illustrated below:

Mean – Mean* = $5,171,833 - $3,842,000 = $1,329,833; while Median – Median* = $585,000.

Clearly this shows that the outlier pushes up the mean price considerably more than the median price, obviating the phrase “skew the median price skyward”.

Cris Wellington