Question

USA Today (May 9,2006 ) published the weekday circulation numbers for the top 20 newspapers in the country. Here are the data for the 6 -month period ending March 31,2006 : \(\begin{array}{rrrrr}2,272,815 & 2,049,786 & 1,142,464 & 851,832 & 724,242 \\\ 708,477 & 673,379 & 579,079 & 513,387 & 438,722 \\ 427,771 & 398,329 & 398,246 & 397,288 & 365,011 \\ 362,964 & 350,457 & 345,861 & 343,163 & 323,031\end{array}\) a. Calculate and interpret the value of the median of this data set. b. Explain why the median is preferable to the mean for describing center for this data set. c. Explain why it would be unreasonable to generalize from this sample of 20 newspapers to the population of daily newspapers in the United States.

Short answer

Median: 433246.5. It is more suitable to use median instead of mean as our data is skewed with few high values. The sample only includes top newspapers by circulation and misses out on a large number of smaller newspapers. Thus, it won't represent an accurate estimate for the population of all U.S. newspapers.

Step-by-step solution

Calculate the Median

Given the dataset is already sorted, find the middle value. Since there are 20 numbers, the median would be the average of the 10th and 11th elements in the series. The 10th and 11th numbers in this collection are 438,722 and 427,771. Average these two to find the median. Median = \(\frac{(438,722 + 427,771)}{2}\).

Interpret the Median

Calculate the value from Step 1. This is the median circulation number which means that half the newspapers have a circulation greater than this number and half have a circulation less than this number.

Understanding Median vs Mean for this data

The dataset seems to have a few high values relative to the others. This makes the dataset skewed. In skewed distributions, the mean, being highly sensitive to extreme values, is not a good measure of central tendency. The median, in such cases, gives a better idea of the central value as it is not affected by extreme values.

Inappropriateness of Generalization

The sample of top 20 newspapers in terms of circulation does not include many small to medium scale newspapers, which are going to have significantly smaller circulation numbers. This sample only represents the larger newspapers with a higher circulation, and cannot be used to provide an accurate estimate of the mean or median circulation for the general population of newspapers in the U.S.