Mean, Median, and Descriptive Statistics

For children under the age of 13, majors is the highest level in Little League Baseball and minors is the next highest level. In the 2023 season, Middletown Little League had enough players to form six majors teams and six minors teams. Instead, they decided to have seven majors teams and five minors teams. What effect did this have on the average quality of the players in the majors and in the minors?

A college reported that its average class size is 14 students per class. However, a survey that asked this college's students the size of the classes they were enrolled in found that the average class size was 38 students and that $70 \%$ of the students' classes had more than 14 students. Provide a statistical explanation for this disparity.

A school estimated the average number of school-age children per family by questioning each child in the school. Why was their estimate much too high? Give an example to illustrate your reasoning.

Explain comedian Will Rogers' joke that, "When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states."

A sports columnist wrote that "Jon Gruden, left [Monday Night Football] to become Oakland's head coach, a move that has been bad for both the broadcast and the team." Is it possible for someone who goes from one organization to another to lower the average quality of both organizations?

A survey found that between 1959 and 1983, the average number of hours that women worked outside the home nearly doubled while the number of hours worked inside the home decreased by only $14 \%$. Do these data show that the total number of hours women worked, outside and inside the home, increased between 1959 and 1983?

A college offers a free book on the history of the college to each graduate whose gift exceeds the median size of the gifts made in the previous year by members of this person's class. Explain why this college either increased or decreased the threshold for a free book by using the median rather than the mean.

Identify the apparent statistical mistake in this commentary:

The median price of a house in Duarte is a whopping $4,276,462, making it the most expensive housing market in the country.... Only 12 homes are currently on the market. So a single high-priced listing (like the mammoth nine-bedroom that's selling for $19.8 million) is enough to skew the median price skyward.

A Prudential Financial advertisement titled, "Let's get ready for a longer retirement," said that

A typical American city. 400 people. And a fascinating experiment. We asked everyday people to show us the age of the oldest person they know by placing a sticker on our chart.

Most of the stickers were between the ages of 90 and 110, which Prudential called, "Living proof that we are living longer. Which means we need more money to live in retirement." Why is this conclusion misleading?

A study divided occupations into five categories and found that women had higher unemployment rates than men in each of the individual categories but, overall, had a lower unemployment rate than men. How is this possible? Use hypothetical data with just two categories to illustrate your reasoning.

The overall unemployment rate of $10.2 \%$ in October 2009 was lower than the $10.8 \%$ unemployment rate in November 1982, which was the peak of the worst recession since the Great Depression. However, when the labor force was divided into four educational categories (college graduate, some college education, high-school graduate, and other), the unemployment rate in each category was higher in 2009 than in 1982. How is this possible? Use hypothetical data with just two educational categories to illustrate your reasoning.

A medical study found that Treatment A was more successful than Treatment B in treating small kidney stones ($90 \%$ versus $80 \%$) and large kidney stones ($70 \%$ versus $60 \%$). Is it possible for Treatment $B$ to be more successful overall, when the data for large and small kidney stones were combined? Explain your reasoning and use a numerical example to illustrate your argument.

A study of the hot hands in basketball included pairs of free throws shot by two players, A and B. When Player A made the first shot, he made the second shot $88 \%$ of the time; when he missed the first shot, he made the second shot $91 \%$ of the time. Player B was also more likely to make the second shot after missing the first shot: $61 \%$ versus $59 \%$. Yet, when the data were combined, the players made $81 \%$ of their second shots after making the first shot and only $73 \%$ after missing the first shot.

Which of these data series has the larger standard deviation?

A researcher found that in the 2009-2010 Premier League season, the soccer teams with the best records tended to win more of their home games than did teams with worse records. He concluded that this demonstrated the importance of a home-field advantage. Why do his data not justify his conclusion?

When it was proposed that Major League Baseball (MLB) institute a pitch clock that allows a maximum of 20 seconds between pitches, a respected sports columnist wrote that, "If MLB were to institute a 20 -second pitch clock and hold players to it, that would, in theory, at this year's pace and pitches per game, have shaved 19 minutes, 48 seconds off the average length of a game." He multiplied 4 seconds (the difference between the 24 -second average time between pitches in 2018 and the 20 -second pitch clock) times the 297 (the average number of pitches per game): $(24-20) 297=1,188$ seconds, which is 19 minutes and 48 seconds. Why is this estimate almost surely wrong? Is the correct time saving most likely larger or smaller than 19 minutes and 48 seconds?

A study of the effect of college education on income used 1990 and 2010 US Census data. The study looked at: (a) women in the 1990 Census who were between 28 and 32 years old and were also the first female in their family to attend college; and (b) women in the 2010 Census who were between 28 and 32 years old and whose mother was the first female in their family to attend college. The average income was $\$ 45,000$ for the women in group (a) and $\$ 40,000$ for the women in group (b), even though average income for the nation as a whole was $50 \%$ higher in 2010 than in 1990.

Is it mathematically possible that every daughter had a higher income than her mother, yet the average daughter income was lower than the average mother income? Give a specific numerical example to illustrate your argument.

A Temple University mathematics professor used these data to show that most Americans have an exaggerated fear of terrorists:

Without some feel for probability, car accidents appear to be a relatively minor problem of local travel while being killed by terrorists looms as a major risk of international travel. While 28 million Americans traveled abroad in 1985, 39 Americans were killed by terrorists that year, a bad year-1 chance in 700,000. Compare that with the annual rates for other modes of travel within the United States-1 chance in 96,000 of dying in a bicycle crash, 1 chance in 37,000 of drowning, and 1 chance in only 5,300 of dying in an automobile accident.

How do you suppose the author calculated the probabilities of dying in a bicycle accident, of drowning, and of dying in a car accident? Do these calculations prove that it is more dangerous to drive to school than to fly to Paris?

Cameron is taking a class where the final grade is an average of the homework and test scores

where H and T are both graded on a scale of 0 to 100. Part way through the semester, Cameron's G was 92. Then Cameron got a score of 90 on a test and the teacher raised Cameron's G to 93.5. Is this possible or did the teacher make a mistake? Use an example to explain your reasoning.

It has been estimated that as many as $50 \%$ of the people infected with COVID-19 were asymptomatic and did not know that they had been infected. If the data on the number of deaths and the number of people who test positive are accurate, but the number of asymptomatic infections is higher than previously estimated, which of the following will be higher, lower, or the same as previously estimated?

Answer this Car Talk puzzler:

The preferred footwear in Townberg is combat boots. Twenty percent of the people have one foot, so they wear one boot. Of the remaining people, half go barefoot and half wear two boots-and there are 20,000 boots worn in Townberg. What's the population of Townberg?

Market Force asked 7,600 adult Americans how likely they would refer the quick-service pizza restaurant they had most recently visited to others. The results

As a statistician, how would you criticize a Yahoo news report that described Blaze pizza as "America's favorite pizza chain"?

A 1994 article in the American Scientist reported the results of a survey of 17 scientific experts about the chances of the earth experiencing catastrophic global warming. Here is a summary of their assessments of the probability of a $3^{\circ } \mathrm{C}$ increase in the global average temperature by 2090. How has this graph been misdrawn?

Suppose that you have a sample of ten observations that has a mean of seven, a median of six, and a standard deviation of three. What are the new values of the following?

Are great discoveries typically made by people who are young and vigorous or old and wise? Below are the ages at which 12 scientists made great discoveries. Use these data to calculate the mean and median ages at which these great discoveries were made.