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Chapter 2: Problem 40

Net worth by degree The Statistical Abstract of the United States reported that in 2004 for those with a college education, the median net worth was \(\$ 226,100\) and the mean net worth was \(\$ 851,300 .\) For those with a high school diploma only, the values were \(\$ 68,700\) and \(\$ 196,800\). a. Explain how the mean and median could be so different for each group. b. Which measure do you think gives a more realistic measure of a typical net worth, the mean or the median. Why?

Short Answer

Expert verified
The mean and median differ due to skewed distributions with outliers. The median gives a more realistic measure of typical net worth in this context.

Step by step solution

01

Understanding Mean and Median

The **mean** of a set of numbers is the average, calculated by adding up all the numbers and dividing by the count of numbers. The **median** is the middle value when the numbers are sorted in order. If the number of values is even, the median is the average of the two middle numbers.
02

Analyzing Differences in Statistics for College-Educated Group

For those with a college education, the mean net worth is much higher (\(\\(851,300\)) than the median net worth (\(\\)226,100\)). This large disparity suggests the presence of a few individuals with extremely high net worths, which are skewing the mean upwards. These high values can inflate the mean significantly while not affecting the median as much because it is resistant to extreme values.
03

Analyzing Differences in Statistics for High-School Group

For individuals with just a high school diploma, the mean net worth is \(\\(196,800\), and the median is \(\\)68,700\). A similar pattern to the college-educated group is present but on a smaller scale, indicating some people have higher net worths but not as dramatically skewed as the previous group. The difference suggests a right-skewed distribution though less pronounced.
04

Comparing Mean and Median as Measures of Central Tendency

The median is generally considered a more realistic measure of central tendency when data is skewed, as it better represents the typical individual within a population by not being affected by extremely high or low values. The mean, conversely, can be heavily influenced by outliers, giving a less accurate picture of an average individual's net worth in skewed distributions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean vs. Median
The difference between mean and median is crucial when interpreting data, especially with varying individual incomes or net worths.
Understanding these two statistical measures can help determine which better represents a typical value in a data set.
  • The mean is the average and is calculated by adding all numbers in the data set and dividing by the number of values. It is useful in providing an overall picture of the data, but it is sensitive to outliers or extremely high or low values.
  • The median is the middle value when the numbers are ordered from least to greatest. It effectively splits the data into two halves. Unlike the mean, the median is not affected by outliers. It provides a more accurate representation of the central point in skewed datasets.
In summary, while the mean gives an overview by totaling values and dividing, the median offers a balanced look at what could be considered 'typical' when data is spread unevenly.
Central Tendency
Central tendency is a statistical measure that identifies the center of a data set or what is typical for the data. It includes both the mean and the median as discussed earlier.
These measures help to give a snapshot of the data in one value.
  • Mean: Best for normal distributed data where values are symmetrically distributed around the center.
  • Median: Best for skewed distributions where some values drastically differ from the rest.
Central tendency gives insight into the overall trends of a dataset and is essential when comparing different groups, such as those with college education versus high-school graduates, as it allows us to note how typical values differ between groups.
Skewed Distribution
Skewed distribution occurs when data is not symmetrically distributed. An uneven spread can significantly affect measures of central tendency such as the mean, but not the median.
  • Right-skewed distribution: Where a few high values pull the mean to the right of the median. This situation was evident in the college-educated group's net worths, where extremely high values led to a mean much greater than the median.

  • Left-skewed distribution: Not as common in personal net worth, but it's where low values pull the mean to the left.
Recognizing a skewed distribution helps in identifying which measure of central tendency to use. In skewed scenarios, the median is usually more reliable for a typical value as it remains unaffected by extremes unlike the mean.

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Most popular questions from this chapter

Public transportation-center The owner of a company in downtown Atlanta is concerned about the large use of gasoline by her employees due to urban sprawl, traffic congestion, and the use of energy inefficient vehicles such as SUVs. She'd like to promote the use of public transportation. She decides to investigate how many miles her employees travel on public transportation during a typical day. The values for her 10 employees (recorded to the closest mile) are \(\begin{array}{llll}0 & 0 & 4 & 0\end{array}\) \(\begin{array}{llll}0 & 0 & 10 & 0\end{array}\) 60 a. Find and interpret the mean, median, and mode. b. She has just hired an additional employee. He lives in a different city and travels 90 miles a day on public transport. Recompute the mean and median. Describe the effect of this outlier.

Sick leave Exercise 2.47 showed data for a company that investigated the annual number of days of sick leave taken by its employees. The data are \(\begin{array}{llllllll}0 & 0 & 4 & 0 & 0 & 0 & 6 & 0\end{array}\) a. The standard deviation is \(2.4 .\) Find and interpret the range. b. The quartiles are \(\mathrm{Q} 1=0,\) median \(=0, \mathrm{Q} 3=2 .\) Find the interquartile range. c. Suppose the 6 was incorrectly recorded and is supposed to be \(60 .\) The standard deviation is then 21.1 but the quartiles do not change. Redo parts \(\mathrm{a}-\mathrm{c}\) with the correct data and describe the effect of this outlier. Which measure of variability, the range, IQR, or standard deviation, is least affected by the outlier? Why?

Vacation days \(\quad\) National Geographic Traveler magazine recently presented data on the annual number of vacation days averaged by residents of eight different countries. They reported 42 days for Italy, 37 for France, 35 for Germany, 34 for Brazil, 28 for Britain, 26 for Canada, 25 for Japan, and 13 for the United States. a. Report the median. b. By finding the median of the four values below the median, report the first quartile. c. Find the third quartile. d. Interpret the values found in parts \(a-c\) in the context of these data.

Shape of the histogram For each of the following variables, indicate whether you would expect its histogram to be symmetric, skewed to the right, or skewed to the left. Explain why. a. Assessed value of houses in a large city (Hint: Would the relatively few homes with extremely high assessed value result in a long right tail or a long left tail?) b. Number of times checking account overdrawn in the past year for the faculty in your school c. IQ for the general population d. The height of female college students

Mean versus median and income A U.S. Federal Reserve study calculated the mean and median incomes for 2007 for each of the different income groups represented in the table. $$ \begin{array}{lrr} \hline \text { Income percentile } & \text { Mean } & \text { Median } \\ \hline \text { Below 20\% } & \$ 10.520 & \$ 8,100 \\ 20 \text { to } 39.9 \% & \$ 134,900 & \$ 37,900 \\ 40 \text { to } 59.9 \% & \$ 209,900 & \$ 88,100 \\ 60 \text { to } 79.9 \% & \$ 375,100 & \$ 204,900 \\ 80 \text { to } 89.9 \% & \$ 606,300 & \$ 356,200 \\ 90 \text { to } 100 \% & \$ 3,306,000 & \$ 1,119,000 \\ \hline \end{array} $$ Why does the disparity between mean and median income get larger as income gets larger?
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