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Chapter 2: Problem 120

In 2014 , the five-number summary statistics for the distribution of statewide number of people (in thousands) without health insurance had a minimum of 31 (Vermont), \(\mathrm{Q} 1=156,\) median \(=418\) \(\mathrm{Q} 3=837,\) and maximum of 5047 (Texas) (Source: 2015 Current Population Survey Annual Social and Economic Supplement - United States). a. Is the distribution symmetric, skewed right, or skewed left? Why? b. The mean of this data is 719 and the range is 5016 . Which is the most plausible value for the standard deviation: \(-160,0,40,1000,\) or \(5000 ?\) Explain what is unrealistic about the other values.

Short Answer

Expert verified
The distribution is skewed right, and a standard deviation of 1000 is most plausible.

Step by step solution

01

Identify skewness from the five-number summary

To determine if the data is skewed, compare the median with the quartiles. If the distance from the median to Q1 is less than from the median to Q3, the data is skewed right. Here, \(Q1 = 156\), median = 418, \(Q3 = 837\). The distance from the median to Q1 is \(418 - 156 = 262\), and from the median to Q3 is \(837 - 418 = 419\). Since 419 is greater than 262, the distribution is skewed right.
02

Consider possible standard deviation values

Check the range and compare it with each given option for standard deviation. The range is 5016 (maximum - minimum = 5047 - 31). Typically, the standard deviation is smaller than the range. Logical values are positive and should reflect spread around the mean. Given options are: -160, 0, 40, 1000, 5000.
03

Eliminate unrealistic standard deviation values

-160 is negative, which is impossible for standard deviation. A standard deviation of 0 implies no variation, which contradicts the wide range of data. 5000 suggests that observations vary nearly as much as the range, unlikely given that mean is 719. The value 40 is too small compared to the range.
04

Select plausible standard deviation

The standard deviation should reasonably reflect the spread of data around the mean. 1000 is a plausible value since it accounts for a reasonable spread considering the range is 5016. Values like 0, -160, and 5000 are unrealistic based on typical data behavior and range.

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Key Concepts

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

Five-number summary
A five-number summary is a concise way to describe a set of data. It consists of five key statistics: minimum, first quartile \((Q_1)\), median, third quartile \((Q_3)\), and maximum. Each of these provides insight into the data set鈥檚 distribution and range.

  • **Minimum**: This is the smallest observation in the dataset. For instance, in the given data about uninsured individuals, Vermont has the minimum value of 31.
  • **First Quartile \((Q_1)\)**: This indicates that 25% of the data is below this value. In our example, \(Q_1\) is 156.
  • **Median**: This represents the middle point of the dataset, where 50% of the data is below and 50% is above. Here, the median is 418.
  • **Third Quartile \((Q_3)\)**: Signifying that 75% of the data is below this point, \(Q_3\) is 837 in the sample data.
  • **Maximum**: The largest value, in this case, is 5047 (Texas).
Each aspect of the five-number summary reveals different parts of the data鈥檚 story. Together, they provide a picture of how the data is spread and where the center and extremities lie.
Skewness in data distribution
Understanding the skewness of data helps reveal its symmetry. Skewness can be visually spotted through a five-number summary by comparing the distances between the median and the quartiles.

  • A **symmetric distribution** means the data is evenly distributed around the median.
  • **Skewed right** indicates a longer tail on the right side. If \(Q_3 - \text{median} > \text{median} - Q_1\), the data is skewed right.
  • **Skewed left** means there's a longer tail on the left, where \(\text{median} - Q_1 > Q_3 - \text{median}\).
In the provided example, the calculation shows that the distance from the median to \(Q_3\) is 419, while from the median to \(Q_1\) it is 262. Since 419 is greater than 262, the dataset is skewed right. This distribution suggests more lower values and a few extremely high values, pulling the mean to the right.
Standard deviation calculation
Standard deviation is a measure of how spread out the numbers in a data set are around the mean. It shows whether the data points are close to the mean or dispersed over a wide range.

**Key aspects of standard deviation:**
  • **Positive Values Only**: By definition, standard deviation can never be negative. It is always a positive value or zero.
  • **Relation to the Mean**: A smaller standard deviation signifies data points closer to the mean, while a larger one indicates wider dispersal.
  • **Compared to Range**: It should typically be smaller than the entire range of the dataset, which in this example is 5016.
Given potential values \(-160, 0, 40, 1000, 5000\), both -160 and 0 are dismissed immediately; negative or zero deviation doesn鈥檛 fit with a varied dataset. Comparing to the range, 5000 is unlikely as the data does not vary nearly as much. 40 is too tiny to reflect data's wide spread. **1000 emerges as a reasonable choice**, adequately capturing the data's variation around the mean of 719, taking into account the range and naturally occurring deviations.

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Most popular questions from this chapter

For an exam given to a class, the students' scores ranged from 35 to \(98,\) with a mean of 74. Which of the following is the most realistic value for the standard deviation: -10,0,3,12,63 ? Clearly explain what's unrealistic about each of the other values.

The workers and the management of a company are having a labor dispute. Explain why the workers might use the median income of all the employees to justify a raise but management might use the mean income to argue that a raise is not needed.

A study reported that in 2007 the mean and median net worth of American families were \(\$ 556,300\) and \(\$ 120,300,\) respectively. a. Is the distribution of net worth for these families likely to be symmetric, skewed to the right, or skewed to the left? Explain. b. During the Great Recession of 2008 , many Americans lost wealth due to the large decline in values of assets such as homes and retirement savings. In \(2009,\) mean and median net worth were reported as \(\$ 434,782\) and \(\$ 91,304\). Why do you think the difference in decline from 2007 to 2009 was larger for the mean than the median?

True or false: a. The mean, median, and mode can never all be the same. b. The mean is always one of the data points. c. When \(n\) is odd, the median is one of the data points. d. The median is the same as the second quartile and the 50 th percentile.

France is most popular holiday spot Which countries are most frequently visited by tourists from other countries? The table shows results according to Travel and Leisure magazine ( 2005\()\). a. Is country visited a categorical or a quantitative variable? b. In creating a bar graph of these data, would it be most sensible to list the countries alphabetically or in the form of a Pareto chart? Explain. c. Does either a dot plot or stem-and-leaf plot make sense for these data? Explain. $$ \begin{array}{lc} \hline {\text { Most Visited Countries, } 2005} \\ \hline \text { Country } & \text { Number of Visits (millions) } \\ \hline \text { France } & 77.0 \\ \text { China } & 53.4 \\ \text { Spain } & 51.8 \\ \text { United States } & 41.9 \\ \text { Italy } & 39.8 \\ \text { United Kingdom } & 24.2 \\ \text { Canada } & 20.1 \\ \text { Mexico } & 19.7 \\ \hline \end{array} $$
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