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Chapter 2: Problem 134

Mode but not median and mean The previous exercise showed how to find the mean and median when a categorical variable has ordered categories. A categorical scale that does not have ordered categories (such as choice of religious affiliation or choice of major in college) is called a nominal scale. For such a variable, the mode (or modal category) applies, but not the mean or median. Explain why.

Short Answer

Expert verified
Mode applies to nominal scales, allowing us to find the most frequent category, while mean and median are not applicable due to the lack of numerical order and ranking.

Step by step solution

01

Understand Nominal Scale

A nominal scale is a type of categorical scale where the data is categorized based on names or labels without any specific order. Examples are religious affiliation or college majors where categories do not have a ranking or numerical significance.
02

Mode on Nominal Scale

The mode refers to the category that appears most frequently in a data set. For nominal data, it is possible to identify which category has the highest frequency, making mode applicable to nominal scales.
03

Limitations of Mean on Nominal Scale

The mean requires numerical values to compute an average. Since nominal scales consist of categorical labels without inherent numerical values, calculating an average is not meaningful.
04

Limitations of Median on Nominal Scale

The median is the middle value of an ordered data set. Since nominal data are not ordered, determining the middle position or central tendency through a median is not applicable.

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Key Concepts

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

Mode for Nominal Scale
A nominal scale is a type of data classification where items are grouped into categories based on qualitative traits like names or labels. These categories lack a natural order or ranking among them. An excellent example could be types of cuisine or smartphone brands. Unlike numerical scales, a nominal scale doesn't involve any inherent measure of magnitude or value.

When working with nominal data, the mode is a suitable statistical measure. The mode is essentially the most common or frequently occurring category within the data set. Let's say you surveyed 100 people on their preferred smartphone brand; finding out which brand gets the most votes is simply determining the mode. In this context, mode works well because it identifies the most popular category without needing the data to be ordered or numerically analyzed, which makes it perfect for nominal scales.
Limitations of Mean
The mean is a measure of central tendency that can effectively summarize numerical data sets by calculating the average. However, for nominal scales, this approach hits a major roadblock. Since nominal data consist of names or categories, there are no numerical values to add or average.

  • Without inherent numerical values, you can't compute a meaningful mean.
  • It doesn't make sense to "average" categories like genres of music or types of animals.
Therefore, trying to apply the mean to nominal data would result not only in misinterpretation but also in a mathematical impracticality. It's crucial to remember that each type of data requires appropriate measures of analysis.
Limitations of Median
The median represents the middle value in an ordered data set. It's especially useful when dealing with skewed numeric data as it provides a central point. However, with nominal scales, this definition loses its potential.

Nominal data do not have a logical order, making the concept of a midpoint nonsensical. For example, with categories like car brands or languages spoken, there is no way to "order" these values meaningfully. Simply put:

  • No logical ranking or order exists for categories in nominal data.
  • Determining a middle value is undefined because no sequence is present.
Without an inherent ordering of the data, the median cannot find its place in nominal data analysis. Therefore, the median is inapplicable for nominal scales.

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Most popular questions from this chapter

Knowing homicide victims The table summarizes responses of 4383 subjects in a recent General Social Survey to the question, "Within the past month, how many people have you known personally that were victims of homicide?" a. To find the mean, it is necessary to give a score to the " 4 or more" category. Find it, using the score 4.5 . (In practice, you might try a few different scores, such as \(4,4.5,5,6,\) to make sure the resulting mean is not highly sensitive to that choice.) b. Find the median. Note that the "4 or more" category is not problematic for it. c. If 1744 observations shift from 0 to 4 or more, how do the mean and median change? d. Why is the median the same for parts \(\mathrm{b}\) and \(\mathrm{c},\) even though the data are so different?

Federal spending on financial aid A 2011 Roper Center survey asked: "If you were making up the budget for the federal government this year (2011), would you increase spending, decrease spending, or keep spending the same for financial aid for college students?" Of those surveyed, \(44 \%\) said to increase spending, \(16 \%\) said to decrease spending, \(37 \%\) said to keep spending the same, and \(3 \%\) either had no opinion or refused to answer. a. Sketch a bar chart to display the survey results. b. Which is easier to sketch relatively accurately, a pie chart or a bar chart? c. What is the advantage of using a graph to summarize the results instead of merely stating the percentages for each response?

Super Bowl tickets \(\quad\) StubHub is a popular Web site where fans can buy and sell tickets to concerts and sporting events. Below are data representing the amounts (in dollars) that buyers using StubHub spent on Super Bowl XLV tickets. $$ \begin{array}{llllllll} 2275 & 3050 & 2800 & 4200 & 7500 & 3500 & 2400 & 2575 \\ 2890 & 2395 & 5000 & 3300 & 2475 & 2195 & 2999 & 3650 \end{array} $$ a. Construct a stem-and-leaf plot. Truncate the data to the first two digits for purposes of constructing the plot. For example, 2275 becomes 22 . b. Summarize what this plot tells you about the data. c. Reconstruct the stem-and-leaf plot in part a using split stems; that is, two stems of \(2,\) two stems of \(3,\) etc. Compare the two stem-and-leaf plots. Explain how one may be more informative than the other.

"On the average day, about how many hours do you personally watch television?" Of 1,324 responses, the mode was 2 , the median was 2 , the mean was \(2.98,\) and the standard deviation was 2.66. Based on these statistics, what would you surmise about the shape of the distribution? Why? (Source: Data from CSM, UC Berkeley.)

True or false: Soccer According to a story in the Guardian newspaper (football.guardian.co.uk), in the United Kingdom the mean wage for a Premiership player in 2006 was \(拢 676,000\). True or false: If the income distribution is skewed to the right, then the median salary was even larger than \(拢 676,000\).
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