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Chapter 2: Problem 134

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
The mode applies because it identifies the most common category, but mean and median require numerical or ordered data, which nominal scales lack.

Step by step solution

01

Understanding Nominal Scale

A nominal scale is used for labeling variables without any quantitative value or order. It categorizes data into distinct classifications without any inherent ranking (e.g., religion, nationality).
02

Mean and its Applicability

Mean is calculated by adding all values and dividing by the number of values. Since nominal variables are qualitative without numerical values or an inherent order, averaging them is meaningless.
03

Median and Ordered Data

The median is the middle value in a data set, which requires an ordering of data. In a nominal scale, there is no logical order, making it impossible to find a median.
04

Mode in Nominal Data

The mode is the most frequently occurring category in a data set. Nominal variables can have a mode, as it simply identifies which category appears most often, requiring no numerical value or order.

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Key Concepts

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

Understanding Categorical Variables
Categorical variables categorize data into distinct groups based on characteristics or qualities, rather than numerical values. They help us label different data points, such as colors, names, or types.
Categorical variables come in two main types:
  • Nominal: These are purely for categorization without any intrinsic order. Examples include gender, language, or brand names.
  • Ordinal: While still categorizing data, these have an order or ranking, like grades, satisfaction levels, or class positions. However, the intervals between the categories are not meaningful.
Nominal variables, as used in the exercise, are perfect for assigning labels where the order does not matter. This makes them very useful for various analytical purposes where qualitative differentiation is key.
Mode and Its Role in Analysis
Mode is a simple yet effective measure of central tendency. It refers to the category or value that appears most frequently in a data set. For categorical variables, particularly nominal ones, the mode is exceptionally useful.
Here's why mode stands out:
  • You don't need to sort the data or deal with numerical calculations, making it straightforward for nominal variables.
  • The mode gives immediate insight into the most popular or common category, like the most preferred color or the most common brand.
Not every data set will have a mode if all categories occur only once, but when they do, it provides a quick snapshot of the data's central tendency.
Mean and Why It Doesn't Apply to Nominal Data
The mean, or average, typically involves numerical data where you sum the values and divide by the count of the values. This measure requires numerical input, which makes it inapplicable for nominal data.
Consider these reasons the mean isn't suitable:
  • Nominal variables are qualitative, without any numeric value or intrinsic order. Summing categories like colors or brands doesn鈥檛 make sense.
  • Even if you assigned arbitrary numbers to categories for mean calculation, it would lead to unreliable and misleading interpretations.
Therefore, mean as a statistical measure is not feasible for analyzing nominal scale data.
Median and Its Limitations with Nominal Data
The median is the middle point separating the higher half from the lower half of an ordered data set. Therefore, median calculations fundamentally require some form of order or ranking.
Why median fails to work for nominal variables:
  • Nominal categories, such as types of cuisine or different species, do not follow a logical sequence or hierarchy.
  • Without a natural order, it's impossible to determine which category lies in the middle.
Consequently, unlike ordered categorical variables, such as those on an ordinal scale, a median is not applicable for insights from nominal data.

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Most popular questions from this chapter

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 \(\begin{array}{llllllllll}0 & 0 & 4 & 0 & 0 & 0 & 10 & 0 & 6 & 0\end{array}\) 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.

During the 2010 Professional Golfers Association (PGA) season, 90 golfers earned at least $$\$ 1$$ million in tournament prize money. Of those, 5 earned at least $$\$ 4$$ million, 11 earned between $$\$ 3$$ million and $$\$ 4$$ million, 21 earned between $$\$ 2$$ million and $$\$ 3$$ million, and 53 earned between $$\$ 1$$ million and $$\$ 2$$ million. a. Would the data for all 90 golfers be symmetric, skewed to the left, or skewed to the right? b. Two measures of central tendency of the golfers' winnings were $$\$ 2,090,012$$ and $$\$ 1,646,853 .$$ Which do you think is the mean and which is the median?

In a study of graduate students who took the Graduate Record Exam (GRE), the Educational Testing Service reported that for the quantitative exam, U.S. citizens had a mean of 529 and standard deviation of \(127,\) whereas the non-U.S. citizens had a mean of 649 and standard deviation of \(129 .\) Which of the following is true? a. Both groups had about the same amount of variability in their scores, but non-U.S. citizens performed better, on the average, than U.S. citizens. b. If the distribution of scores was approximately bell shaped, then almost no U.S. citizens scored below 400 . c. If the scores range between 200 and \(800,\) then probably the scores for non-U.S. citizens were symmetric and bell shaped. d. A non-U.S. citizen who scored 3 standard deviations below the mean had a score of 200 .

European fertility The European fertility rates (mean number of children per adult woman) from Exercise 2.18 are shown again in the table. a. Find the median of the fertility rates. Interpret. b. Find the mean of the fertility rates. Interpret. c. For each woman, the number of children is a whole number, such as 2 or 3 . Explain why it makes sense to measure a mean number of children per adult woman (which is not a whole number) to compare fertility levels, such as the fertility levels of 1.5 in Canada and 2.4 in Mexico. $$\begin{array}{lclc}\hline \text { Country } & \text { Fertility } & \text { Country } & \text { Fertility } \\ \hline \text { Austria } & 1.4 & \text { Netherlands } & 1.7 \\ \text { Belgium } & 1.7 & \text { Norway } & 1.8 \\ \text { Denmark } & 1.8 & \text { Spain } & 1.3 \\ \text { Finland } & 1.7 & \text { Sweden } & 1.6 \\ \text { France } & 1.9 & \text { Switzerland } & 1.4 \\ \text { Germany } & 1.3 & \text { United Kingdom } & 1.7 \\ \text { Greece } & 1.3 & \text { United States } & 2.0 \\ \text { Ireland } & 1.9 & \text { Canada } & 1.5 \\ \text { Italy } & 1.3 & \text { Mexico } & 2.4 \\ \hline\end{array}$$

In parts a and b, what shape do you expect for the distributions of electricity use and water use in a recent month in Gainesville, Florida? Why? (Data supplied by N. T. Kamhoot, Gainesville Regional Utilities.) a. Residential electricity used had mean \(=780\) and standard deviation \(=506\) kilowatt hours \((\mathrm{Kwh})\). The minimum usage was \(3 \mathrm{Kwh}\) and the maximum was \(9390 \mathrm{Kwh}\) b. Water consumption had mean \(=7100\) and standard deviation \(=6200\) (gallons).
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