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Chapter 1: Problem 29

Identify the level of measurement of the data as nominal, ordinal, interval, or ratio. Also, explain what is wrong with the given calculation. The first Super Bowl attended by the author was Super Bowl XLVIII. On the first play of the game, the Seatile defense scored on a safety. The defensive players wore jerseys numbered \(31,28,41,56,25,54,69,50,91,72,29,\) and the average (mean) of those numbers is 49.6.

Short Answer

Expert verified
The data is nominal. Calculating the mean for nominal data is inappropriate because it lacks quantitative value.

Step by step solution

01

Identify the Level of Measurement

The numbers on the jerseys are identifiers for the players and do not represent any measurable quantity. Therefore, these numbers are nominal. Nominal data is used for labeling variables without any quantitative value.
02

Explain the Purpose of Jersey Numbers

Jersey numbers are used to identify players on the field. They do not have a meaningful order or ranking, nor do they represent an inherent numerical value that can be used for calculation.
03

Review the Calculation

The mean (average) of the jersey numbers is given as 49.6. However, calculations like the mean are not valid for nominal data because nominal measurements do not represent magnitude or order. Calculating an average implies a level of measurement that carries quantitative meaning, which nominal data does not.
04

Conclusion

The level of measurement for jersey numbers is nominal. The calculation of the mean jersey number is not meaningful because it applies quantitative analysis to qualitative data.

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Key Concepts

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

Nominal Data
Nominal data refers to variables that are used to label or categorize without implying any quantitative value. Think of it as naming things. Examples of nominal data include:
  • Colors - like red, blue, green
  • Names - like John, Maria, Alex
  • Jersey numbers - like 31, 28, 41
In the given exercise, the jersey numbers of the players are nominal. These numbers help identify the players, but they do not indicate order, rank, or any measurable quantity.

This makes it clear why calculating the mean of jersey numbers is incorrect. Average calculations are meaningful only when data carries quantitative value.
Ordinal Data
Ordinal data represents categories with a meaningful order or ranking but does not quantify the difference between these rankings. Examples include:
  • Survey answers: 'very satisfied,' 'satisfied,' 'neutral,' 'unsatisfied,' 'very unsatisfied'
  • Education levels: high school, bachelor's, master's, PhD
  • Race positions: 1st, 2nd, 3rd
For ordinal data, the sequence matters, but the actual distance between ranks doesn't.

Unlike nominal data, with ordinal data, you can say which rank is higher or lower, but you cannot compute an average meaningfully.
Interval Data
Interval data has meaningful differences between measurements but no true zero point. This means you can perform arithmetic operations, but ratios are not meaningful. Examples include:
  • Temperatures in Celsius or Fahrenheit
  • Years on a calendar
  • IQ scores
For example, the difference between 20掳C and 30掳C is the same as between 30掳C and 40掳C, making the computations of intervals valid.

However, saying that 40掳C is 'twice as hot' as 20掳C is incorrect because an interval scale does not have a true zero point.
Ratio Data
Ratio data is the most informative type of measurement, including a true zero point, allowing for the calculation of meaningful ratios. Examples include:
  • Height in centimeters
  • Weight in kilograms
  • Time in seconds
With ratio data, you can accurately say that 4 meters is twice as long as 2 meters or that someone weighing 50 kg is half as heavy as someone at 100 kg.

This type of data supports all arithmetical operations, including mean, median, and mode, as well as the computation of ratios and percentages, making it the most versatile level of measurement.

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Most popular questions from this chapter

Identify which of these designs is most appropriate for the given experiment: completely randomized design, randomized block design, or matched pairs design. The HIV Trials Network is conducting a study to test the effectiveness of two different experimental HIV vaccines. Subjects will consist of 80 pairs of twins. For each pair of twins, one of the subjects will be treated with the DNA vaccine and the other twin will be treated with the adenoviral vector vaccine.

Identify which of these types of sampling is used: random, systematic, convenience, stratified, or cluster. The author surveyed a sample from the population of his statistics class by identifying groups of males and females, then randomly selecting five students from each of those two groups.

In a study designed to test the effectiveness of paracetamol (or acetaminophen) as a treatment for lower back pain, 1643 patients were randomly assigned to one of three groups: (1) the 547 subjects in the placebo group were given pills containing no medication; (2) 550 subjects were in a group given pills with paracetamol taken at regular intervals; (3) 546 subjects were in a group given pills with paracetamol to be taken when needed for pain relief. (See "Efficacy of Paracetamol for Acute Low-Back Pain," by Williams, et al., Lancet, doi:10.1016/S0140-6736(14)60805-9.) Is this study an experiment or an observational study? Explain.

Determine which of the four levels of measurement (nominal, ordinal, interval, ratio is most appropriate. In a study of fast food service times, a researcher records the time intervals of drive-up customers beginning when they place their order and ending when they receive their order.

Identify which of these types of sampling is used: random, systematic, convenience, stratified, or cluster. In a study of treatments for back pain, 641 subjects were randomly assigned to the four different treatment groups of individualized acupuncture, standardized acupuncture, simulated acupuncture, and usual care (based on data from "A Randomized Trial Comparing Acupuncture, Simulated Acupuncture, and Usual Care for Chronic Low Back Pain," by Cherkin et al., Archives of Internal Medicine, Vol. 169, No. 9).
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