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Introductory Business Statistics 2e

1.3 Levels of Measurement

Introductory Business Statistics 2e1.3 Levels of Measurement

Table of contents
  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. Key Terms
    7. Chapter Review
    8. Homework
    9. References
    10. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Display Data
    3. 2.2 Measures of the Location of the Data
    4. 2.3 Measures of the Center of the Data
    5. 2.4 Sigma Notation and Calculating the Arithmetic Mean
    6. 2.5 Geometric Mean
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. Key Terms
    10. Chapter Review
    11. Formula Review
    12. Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables and Probability Trees
    6. 3.5 Venn Diagrams
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Bringing It Together: Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Hypergeometric Distribution
    3. 4.2 Binomial Distribution
    4. 4.3 Geometric Distribution
    5. 4.4 Poisson Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Properties of Continuous Probability Density Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution
    5. Key Terms
    6. Chapter Review
    7. Formula Review
    8. Practice
    9. Homework
    10. References
    11. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Estimating the Binomial with the Normal Distribution
    5. Key Terms
    6. Chapter Review
    7. Formula Review
    8. Practice
    9. Homework
    10. References
    11. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means
    3. 7.2 Using the Central Limit Theorem
    4. 7.3 The Central Limit Theorem for Proportions
    5. 7.4 Finite Population Correction Factor
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Confidence Interval When the Population Standard Deviation Is Known or Large Sample Size
    3. 8.2 A Confidence Interval When the Population Standard Deviation Is Unknown and Small Sample Case
    4. 8.3 A Confidence Interval for A Population Proportion
    5. 8.4 Calculating the Sample Size n: Continuous and Binary Random Variables
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Probability Distribution Needed for Hypothesis Testing
    5. 9.4 Full Hypothesis Test Examples
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Comparing Two Independent Population Means
    3. 10.2 Cohen's Standards for Small, Medium, and Large Effect Sizes
    4. 10.3 Test for Differences in Means: Assuming Equal Population Variances
    5. 10.4 Comparing Two Independent Population Proportions
    6. 10.5 Two Population Means with Known Standard Deviations
    7. 10.6 Matched or Paired Samples
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Test of a Single Variance
    4. 11.3 Goodness-of-Fit Test
    5. 11.4 Test of Independence
    6. 11.5 Test for Homogeneity
    7. 11.6 Comparison of the Chi-Square Tests
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  13. 12 F Distribution and One-Way ANOVA
    1. Introduction
    2. 12.1 Test of Two Variances
    3. 12.2 One-Way ANOVA
    4. 12.3 The F Distribution and the F-Ratio
    5. 12.4 Facts About the F Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  14. 13 Linear Regression and Correlation
    1. Introduction
    2. 13.1 The Correlation Coefficient r
    3. 13.2 Testing the Significance of the Correlation Coefficient
    4. 13.3 Linear Equations
    5. 13.4 The Regression Equation
    6. 13.5 Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation
    7. 13.6 Predicting with a Regression Equation
    8. 13.7 How to Use Microsoft Excel® for Regression Analysis
    9. Key Terms
    10. Chapter Review
    11. Practice
    12. Solutions
  15. A | Statistical Tables
  16. B | Mathematical Phrases, Symbols, and Formulas
  17. Index

Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible.

Levels of Measurement

The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are (from lowest to highest level):

  • Nominal scale level
  • Ordinal scale level
  • Interval scale level
  • Ratio scale level

Data that is measured using a nominal scale is qualitative (categorical). Categories, colors, names, labels and favorite foods along with yes or no responses are examples of nominal level data. Nominal scale data are not ordered. For example, trying to classify people according to their favorite food does not make any sense. Putting pizza first and sushi second is not meaningful.

Smartphone companies are another example of nominal scale data. The data are the names of the companies that make smartphones, but there is no agreed upon order of these brands, even though people may have personal preferences. Nominal scale data cannot be used in calculations.

Data that is measured using an ordinal scale is similar to nominal scale data but there is a big difference. The ordinal scale data can be ordered. An example of ordinal scale data is a list of the top five national parks in the United States. The top five national parks in the United States can be ranked from one to five but we cannot measure differences between the data.

Another example of using the ordinal scale is a cruise survey where the responses to questions about the cruise are “excellent,” “good,” “satisfactory,” and “unsatisfactory.” These responses are ordered from the most desired response to the least desired. But the differences between two pieces of data cannot be measured. Like the nominal scale data, ordinal scale data cannot be used in calculations.

Data that is measured using the interval scale is similar to ordinal level data because it has a definite ordering but there is a difference between data. The differences between interval scale data can be measured though the data does not have a starting point.

Temperature scales like Celsius (C) and Fahrenheit (F) are measured by using the interval scale. In both temperature measurements, 40° is equal to 100° minus 60°. Differences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like -10° F and -15° C exist and are colder than 0.

Interval level data can be used in calculations, but one type of comparison cannot be done. 80° C is not four times as hot as 20° C (nor is 80° F four times as hot as 20° F). There is no meaning to the ratio of 80 to 20 (or four to one).

Data that is measured using the ratio scale takes care of the ratio problem and gives you the most information. Ratio scale data is like interval scale data, but it has a 0 point and ratios can be calculated. For example, four multiple choice statistics final exam scores are 80, 68, 20 and 92 (out of a possible 100 points). The exams are machine-graded.

The data can be put in order from lowest to highest: 20, 68, 80, 92.

The differences between the data have meaning. The score 92 is more than the score 68 by 24 points. Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20.

Frequency

Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3.

Table 1.7 lists the different data values in ascending order and their frequencies.

Data value Frequency
2 3
3 5
4 3
5 6
6 2
7 1
Table 1.7 Frequency Table of Student Work Hours

A frequency is the number of times a value of the data occurs. According to Table 1.7, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.

A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.

Data value Frequency Relative frequency
2 3 3 20 3 20 or 0.15
3 5 5 20 5 20 or 0.25
4 3 3 20 3 20 or 0.15
5 6 6 20 6 20 or 0.30
6 2 2 20 2 20 or 0.10
7 1 1 20 1 20 or 0.05
Table 1.8 Frequency Table of Student Work Hours with Relative Frequencies

The sum of the values in the relative frequency column of Table 1.8 is 20 20 20 20 , or 1.

Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table 1.9.

Data value Frequency Relative frequency Cumulative relative frequency
2 3 3 20 3 20 or 0.15 0.15
3 5 5 20 5 20 or 0.25 0.15 + 0.25 = 0.40
4 3 3 20 3 20 or 0.15 0.40 + 0.15 = 0.55
5 6 6 20 6 20 or 0.30 0.55 + 0.30 = 0.85
6 2 2 20 2 20 or 0.10 0.85 + 0.10 = 0.95
7 1 1 20 1 20 or 0.05 0.95 + 0.05 = 1.00
Table 1.9 Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.

NOTE

Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.

Table 1.10 represents the heights, in inches, of a sample of 100 semiprofessional soccer players.

Heights (inches) Frequency Relative frequency Cumulative relative frequency
59.95–61.95 5 5 100 5 100 = 0.05 0.05
61.95–63.95 3 3 100 3 100 = 0.03 0.05 + 0.03 = 0.08
63.95–65.95 15 15 100 15 100 = 0.15 0.08 + 0.15 = 0.23
65.95–67.95 40 40 100 40 100 = 0.40 0.23 + 0.40 = 0.63
67.95–69.95 17 17 100 17 100 = 0.17 0.63 + 0.17 = 0.80
69.95–71.95 12 12 100 12 100 = 0.12 0.80 + 0.12 = 0.92
71.95–73.95 7 7 100 7 100 = 0.07 0.92 + 0.07 = 0.99
73.95–75.95 1 1 100 1 100 = 0.01 0.99 + 0.01 = 1.00
Total = 100 Total = 1.00
Table 1.10 Frequency Table of Soccer Player Height

The data in this table have been grouped into the following intervals:

  • 59.95 to 61.95 inches
  • 61.95 to 63.95 inches
  • 63.95 to 65.95 inches
  • 65.95 to 67.95 inches
  • 67.95 to 69.95 inches
  • 69.95 to 71.95 inches
  • 71.95 to 73.95 inches
  • 73.95 to 75.95 inches

In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints.

Example 1.14

Problem

From Table 1.10, find the percentage of heights that are less than 65.95 inches.

Try It 1.14

Table 1.11 shows the amount, in inches, of annual rainfall in a sample of towns.

Rainfall (inches) Frequency Relative frequency Cumulative relative frequency
2.95–4.976 6 50 6 50 = 0.12 0.12
4.97–6.997 7 50 7 50 = 0.14 0.12 + 0.14 = 0.26
6.99–9.0115 15 50 15 50 = 0.30 0.26 + 0.30 = 0.56
9.01–11.038 8 50 8 50 = 0.16 0.56 + 0.16 = 0.72
11.03–13.059 9 50 9 50 = 0.18 0.72 + 0.18 = 0.90
13.05–15.075 5 50 5 50 = 0.100.90 + 0.10 = 1.00
Total = 50Total = 1.00
Table 1.11

From Table 1.11, find the percentage of rainfall that is less than 9.01 inches.

Example 1.15

Problem

From Table 1.10, find the percentage of heights that fall between 61.95 and 65.95 inches.

Try It 1.15

From Table 1.11, find the percentage of rainfall that is between 6.99 and 13.05 inches.

Example 1.16

Problem

Use the heights of the 100 semiprofessional soccer players in Table 1.10. Fill in the blanks and check your answers.

  1. The percentage of heights that are from 67.95 to 71.95 inches is: ____.
  2. The percentage of heights that are from 67.95 to 73.95 inches is: ____.
  3. The percentage of heights that are more than 65.95 inches is: ____.
  4. The number of players in the sample who are between 61.95 and 71.95 inches tall is: ____.
  5. What kind of data are the heights?
  6. Describe how you could gather this data (the heights) so that the data are characteristic of all semiprofessional soccer players.

Remember, you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.

Try It 1.16

From Table 1.11, find the number of towns that have rainfall between 2.95 and 9.01 inches.

Collaborative Exercise

In your class, have someone conduct a survey of the number of siblings each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions:

  1. What percentage of the students in your class have no siblings?
  2. What percentage of the students have from one to three siblings?
  3. What percentage of the students have fewer than three siblings?

Example 1.17

Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows: 2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10. Table 1.12 was produced:

Data Frequency Relative frequency Cumulative relative frequency
3 3 3 19 3 19 0.1579
4 1 1 19 1 19 0.2105
5 3 3 19 3 19 0.1579
7 2 2 19 2 19 0.2632
10 3 4 19 4 19 0.4737
12 2 2 19 2 19 0.7895
13 1 1 19 1 19 0.8421
15 1 1 19 1 19 0.8948
18 1 1 19 1 19 0.9474
20 1 1 19 1 19 1.0000
Table 1.12 Frequency of Commuting Distances

Problem

  1. Is the table correct? If it is not correct, what is wrong?
  2. True or False: Three percent of the people surveyed commute three miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections.
  3. What fraction of the people surveyed commute five or seven miles?
  4. What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)?

Try It 1.17

Table 1.11 represents the amount, in inches, of annual rainfall in a sample of towns. What fraction of towns surveyed get between 11.03 and 13.05 inches of rainfall each year?

Example 1.18

Table 1.13 contains data for the number of years of service for 70 federal employees.

Number of Years of Service Number of Federal Employees
24 2
25 1
26 3
27 0
28 4
29 6
30 11
31 12
32 7
33 8
34 6
35 10
Table 1.13

Problem

Answer the following questions.

  1. What is the cumulative frequency for years of service between 30 and 35 (inclusive)?
  2. What is the relative frequency for 30 years of service?
  3. What is the relative frequency for 30 years of service or less?
  4. What is the relative frequency for 25 years of service or more?

Try It 1.18

Table 1.14 contains the total number of fatal motor vehicle traffic crashes in the United States for a period of 18 years.

Year Total number of crashes Year Total number of crashes
Year 136,254 Year 1138,444
Year 2 37,241 Year 12 39,252
Year 3 37,494 Year 13 38,648
Year 4 37,324 Year 14 37,435
Year 5 37,107 Year 15 34,172
Year 6 37,140 Year 16 30,862
Year 7 37,526 Year 17 30,296
Year 8 37,862 Year 18 29,757
Year 9 38,491 Total 653,782
Year 10 38,477
Table 1.14

Answer the following questions.

  1. What is the frequency of deaths measured from Year 7 through Year 11?
  2. What percentage of deaths occurred after Year 13?
  3. What is the relative frequency of deaths that occurred in Year 7 or before?
  4. What is the percentage of deaths that occurred in Year 18?
  5. What is the cumulative relative frequency for Year 13? Explain what this number tells you about the data.
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