Section 1.2 Data Classification
Objectives Classify data as qualitative or quantitative; as discrete, continuous, or neither; and by the level of measurement.
Example 1.4: Classifying Data as Qualitative or Quantitative Classify the following data as either qualitative or quantitative. a. Shades of red paint in a home improvement store b. Rankings of the most popular paint colors for the season c. Amount of red primary dye necessary to make one gallon of each shade of red paint d. Numbers of paint choices available at several stores
Example 1.4: Classifying Data as Qualitative or Quantitative (cont.) Solution a. Shades of paint are descriptions and cannot be measured, so these are qualitative data. b. Rankings are numeric but not measurements or counts, so these are qualitative data. c. The amounts of dye needed are measured and therefore are quantitative data. d. The numbers of paint choices must be counted, so they are quantitative data as well.
Example 1.5: Classifying Data as Continuous or Discrete Determine whether the following data are continuous or discrete. a. Temperatures in Fahrenheit of cities in South Carolina b. Numbers of houses in various neighborhoods in a city c. Numbers of elliptical machines in every YMCA in your state d. Heights of doors
Example 1.5: Classifying Data as Continuous or Discrete (cont.) Solution a. Temperatures could be measured to any level of precision based on the thermometer used, so these are continuous data. b. Numbers of houses are discrete data because houses are counted in whole numbers. A house under construction is still a house. c. The numbers of elliptical machines are counts, so these are discrete data.
Example 1.5: Classifying Data as Continuous or Discrete (cont.) d. Heights are measurements and again, depending on the ruler, the heights could be measured to any level of precision, so they are continuous data.
Example 1.6: Understanding the Nominal Level of Measurement a. Suppose all students in a statistics class were asked what pizza topping is their favorite. Explain why these data are at the nominal level of measurement. b. Suppose instead that you wish to know the number of students whose favorite pizza topping is sausage. Explain why this data value is not at the nominal level of measurement.
Example 1.6: Understanding the Nominal Level of Measurement (cont.) Solution a. These data are nominal because the data simply describe or label the different toppings of the pizza. b. In the second scenario, the data value is a count of students who prefer sausage. This data value is quantitative, not qualitative, so it is not a label and would not be considered to be at the nominal level of measurement.
Example 1.7: Classifying Data as Nominal or Ordinal Determine whether the data are nominal or ordinal. a. The seat numbers on your concert tickets, such as A23 and A24 b. The genres of the music performed at the 2013 Grammys
Example 1.7: Classifying Data as Nominal or Ordinal (copy) Solution a. Seat numbers are ordinal because there is a meaningful order to the data, namely, the position in the theater. b. Despite the fact that you may have your own personal preference for specific genres of music, there is no standard order. Therefore, music genres are nominal data.
Example 1.8: Classifying Data by the Level of Measurement The birth years of your classmates are collected. What level of measurement are these data? Solution Birth years can be ordered. It is also meaningful to subtract years to determine the difference in age. However, the year 0 A.D. does not mean the beginning of time. Therefore, birth years are at the interval level of measurement.
Levels of Measurement Qualitative data consist of labels or descriptions of traits. Quantitative data consist of counts or measurements. Continuous data are quantitative data that can take on any value in a given interval and are usually measurements. Discrete data are quantitative data that can take on only particular values and are usually counts.
Levels of Measurement Data at the nominal level of measurement are qualitative data consisting of labels or names. Data at the ordinal level of measurement are qualitative data that can be arranged in a meaningful order, but calculations such as addition or division do not make sense.
Levels of Measurement Data at the interval level of measurement are quantitative data that can be arranged in a meaningful order, and differences between data entries are meaningful. Data at the ratio level of measurement are quantitative data that can be ordered, differences between data entries are meaningful, and the zero point indicates the absence of something.
Example 1.9: Classifying Data by the Level of Measurement Consider the ages in whole years of US presidents when they were inaugurated. What level of measurement are these data? Solution The ages of US presidents are measurable, can be ordered, and an age of zero indicates the absence of life. Therefore, ages are at the ratio level of measurement. In contrast to Example 1.8, involving birth years, you can be twice as old as someone else.
Example 1.10: Classifying Data Determine the following classifications for the given data sets: qualitative or quantitative; discrete, continuous, or neither; and level of measurement. a. Finishing times for runners in the Labor Day 10K race b. Colors contained in a box of crayons c. Boiling points (on the Celsius scale) for various caramel candies d. The top ten Spring Break destinations as ranked by MTV
Example 1.10: Classifying Data (cont.) Solution a. The amount of time it takes for each runner to run the race is quantitative since calculations performed on these data are meaningful. A finishing time is a measurement, therefore the data are continuous. Differences between finishing times are meaningful, and a time of zero represents the absence of racing. We could also say that Andrew finished the race in half of Peyton’s time; thus, the data are at the ratio level of measurement.
Example 1.10: Classifying Data (cont.) b. Colors are labels, so these data are qualitative. Qualitative data are neither discrete nor continuous. There are many ways to order colors, such as alphabetically or based on the color spectrum. However, when discussing colors of crayons, order is not the primary factor, as opposed to data such as rankings, in which order is important. Therefore, the data are at the nominal level of measurement.
Example 1.10: Classifying Data (cont.) c. Calculations can be performed on boiling points because they are measurements, making these data quantitative. Temperatures are measurements, so the data are continuous. For the Celsius scale, a temperature of zero degrees is simply a placeholder and does not indicate the absence of heat. Therefore, data from the Celsius scale are always at the interval level of measurement.
Example 1.10: Classifying Data (cont.) d. Since the rankings cannot be meaningfully added or subtracted, the data must be qualitative. Qualitative data are neither discrete nor continuous. The rankings are in a specific order, so the data are at the ordinal level of measurement.