1. 2.1 Data Types and Levels of Measurement
LEARNING GOAL
Be able to identify data as qualitative or quantitative, to identify quantitative data as discrete or continuous, and to assign data a level of measurement (nominal, ordinal, interval, or ratio).
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2. Data Types
Data Types
Qualitative (or categorical) data consist of values that can be placed into nonnumerical categories.
Quantitative data consist of values representing counts or measurements.
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3. Brand names of shoes in a consumer survey
EXAMPLE 1 Data Types
Classify each of the following sets of data as qualitative or quantitative.
Brand names of shoes in a consumer survey
Scores on a multiple-choice exam
Solution:
Brand names are categorical and therefore represent qualitative data.
Scores on a multiple-choice exam are quantitative because they represent a count of the number of correct answers.
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4. Discrete versus Continuous Data
Continuous data can take on any value in a given interval.
Discrete data can take on only particular, distinct values and not other values in between.
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5. Measurements of the time it takes to walk a mile
EXAMPLE 2 Discrete or Continuous?
For each data set, indicate whether they data are discrete or continuous.
Measurements of the time it takes to walk a mile
The number of calendar years (such as 2007, 2008,
2009)
Solution:
Time can take on any value, so measurements of time are continuous.
The number of calendar years are discrete because they cannot have fractional values. For example, on New Years Eve of 2009, the year will change from 2009 to 2010; we’ll never say the year is 2009½ .
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6. Levels of Measurement
The simplest level of measurement applies to variables that can be described solely by names, labels, or categories. We say that such data are at a nominal level of measurement.
When we describe data with a ranking or ordering scheme, such as star ratings of movies or restaurants, we are using an ordinal level of measurement.
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7. TIME OUT TO THINK
Consider a survey that asks “What’s your favorite flavor of ice cream?” We've said that ice cream flavors represent data at the nominal level of measurement. But suppose that, for convenience the researchers enter the survey data into a computer by assigning numbers the different flavors. For example, they assign 1 = vanilla, 2 = chocolate, 3 = cookies and cream, and so on. Does this change the ice cream flavor from nominal to ordinal?
Why or why not?
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8. Levels of Measurement
The ordinal level of measurement provides a ranking system, but it does not allow us to determine precise differences between measurements.
If intervals are meaningful but ratios are not, we say that the data are at the interval level of measurement.
When both intervals and ratios are meaningful, we say that data are at the ratio level of measurement.
Page 56. Use movie rankings versus temperatures to compare the ordinal level of measurement to the interval level of measurement. Then use temperatures versus distances to compare the interval level of measurement to the ratio level of measurement.
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9. Figure 2.1 summarizes the possible data types and levels of measurement.
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Figure 2.1 Data types and levels of measurement.
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10. Levels of Measurement
The nominal level of measurement is characterized by data that consist of names, labels, or categories only. The data are qualitative and cannot be ranked or ordered.
The ordinal level of measurement applies to qualitative data that can be arranged in some order (such as low to high). It generally does not make sense to do computations with data at the ordinal level of measurement.
The interval level of measurement applies to quantitative data in which intervals are meaningful, but ratios are not. Data at this level have an arbitrary zero point.
The ratio level of measurement applies to quantitative data in which both intervals and ratios are meaningful. Data at this level have a true zero point.
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11. By the Way ...
Scientists often measure temperatures on the Kelvin scale. Data on the Kelvin scale are at the ratio level of measurement, because the Kelvin scale has a true zero. A temperature of 0 Kelvin really is the coldest possible temperature. Called absolute zero, 0 K is equivalent to about °C or °F.
(The degree symbol is not used for Kelvin temperatures.)
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12. Numbers on uniform that identify players on a basketball team
EXAMPLE 3 Levels of Measurement
Identify the level of measurement (nominal, ordinal, interval, ratio) for each of the following sets of data.
Numbers on uniform that identify players on a
basketball team
Solution:
Numbers on uniforms are at the nominal level of measurement because they are labels and do not imply any kind of ordering.
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13. b. Student rankings of cafeteria food as excellent, good,
EXAMPLE 3 Levels of Measurement
Identify the level of measurement (nominal, ordinal, interval, ratio) for each of the following sets of data.
b. Student rankings of cafeteria food as excellent, good,
fair, or poor
Solution:
b. A set of rankings represents data at the ordinal level of measurement because the categories (excellent food, fair, or poor) have a definite order.
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14. c. Calendar years of historic events, such as 1776, 1945, or 2001
EXAMPLE 3 Levels of Measurement
Identify the level of measurement (nominal, ordinal, interval, ratio) for each of the following sets of data.
c. Calendar years of historic events, such as 1776, 1945,
or 2001
Solution:
c. An interval of one calendar year always has the same meaning. But ratios of calendar years do not make sense because the choice of the year 0 is arbitrary and does not mean “the beginning of time.” Calendar years are therefore at the interval level of measurement.
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15. d. Temperatures on the Celsius scale
EXAMPLE 3 Levels of Measurement
Identify the level of measurement (nominal, ordinal, interval, ratio) for each of the following sets of data.
d. Temperatures on the Celsius scale
Solution:
d. Like Fahrenheit temperatures, Celsius temperatures are at the interval level of measurement. An interval of 1°C always has the same meaning, but the zero point (0°C = freezing point of water) is arbitrary and does not mean “no heat.”
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16. e. Runners’ times in the Boston Marathon
EXAMPLE 3 Levels of Measurement
Identify the level of measurement (nominal, ordinal, interval, ratio) for each of the following sets of data.
e. Runners’ times in the Boston Marathon
Solution:
e. Marathon times have meaningful ratios—for example, a time of 6 hours really is twice as long as a time of 3 hours—because they have a true zero point at a time of 0 hours.
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17. The End
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