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Variables and their types

Learning goals

Identifies observational units and variables in a data set.
Classifies variables as categorical or quantitative.
Recognizes when numerical labels are categorical rather than quantitative.
Connects variable type to appropriate displays and summaries.

Difficulty: 1 – 5

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Variables and their types 1

A study records the commute method of each selected employee. Identify the observational units and the variable, and classify the variable as nominal categorical, ordinal categorical, or quantitative.

Hints

- Ask what item or individual supplies each datum. - Decide whether the category labels have a natural ranking.

Solution

1. The observational units are the selected employees. 2. The variable is commute method. 3. Commute methods such as car, bus, bicycle, and walking are category labels with no inherent ranking, so the variable is nominal categorical.

Answer

Observational units: selected employees. Variable: commute method. Type: nominal categorical.

Variables and their types 2

A survey records the number of concerts each respondent attended last year. Classify the variable and explain why.

Hints

- Decide whether the possible values are countable. - Ask whether a value between two consecutive whole-number counts would be meaningful.

Solution

1. The variable is a numerical count. 2. It can take countable whole-number values, so it is a discrete quantitative variable.

Answer

Discrete quantitative, because it is a count.

Variables and their types 3

A botanist counts the number of petals on each flower and measures the flower’s diameter. Which variable can take countably many values, and which can take any value in an interval?

Hints

- Separate the count from the measurement. - A count advances in separate whole-number steps, whereas a measurement can vary between them.

Solution

1. Petal count can take countably many whole-number values, so it is discrete. 2. Diameter can take any measured value in an interval, so it is continuous.

Answer

Petal count is discrete; flower diameter is continuous.

Variables and their types 4

A traffic study records whether each passing vehicle has an out-of-state license plate and also records its speed. Identify one categorical variable and one continuous quantitative variable.

Hints

- Classify each recorded characteristic separately. - Test whether each variable has meaningful numerical differences and measurement units.

Solution

1. Out-of-state plate status has category values such as yes and no, so it is categorical. 2. Speed is measured and can vary continuously, so it is continuous quantitative.

Answer

Categorical: out-of-state plate status. Continuous quantitative: vehicle speed.

Variables and their types 5

A language survey records each respondent’s primary language and the number of languages the respondent can hold a conversation in. Classify both variables.

Hints

- Decide whether each value is a label or a count. - Check whether the language labels have a natural ranking. - Consider the possible numerical values of the count.

Solution

1. Primary language consists of category labels with no inherent ranking, so it is nominal categorical. 2. Number of languages is a whole-number count, so it is discrete quantitative.

Answer

Primary language is nominal categorical. Number of conversational languages is discrete quantitative.

Variables and their types 6

A reading log has these rows: <table><thead><tr><th>Book</th><th>Pages</th></tr></thead><tbody><tr><td>Oak Street</td><td>\(180\)</td></tr><tr><td>Red Moon</td><td>\(245\)</td></tr><tr><td>Long River</td><td>\(210\)</td></tr></tbody></table> What does one row represent? Name the quantitative variable.

Hints

- Notice what changes from one row to the next. - Which column contains values that count something?

Solution

1. Each row describes one book. 2. The column headed “Pages” gives a numerical count for that book.

Answer

One row represents one book. The quantitative variable is the number of pages.

Variables and their types 7

A school gives lockers numbers such as \(205\), \(206\), and \(207\). It also records how many times each locker was opened yesterday. Which variable is categorical even though it uses numbers, and why?

Hints

- Ask whether adding two locker numbers would mean anything useful. - Compare a label that identifies something with a count of events.

Solution

1. The locker number identifies a particular locker; its numerical size is not a measured amount. 2. The number of openings is a count and is quantitative. 3. Therefore the locker number is categorical.

Answer

Locker number is categorical because the numbers are labels. Number of openings is quantitative.

Variables and their types 8

A marine lab records the body length of each captured horseshoe crab to the nearest \(0.1\,\text{cm}\). Classify the variable as categorical or quantitative, and as discrete or continuous.

Hints

- Separate the possible values of the underlying characteristic from the precision used to record it. - Ask whether values between two adjacent recorded readings could exist before rounding.

Solution

1. Body length is numerical and has units, so it is quantitative. 2. Length can take any value in an interval before rounding, so the underlying variable is continuous. 3. Recording to the nearest \(0.1\,\text{cm}\) limits precision; it does not turn the measured continuum into a count variable.

Answer

Quantitative and continuous; rounding the recorded measurement does not make the underlying variable discrete.

Variables and their types 9

A data file uses the numbers \(1\), \(2\), and \(3\) to code eye color as brown, blue, and green. Is the variable quantitative because its recorded values are numbers? If not, classify it as nominal or ordinal categorical.

Hints

- Consider whether numerical operations on the recorded codes would be meaningful. - Ask whether the categories have any inherent ranking.

Solution

1. The numbers are codes for category labels. 2. Arithmetic and numerical order on the codes have no meaningful interpretation for eye color. 3. Therefore, eye color is nominal categorical.

Answer

No. Eye color is nominal categorical; the numbers are only labels.

Variables and their types 10

A school study records each student’s locker number and the mass of the student’s backpack. Classify both variables.

Hints

- A number can be an identifier rather than a measured quantity. - Classify each column by its role, not simply by whether digits appear.

Solution

1. Locker number identifies a locker and functions as a label, so it is categorical. 2. Backpack mass is numerical with units and is measured on a continuum, so it is continuous quantitative.

Answer

Locker number: categorical. Backpack mass: continuous quantitative.

Variables and their types 11

A researcher records the dominant sound heard during each \(30\)-second interval in a forest as birds, insects, wind, water, or human activity. Identify the observational units and classify the variable.

Hints

- The observational unit need not be a person or physical object. - Decide whether the sound categories have a natural ranking.

Solution

1. The observational units are the \(30\)-second intervals. 2. The variable is the dominant sound heard during each interval. 3. The values are category labels with no natural order, so the variable is nominal categorical.

Answer

Observational units: the \(30\)-second intervals. Variable: dominant sound. Type: nominal categorical.

Variables and their types 12

For each variable, state whether it is discrete quantitative, continuous quantitative, or categorical: a) Number of text messages received in one day b) Water temperature in a lake c) Type of internet connection

Hints

- Ask whether each value is a label, a count, or a measurement. - Counts are discrete, measured quantities are continuous, and names or types are categorical.

Solution

1. a) is a count, so it is discrete quantitative. 2. b) is measured on a continuum, so it is continuous quantitative. 3. c) consists of group labels, so it is categorical.

Answer

a) Discrete quantitative b) Continuous quantitative c) Categorical

Variables and their types 13

A survey asks respondents to rate service as poor, fair, good, or excellent. Maëlys calls the variable continuous because there are many possible opinions between the labels. Correct Maëlys's reasoning and classify the recorded variable as nominal or ordinal categorical.

Hints

- Use only the response options that enter the data set. - Ask whether the categories have a natural ranking.

Solution

1. The recorded values are four category labels, not measurements on a numerical continuum. 2. The categories have a natural order from poor through excellent. 3. Therefore, the recorded variable is ordinal categorical.

Answer

The recorded variable is ordinal categorical because the response categories are ordered labels.

Variables and their types 14

A sensor takes a photograph of cloud cover every hour. A computer later classifies each photograph as clear, partly cloudy, mostly cloudy, or overcast. What variable is used in the analysis, and is it nominal or ordinal categorical?

Hints

- Distinguish the form in which data are captured from the variable ultimately analyzed. - Order the categories by increasing cloud cover to determine the categorical subtype.

Solution

1. The photographs are raw data collected from hourly observational units. 2. The analyzed variable is cloud-cover category. 3. The categories have a natural order from less cloud cover to more cloud cover, so the variable is ordinal categorical.

Answer

The analyzed variable is cloud-cover category, and it is ordinal categorical.

Variables and their types 15

A college records each applicant’s five-digit ZIP code. Classify the variable and justify your answer.

Hints

- Ask whether adding or averaging the values would have a useful interpretation. - Check whether numerical magnitude creates any meaningful ranking among ZIP codes.

Solution

1. ZIP codes identify geographic categories. 2. Their numerical order and arithmetic differences are not meaningful measurements, and the categories have no inherent ranking. 3. Therefore, ZIP code is nominal categorical.

Answer

Nominal categorical, because a ZIP code is an identifier rather than a measured or ordered quantity.

Variables and their types 16

A report states, “The median rent among the \(75\) sampled apartments is \(\$1460\).” Identify the variable and the statistic.

Hints

- Do not confuse the characteristic measured on each unit with the summary of all sampled values. - First name what is recorded for one apartment, then name the summary calculated from the sample.

Solution

1. The variable is monthly rent for an apartment, a quantitative variable measured in dollars. 2. The statistic is the sample median, \(\$1460\).

Answer

Variable: monthly apartment rent, quantitative. Statistic: the sample median of \(\$1460\).

Variables and their types 17

A database has one row for each county and records the county’s population, region, and number of public libraries. Identify the observational units and classify all three variables.

Hints

- Use the rows to identify the observational units. - After identifying what one row represents, classify each column independently.

Solution

1. The observational units are counties. 2. Population is a discrete quantitative count. 3. Region is categorical. 4. Number of public libraries is a discrete quantitative count.

Answer

Observational units: counties. Population: discrete quantitative. Region: categorical. Number of public libraries: discrete quantitative.

Variables and their types 18

A survey file has a column named “time.” Its numbers might be seconds spent completing the survey or times of day when answers were submitted. Why should the values not be analyzed until the meaning is known? Name two details the survey team should provide.

Hints

- Ask what one number in the column tells you about a response. - Does a value such as \(45\) mean the same thing under both interpretations? - Think about what units or time labels would distinguish the two possibilities.

Solution

1. A duration and a clock time answer different questions. 2. The same number could mean different things under the two definitions. 3. The team must say what is being recorded and give the unit or clock convention, including how the values are recorded.

Answer

The column could describe either elapsed time or time of day. The team must define the quantity and explain its unit or clock format.

Variables and their types 19

A file of race-completion times contains the values \(142\), \(151\), \(163\), and \(170\), but the file does not state a unit. Explain why the data cannot be interpreted correctly until the unit is known. Give two possible interpretations of the value \(142\).

Hints

- Ask what information gives a number its real-world scale. - Imagine attaching two plausible time units to the same recorded value. - Consider whether conclusions would change under those interpretations.

Solution

1. A numerical datum needs a measurement unit to connect it to the real-world variable. 2. The value \(142\) could represent \(142\) seconds, or \(142\) minutes, among other possibilities. 3. Those interpretations describe very different completion times, so the unit must be recovered before analysis is reported.

Answer

The values are incomplete without a unit. For example, \(142\) could mean \(142\) seconds or \(142\) minutes, which are not equivalent race times.

Variables and their types 20

A museum membership file records membership level as Basic, Plus, Premium, or Patron and records the annual fee paid. Classify the two variables, including whether the membership-level categories are nominal or ordinal.

Hints

- Decide whether each value is a label or a measured amount. - Check whether the category labels have a meaningful progression. - Classify the variables independently even though they may be related.

Solution

1. Membership level is categorical because its values are labels. 2. The levels have a natural progression in benefits, so membership level is ordinal categorical. 3. Annual fee is quantitative because it measures an amount of money.

Answer

Membership level is ordinal categorical. Annual fee is quantitative.

Variables and their types 21

A medical file records blood type as A, B, AB, or O. Yating says the variable is ordinal because the labels can be arranged alphabetically. Correct Yating's reasoning and explain what kind of order is required for an ordinal variable.

Hints

- Distinguish a convenient sorting rule from a meaningful ranking. - Ask whether one category represents more or less of the measured characteristic. - Classify the variable from the relationship among its categories.

Solution

1. Blood types are category labels, so the variable is categorical. 2. Alphabetical order is only a display convention and does not rank the categories by more or less of a characteristic. 3. The variable is nominal categorical because it has no meaningful intrinsic ordering.

Answer

Blood type is nominal categorical. An ordinal variable needs a meaningful rank based on the characteristic, not merely an alphabetical arrangement of labels.

Variables and their types 22

At a randomly chosen moment, a browser study records the number of open tabs on each participant’s computer. Classify the variable as discrete or continuous and explain why there is no need for a fixed upper limit in the definition.

Hints

- Determine whether the observation is counted or measured. - Consider the form of all possible values rather than how large they might become. - Recall that countability, not a fixed maximum, defines this distinction.

Solution

1. Open tabs are counted, so the possible values are whole numbers. 2. A discrete variable can have a finite or countably infinite set of possible values. 3. Therefore, the variable is discrete quantitative even if the study does not specify a maximum possible count.

Answer

The number of open tabs is discrete quantitative because it takes countable whole-number values; a discrete variable need not have a stated finite maximum.

Variables and their types 23

A weather file records the overnight low temperature and a second variable indicating whether the low was below freezing. Classify both variables and describe the relationship between them.

Hints

- Classify the measured value and the Yes-or-No result separately. - Identify whether each variable records an amount or group membership. - Explain how one variable can be created from the other.

Solution

1. Overnight low temperature is a continuous quantitative variable. 2. Below-freezing status has two categories, Yes and No, so it is binary categorical. 3. The categorical variable is derived by comparing the quantitative temperature with the freezing threshold.

Answer

Overnight low temperature is continuous quantitative. Below-freezing status is binary categorical and is determined from the temperature by a threshold.

Variables and their types 24

A reservoir file records the daily change in water level in inches. Values may be negative, zero, or positive. Classify the variable and explain why negative values do not make it categorical.

Hints

- Ask whether the sign is part of a measurement or merely a label. - Consider whether arithmetic with the values has a meaningful unit. - A quantitative scale may include values below zero.

Solution

1. Daily change in water level measures a numerical amount and direction of change. 2. Differences and averages of the values have meaningful interpretations in inches. 3. The variable is continuous quantitative; negative values indicate decreases rather than category labels.

Answer

Daily water-level change is continuous quantitative. A negative value is a measured decrease, not a categorical code.

Variables and their types 25

A course file records each student’s percentage score and final letter grade. Classify both variables and explain why the letter grade is ordered but not a quantitative measurement.

Hints

- Decide which value measures an amount and which assigns a label. - Check whether the labels can be ranked. - Ask whether adjacent category gaps have a fixed numerical size.

Solution

1. Percentage score is a quantitative variable because it measures the share of available points earned. 2. Letter grade is ordinal categorical because grades such as A, B, C, D, and F have a natural order. 3. The differences between adjacent letter grades are not defined as equal numerical intervals.

Answer

Percentage score is quantitative. Final letter grade is ordinal categorical because it has order without a defined equal-interval scale.

Variables and their types 26

A phone-service file records country calling code, number of calls made in a month, and mean call duration. Classify all three variables.

Hints

- Determine whether each numerical-looking value is a label, count, or measurement. - Classify the variables independently. - Consider the possible values each variable can take.

Solution

1. Country calling code identifies a country or calling region, so it is categorical. 2. Number of calls is a whole-number count, so it is discrete quantitative. 3. Mean call duration is a measured numerical average and is continuous quantitative.

Answer

Country calling code: categorical. Number of calls: discrete quantitative. Mean call duration: continuous quantitative.

Variables and their types 27

A flood model gives each location an estimated annual flood probability and a risk label of Low, Moderate, High, or Very High. Classify both variables and describe how the label differs from the probability.

Hints

- Identify which variable records a numerical chance and which assigns a category. - Check whether the category labels have a natural order. - Compare the precision retained by the two representations.

Solution

1. Estimated annual flood probability is a quantitative variable measured on a numerical scale from \(0\) to \(1\). 2. Risk label is ordinal categorical because its categories have a natural increasing order. 3. The label groups ranges of probabilities and therefore contains less numerical detail than the probability itself.

Answer

Flood probability is quantitative. Risk label is ordinal categorical and is a grouped summary of the probability.

Variables and their types 28

A lab measures lead concentration in water. If the amount is too small for its instrument to measure, the lab records “ND” instead of a number. What type of variable is lead concentration? Why does “ND” not mean the amount is zero?

Hints

- Think about what the instrument is attempting to measure. - Can a measurement be too small for the instrument without being zero? - Decide whether “ND” gives an exact number or only limited information about it.

Solution

1. Lead concentration is a continuous quantitative variable. 2. “ND” means the amount was below what the instrument could measure, not that no lead was present. 3. The precise concentration of an “ND” sample is unknown, so the column is not a complete list of exact measurements.

Answer

Lead concentration is continuous quantitative. “ND” means too small to measure with this instrument; it does not prove the concentration is zero.

Variables and their types 29

A tool-rental shop checks each rented electric drill for several days. For each drill it records the days checked and whether it stopped working or was still working on the last day. What does one row represent? Classify both variables. Why aren't the days checked always the drill's time until failure?

Hints

- What was followed through the days at the shop? - Does each recorded detail give a number or one of two choices? - What can you say about when a working drill will eventually stop?

Solution

1. One row represents one rented electric drill. 2. Days checked is quantitative; stopped working versus still working is binary categorical. 3. For a drill still working on the last day, the actual time until failure is unknown.

Answer

One row is one drill. Days checked is quantitative, and working status is binary categorical. The failure time is unknown for a drill still working at the end.

Variables and their types 30

An airline file records airport gate, boarding group, and number of checked bags for each passenger. Classify all three variables, including whether either categorical variable is ordinal.

Hints

- Decide whether each value is a location label, ordered category, or count. - Check which category labels determine a sequence. - Classify the numerical variable by how it is obtained.

Solution

1. Airport gate is nominal categorical because it identifies a location without a meaningful ranking. 2. Boarding group is ordinal categorical because the groups determine a boarding sequence. 3. Number of checked bags is discrete quantitative because it is a whole-number count.

Answer

Airport gate: nominal categorical. Boarding group: ordinal categorical. Number of checked bags: discrete quantitative.

Variables and their types 31

A transit center records one main travel mode per visit. The rule is to use the mode covering the longest distance; if two modes tie, record “Mixed.” Three visitors report: Rosa: walked \(0.4\) mile and rode a bus \(3\) miles. Jin: biked \(1\) mile and rode a bus \(1\) mile. Nia: walked \(2\) miles. What single category should be recorded for each visitor? Why would recording both of Jin's modes in the same category column break the rule?

Hints

- Compare the distances within each person's trip, not between different people. - What should happen when two distances are equal? - Check that each completed row has exactly one category.

Solution

1. Rosa traveled farther by bus than on foot, so her category is Bus. 2. Jin traveled the same distance by bike and bus, so the category is Mixed. 3. Nia only walked, so her category is Walk. 4. Writing Bike and Bus together for Jin would place two category values in a column meant to store one value per visit.

Answer

Rosa: Bus. Jin: Mixed. Nia: Walk. Jin cannot be entered as both Bike and Bus because the column must hold one category per visit; “Mixed” handles the tie.

Variables and their types 32

A club survey lets students check any of three activities: hiking, music, and chess. One student checks hiking and chess but not music. To keep one row per student, what three variables should the club record, and what values should this student's row contain?

Hints

- Would a single category tell the whole story if students can check two activities? - Give every activity its own recorded answer. - Check which boxes this student marked.

Solution

1. Each activity needs its own yes/no variable because students can choose more than one. 2. Use the columns “Hiking selected,” “Music selected,” and “Chess selected.” 3. The student's row is Yes, No, Yes in that order.

Answer

Three categorical yes/no variables: Hiking selected = Yes, Music selected = No, Chess selected = Yes.

Variables and their types 33

A cafeteria collects these comments in order: “More apples,” “Clean the tables,” “Add bananas,” and “More berries.” It uses only two codes: Food choices (requests for foods) and Cleanliness (requests about cleaning). Code all four comments and name the type of the new variable.

Hints

- Use the same stated coding rule for every comment. - Decide whether each comment asks for food or cleaning. - Are the codes measurements or group labels?

Solution

1. “More apples,” “Add bananas,” and “More berries” request foods, so they get Food choices. 2. “Clean the tables” gets Cleanliness. 3. The coded topic is a categorical variable because its values are category labels.

Answer

In order: Food choices, Cleanliness, Food choices, Food choices. The new variable is categorical.

Variables and their types 34

A hospital records a patient’s age in completed years. Is this recorded variable discrete or continuous? Distinguish it from exact age.

Hints

- Pay attention to the exact operational definition of the recorded field. - Ask whether the field is produced by counting completed units or measuring elapsed time. - Compare the possible values of the recorded field with the possible values of exact age.

Solution

1. Age in completed years counts fully elapsed years and takes whole-number values, so the recorded variable is discrete. 2. Exact age is elapsed time since birth and can take any value over an interval, so exact age is continuous. 3. The distinction comes from the operational definition of the variable, not merely from rounding a continuous measurement.

Answer

Recorded age in completed years is discrete; exact age is continuous.

Variables and their types 35

A runner’s heart rate is recorded by a device as an integer number of beats per minute. Is heart rate inherently discrete or continuous for this study? Explain the role of the device’s rounding.

Hints

- Classify the underlying quantity rather than relying only on the display format. - Decide whether the integer reading comes from counting separate objects or rounding a measured rate. - Compare this recording rule with a variable explicitly defined as completed whole units.

Solution

1. Heart rate as a rate can vary continuously over an interval. 2. The device reports rounded integer values, but rounding does not change the underlying measured variable from continuous to discrete. 3. Unlike a variable defined as a whole-number count, the integer display is only a limited-precision reading of the underlying rate.

Answer

It is treated as continuous; the integer display is a rounded measurement rather than a discrete count.

Variables and their types 36

Two airports provide monthly files labeled “canceled flights.” Airport A counts only flights canceled on the day of departure. Airport B also counts flights canceled several days in advance. An analyst wants to compare the airports' monthly cancellation counts. Explain why the counts are not yet comparable and what must happen before the comparison is made.

Hints

- Compare the exact rule each airport uses to decide whether a flight belongs in the file. - Ask whether the same label guarantees that the same variable was measured in the same way. - Identify what must be standardized before the numerical counts can be compared.

Solution

1. The two airports use different operational definitions for the variable “canceled flight.” 2. Because the inclusion rules differ, the reported monthly counts do not measure the same thing. 3. The airports must apply one common cancellation definition and recode or recollect the monthly data consistently before comparing the counts.

Answer

The counts use different definitions of “canceled flight,” so they are not directly comparable. The same operational definition must be applied to both airports before the counts are compared.

Variables and their types 37

A fitness app stores \(0\) in the “daily exercise minutes” field both when a user completed no exercise and when the device failed to sync. Explain why the distribution cannot be interpreted correctly from this field alone and identify the additional variable the data set needs.

Hints

- Distinguish a measured value of zero from the absence of a measurement. - Consider how combining the two meanings changes the graph at zero. - Identify a separate field that would resolve the ambiguity.

Solution

1. A true value of \(0\) is an observed outcome, while a sync failure means the exercise value is unknown. 2. Using the same code for both conditions mixes valid zeros with missing data and can inflate the apparent frequency at \(0\). 3. The file needs a separate synchronization or measurement-status variable so true zeros can be distinguished from missing observations.

Answer

The value \(0\) has two incompatible meanings, so the zero frequency is ambiguous. Add a status indicator that distinguishes a valid zero from a failed or missing measurement.

Variables and their types 38

Halfway through a customer study, a survey changed its satisfaction scale from \(1\)–\(5\) to \(1\)–\(10\), but the final file places all ratings in one column without identifying the scale used. Explain why the raw ratings should not be analyzed together and what information is needed.

Hints

- Compare what a particular numerical rating means under each scale. - Decide whether equal recorded numbers represent equal satisfaction levels. - Identify the contextual label needed for each response.

Solution

1. The same recorded number has different meanings on the two scales. 2. Combining the raw values would treat non-equivalent ratings as though they were measured identically. 3. Each record needs a scale-version label, after which the ratings can be analyzed separately or converted using a justified common definition.

Answer

The ratings are not directly comparable because they come from different scales. Each response must be linked to its scale version before the data are separated or validly converted.

Variables and their types 39

A vehicle-speed file combines records from two countries. Some speeds are in miles per hour and others are in kilometers per hour, but the unit label was lost. Explain why the numerical values cannot be safely combined and what must be recovered before analysis.

Hints

- Ask whether the same numerical value represents the same physical speed in both systems. - Identify the information needed to convert a measurement. - State the consistency required before forming one distribution.

Solution

1. Miles per hour and kilometers per hour use different numerical scales for the same physical speed. 2. Without a unit label, a recorded value cannot be interpreted or converted correctly. 3. The unit for each observation must be recovered, and all speeds must then be converted to one common unit.

Answer

The values cannot be safely combined because their units are unknown. Each observation’s unit must be recovered before all speeds are converted to a common scale.

Variables and their types 40

A climate archive records the month in which the first freezing temperature occurred each year. Explain why the variable’s classification depends on how the analysis defines the annual cycle. Give one categorical representation and one quantitative representation.

Hints

- Decide whether the recorded values are names or measured elapsed amounts. - Consider why December-to-January makes an ordinary linear ranking problematic. - Think about how choosing a fixed start date can turn seasonal timing into a numerical measurement.

Solution

1. Month names can be treated as categorical labels when the analysis compares named months. Because the months form a cycle, calling them simply “ordinal” is incomplete unless a starting point for the cycle is specified. 2. If the study defines a season that begins on a fixed date, such as July \(1\), the month labels can be ordered within that season. 3. A quantitative representation would record the number of days from the fixed starting date to the first freeze, which gives meaningful numerical differences.

Answer

The raw month names can be treated as categorical labels, but their order is cyclic rather than a single permanent lowest-to-highest order. After defining a seasonal starting point, the months may be ordered within that cycle. Recording days since that starting point produces a quantitative variable.

Variables and their types 41

A bookstore saves each customer's written suggestion and the number of words in that suggestion. Can the raw comments be used as meaningful categories in a frequency table? Explain what the store should do first, and classify the word-count variable.

Hints

- Would listing every different sentence make an informative category table? - Think about how two comments on the same subject could share a label. - Decide whether word count is a label or a numerical amount obtained by counting.

Solution

1. Free-response comments are unstructured text; each unique sentence is not automatically a useful category. 2. The store should choose clear categories, such as requests about hours, prices, or book choices, and apply the same coding rule to every comment. 3. The coded topic is categorical. Word count is discrete quantitative because it counts words.

Answer

Not without grouping the comments by a consistent rule. The coded comment topic would be categorical; the number of words is discrete quantitative.

Variables and their types 42

A race record lists each runner’s finishing place and exact elapsed time. Classify finishing place and elapsed time. Explain why finishing place should not be treated as a quantitative measurement even though it is written with numbers.

Hints

- Separate ordering information from measured magnitude. - Ask whether equal numerical gaps have a consistent real-world meaning. - Classify the time variable by the values it could take before rounding.

Solution

1. Finishing place gives an ordered rank, so it is ordinal categorical. 2. Differences between ranks do not measure equal differences in performance; the time gap from first to second may differ from the gap from second to third. 3. Exact elapsed time is a continuous quantitative variable.

Answer

Finishing place is ordinal categorical, while exact elapsed time is continuous quantitative. Rank numbers show order but not equal measured intervals.

Variables and their types 43

For each weather station, a monthly variable is the proportion of \(30\) observed days on which rain occurred. Should this recorded proportion be classified as discrete or continuous quantitative? Explain using its possible values.

Hints

- Identify the count used to form the proportion. - List the smallest possible change in the recorded value. - Do not classify a variable as continuous merely because it is written as a decimal.

Solution

1. The numerator is a count from \(0\) through \(30\). 2. With a fixed denominator of \(30\), the possible proportions are \(0,\frac{1}{30},\frac{2}{30},\ldots,1\). 3. Because the recorded variable has a finite set of possible values, it is discrete quantitative even though it may be written as a decimal or percentage.

Answer

The variable is discrete quantitative. Its possible values are multiples of \(\frac{1}{30}\), not every value in the interval from \(0\) to \(1\).

Variables and their types 44

A historic-building file records the calendar year each building was completed and also assigns each building to an era labeled Colonial, Industrial, or Modern. Classify both variables and explain why calendar year is not merely a category label.

Hints

- Ask whether subtracting two recorded values has a meaningful interpretation. - Separate a measured time location from a named historical grouping. - Consider the set of possible year values.

Solution

1. Completion year is a discrete quantitative variable because it takes countable year values and numerical differences represent elapsed time. 2. Era is a categorical variable because it assigns each building to a named group. 3. Arithmetic comparisons between years have a time interpretation, unlike arithmetic with the era labels.

Answer

Completion year is discrete quantitative. Era is categorical. Differences between completion years measure elapsed time, so year is more than an identifier.

Variables and their types 45

A survey asks for primary commuting method. Respondents who work entirely from home select “Not applicable,” while a blank cell means the respondent skipped the question. Explain why “Not applicable” is a valid categorical value but a blank cell is missing data.

Hints

- Ask whether each entry communicates known information about the respondent. - Distinguish a response option from the absence of a response. - Consider how merging the two would affect the frequency table.

Solution

1. “Not applicable” communicates a known status about the respondent and belongs in the set of response categories. 2. A blank cell provides no observed value for commuting method, so the response is unknown. 3. Combining the two would confuse a meaningful category with item nonresponse and would change category frequencies.

Answer

“Not applicable” is an observed categorical response. A blank is missing because the respondent’s value is unknown. They must be stored and analyzed separately.

Variables and their types 46

A transit study stores arrival time in two ways: time of arrival measured to the nearest minute after midnight and a label of Morning, Afternoon, Evening, or Overnight. Classify both variables. Explain why the label version loses quantitative detail.

Hints

- Distinguish the underlying quantity from the precision used to record it. - Compare a measured time with a grouped label derived from that time. - Ask whether numerical differences between values remain available after grouping.

Solution

1. Arrival time is an underlying continuous quantitative variable, even though it is recorded to the nearest minute. 2. The four day-part labels form a categorical variable. 3. Converting measured times to labels groups many different times together and removes the numerical differences within each day part.

Answer

Arrival time is continuous quantitative, recorded to the nearest minute. Day-part label is categorical. The labels discard the precise time and differences among arrivals within the same category.

Variables and their types 47

A laboratory dries each of \(40\) soil samples for exactly \(24\) hours and records the mass lost by each sample. Identify the observational units and the variable. Explain why “\(24\) hours” is a fixed study condition rather than a variable in this data set.

Hints

- Determine what one row of the data table would represent. - Identify which quantity can differ from one row to another. - A number mentioned in a study is not a variable unless it varies across units.

Solution

1. The observational units are the \(40\) soil samples. 2. The variable is mass lost during drying, a continuous quantitative measurement recorded separately for each sample. 3. Every sample is dried for the same \(24\)-hour period, so drying time does not vary across observational units and is a controlled condition rather than a variable.

Answer

The units are soil samples, and the variable is mass lost. The \(24\)-hour duration is constant for every sample, so it is not a variable in this data set.

Variables and their types 48

A music-recommendation file has one row for each pair of songs. It records a similarity score from \(0\) to \(1\) and whether the two songs have the same primary genre. Identify the observational units and classify both variables.

Hints

- Identify exactly what one row describes. - Notice that one object can participate in several observational units. - Separate a measured score from a group-membership indicator.

Solution

1. Each row describes a pair of songs, so song pairs are the observational units; an individual song may appear in many units. 2. Similarity score is quantitative because it measures the degree of similarity on a numerical scale. 3. Same primary genre is binary categorical because its values are Yes and No.

Answer

Observational units: pairs of songs. Similarity score: quantitative. Same primary genre: binary categorical.

Variables and their types 49

A survey asks each respondent to select every news source used during the past week: television, radio, newspaper, social media, and podcasts. Explain why the response should be represented as five binary categorical variables rather than one categorical variable with five mutually exclusive categories.

Hints

- Check whether one observational unit can belong to several listed options. - Recall the requirement for categories of a single categorical variable. - Reframe each option as its own recorded condition.

Solution

1. A respondent may select more than one source, so the five source labels are not mutually exclusive outcomes of one variable. 2. For each source, the respondent has a Yes or No value indicating whether that source was used. 3. The response is therefore represented by five binary categorical variables.

Answer

Because respondents may choose several sources, the categories are not mutually exclusive. Use one Yes-or-No categorical variable for each source.

Variables and their types 50

An image-analysis system records each pixel’s grayscale intensity as an integer from \(0\) to \(255\) and also labels the pixel Dark, Medium, or Bright. Classify both variables and explain how the label is derived from the numerical variable.

Hints

- Classify the stored numerical scale and the grouped labels separately. - Check whether the labels have a natural order. - Identify what information is lost when many numbers receive the same label.

Solution

1. Grayscale intensity is discrete quantitative because the stored value is one of the integers from \(0\) to \(255\). 2. Dark, Medium, or Bright is an ordinal categorical variable because the labels have a natural order. 3. The categorical label is created by grouping ranges of intensity values, so it retains order but loses exact numerical detail.

Answer

Grayscale intensity is discrete quantitative. Brightness label is ordinal categorical and is formed by grouping intensity values into ordered ranges.

Variables and their types 51

A vending-machine file records item price, and every possible price is a multiple of \(\$0.05\). Should the recorded price variable be classified as discrete or continuous quantitative? Explain why displaying prices with decimal points does not determine the classification.

Hints

- Identify the smallest possible change in the recorded value. - Ask whether every amount between two listed prices can occur. - Classify from the set of possible values, not the number format.

Solution

1. Price measures an amount of money, so it is quantitative. 2. Under the machine’s pricing rule, only separated values such as \(\$1.00\), \(\$1.05\), and \(\$1.10\) can occur. 3. The recorded variable is discrete quantitative because its possible values advance in fixed \(\$0.05\) increments.

Answer

The recorded price is discrete quantitative. Decimal notation does not make a variable continuous; the allowed values are separated by \(\$0.05\).

Variables and their types 52

An app has one row for every login. For each user, an analyst creates a new variable equal to the number of distinct calendar days on which that user logged in during a month. Identify the observational units of the new data set and classify the new variable. Explain why multiple logins on one day contribute only \(1\).

Hints

- Determine what one value in the derived data set represents. - Separate login events from the calendar days containing them. - Decide whether the final values arise from counting or measuring.

Solution

1. The derived data set has one value for each user, so users are the observational units. 2. The variable is a count of distinct active days and is therefore discrete quantitative. 3. The definition counts calendar days, not login events, so any number of logins by one user on the same day contributes one active day.

Answer

The observational units are users. The number of distinct login days is discrete quantitative, and repeated logins on the same day count once because the variable counts days rather than events.

Variables and their types 53

A baseball tracking file has one row for each pitch and records pitcher name, pitch type, and pitch speed. Identify the observational units and classify all three variables. Explain why pitchers are not the observational units even though each pitcher appears in many rows.

Hints

- Determine what one row of the file represents. - Classify names and types separately from a measured speed. - Distinguish the source associated with repeated rows from the event measured in each row.

Solution

1. Each row describes one pitch, so individual pitches are the observational units. 2. Pitcher name and pitch type are categorical variables. 3. Pitch speed is continuous quantitative. 4. A pitcher can contribute many pitch observations, so the person named in a row is not the row-level observational unit.

Answer

Observational units: individual pitches. Pitcher name and pitch type: categorical. Pitch speed: continuous quantitative.

Variables and their types 54

An elevator log records the destination floor as B2, B1, Lobby, \(2\), \(3\), and so on. For describing riders’ destination choices, should this variable be treated as quantitative or ordinal categorical? Explain the choice and state when a quantitative representation would be more appropriate.

Hints

- Identify whether the analysis concerns named destinations or measured travel distance. - Separate meaningful order from equal numerical spacing. - Consider what additional measurement would support arithmetic comparisons between floors.

Solution

1. For describing destination choices, the recorded floor labels identify ordered locations, so ordinal categorical is the more useful classification. 2. Labels such as B2, B1, and Lobby do not by themselves guarantee equal numerical spacing, and building floor labels may skip levels. 3. If the analysis concerns vertical distance traveled, the researcher should record a quantitative measurement such as elevation or height above a reference level rather than treat the labels themselves as measurements.

Answer

For destination choices, floor is best treated as ordinal categorical because the locations have a natural order but the labels do not guarantee equal spacing. A quantitative representation is appropriate when the study records actual elevation or vertical distance.

Variables and their types 55

A water sample's pH is measured on a bounded numerical scale and reported to two decimal places. Katarína says pH must be discrete because the scale has limits and the instrument displays only hundredths. Evaluate Katarína's claim by distinguishing the underlying variable from the recorded values.

Hints

- Separate the quantity being measured from the instrument's display rule. - Ask whether physical values between two displayed hundredths can exist. - A limited range does not by itself determine whether a variable is discrete.

Solution

1. Underlying pH is a measured quantity that can vary continuously within its physical range. 2. The instrument rounds that measurement to a finite set of displayed hundredth values. 3. The underlying variable is continuous quantitative, while the stored display is discretized by measurement precision.

Answer

The claim is incorrect. Boundedness does not make a variable discrete. Underlying pH is continuous quantitative, although the recorded values occur in hundredth-unit increments.

Variables and their types 56

A meeting file stores both a time-zone abbreviation, such as EST or CST, and the corresponding UTC offset in hours, such as \(-5\) or \(-6\). Classify the two variables and explain why the numerical offset is quantitative rather than merely another code.

Hints

- Ask whether each value names a group or measures a difference. - Test whether subtracting two recorded numbers has a meaningful unit. - Numerical storage can represent either a code or a measurement depending on meaning.

Solution

1. Time-zone abbreviation is nominal categorical because it names a zone. 2. UTC offset is discrete quantitative because it measures a signed difference from UTC in hours. 3. A difference of \(1\) between offsets represents an actual one-hour difference, so arithmetic with the offsets has a time interpretation.

Answer

Time-zone abbreviation is nominal categorical. UTC offset is discrete quantitative because its numerical differences measure differences in hours.

Variables and their types 57

A computer log records each event time as an integer number of seconds since a fixed reference instant. Should this timestamp be treated as a categorical identifier or a quantitative variable? Explain which arithmetic operation has a meaningful interpretation.

Hints

- Ask whether the stored number locates an observation on a measurement scale. - Test the meaning of subtracting two recorded values. - Distinguish a large numerical measurement from an arbitrary ID.

Solution

1. The timestamp locates an event on a numerical time scale, so it is quantitative. 2. Because the stored values are whole seconds, the recorded variable is discrete quantitative. 3. Subtracting two timestamps gives elapsed time in seconds, which is meaningful; the large integer is not merely an arbitrary label.

Answer

The timestamp is discrete quantitative. The difference between two timestamps measures elapsed seconds, so the values have quantitative meaning.

Variables and their types 58

A workforce report stores exact job tenure in years and also places each employee in one of these groups: Less than \(1\) year, \(1\)–\(4\) years, \(5\)–\(9\) years, or \(10\) or more years. Classify both variables and explain the effect of the open-ended final category.

Hints

- Separate the measured value from the category formed from it. - Check whether the categories have a meaningful order. - Determine what exact information is lost in the last group.

Solution

1. Exact job tenure is continuous quantitative. 2. Tenure group is ordinal categorical because the categories have a natural increasing order. 3. The final category gives only a lower bound, so employees with different tenures of \(10\) years or more are indistinguishable in the grouped variable.

Answer

Exact tenure is continuous quantitative. Tenure group is ordinal categorical, and the open-ended final group hides the exact amount above \(10\) years.

Variables and their types 59

A city asked \(150\) residents which emergency alerts they would use. Each person could choose more than one method. <table><thead><tr><th>Method</th><th>Number choosing it</th></tr></thead><tbody><tr><td>Email alert</td><td>\(92\)</td></tr><tr><td>Text alert</td><td>\(71\)</td></tr><tr><td>App alert</td><td>\(46\)</td></tr><tr><td>Phone call</td><td>\(33\)</td></tr></tbody></table> Name the yes/no variables needed to record each resident's choices. Find the percentage choosing each method, rounded to one decimal place. Which method was chosen by a majority? Explain why the four percentages do not have to add to \(100\%\).

Hints

- How would you record the choices of one person who selected more than one method? - When finding the percentage for a method, how many residents were asked that question? - Does each resident's response contribute to just one row, or could it affect several rows?

Solution

1. Record four yes/no variables for each resident: email chosen, text chosen, app chosen, and phone call chosen. 2. Divide each count by \(150\): email \(\frac{92}{150}\cdot100\%\approx61.3\%\), text \(\frac{71}{150}\cdot100\%\approx47.3\%\), app \(\frac{46}{150}\cdot100\%\approx30.7\%\), and phone calls \(\frac{33}{150}\cdot100\%=22.0\%\). 3. Email alerts were chosen by a majority because their percentage is above \(50\%\). 4. One person could choose several methods, so the four selection rates are not parts of one exclusive set of categories and need not sum to \(100\%\).

Answer

Yes/no variables: email chosen, text chosen, app chosen, phone call chosen. Selection rates: email \(61.3\%\), text \(47.3\%\), app \(30.7\%\), phone calls \(22.0\%\). Email alerts were chosen by a majority. The rates need not total \(100\%\) because residents could choose several methods.

Variables and their types 60

A community garden asks four volunteers—Amina, Luis, Mei, and Petra—which days they can help: Saturday, Sunday, both, or neither. Two volunteers choose Saturday, and two choose Sunday. Give two different sets of yes/no responses, one for each volunteer and day, that fit these totals but have different numbers of volunteers available on both days. What do the totals alone leave unknown?

Hints

- Write a yes/no pair for each volunteer and count the Yes entries for each day. - After one set of choices works, could you change some individual choices without changing either daily total? - Check whether the number of volunteers who chose both days changed in your second set.

Solution

1. For one possible record, Amina and Luis choose both days, while Mei and Petra choose neither. Each day has \(2\) volunteers, and \(2\) choose both days. 2. For another record, Amina and Luis choose Saturday only, while Mei and Petra choose Sunday only. Each day still has \(2\) volunteers, but nobody chooses both days. 3. The two totals alone cannot tell how many volunteers chose both days.

Answer

One possible record: Amina Yes/Yes, Luis Yes/Yes, Mei No/No, Petra No/No (Saturday/Sunday); \(2\) chose both. Another: Amina Yes/No, Luis Yes/No, Mei No/Yes, Petra No/Yes; none chose both. The separate daily totals do not determine the number who chose both days.

Variables and their types 61

A vessel-tracking file records longitude in degrees from \(-180\) to \(180\) and also records hemisphere as East or West. Classify both variables. Explain why longitude is quantitative even though ordinary numerical distance can be misleading near the \(-180\)-to-\(180\) boundary.

Hints

- Decide which field measures position and which field assigns a label. - Consider what happens at the two endpoints of the longitude scale. - A quantitative variable can still require a special interpretation of differences.

Solution

1. Longitude is a continuous quantitative measurement of angular position. 2. East or West is a binary categorical label. 3. The longitude scale wraps around, so values such as \(-179\) and \(179\) are geographically close even though their ordinary numerical difference is large.

Answer

Longitude is continuous quantitative, and hemisphere is binary categorical. Longitude remains quantitative, but its circular boundary must be considered when comparing or summarizing values.

Variables and their types 62

A spreadsheet column called “grade” contains \(9,10,11,12\). Give one meaning that makes these numbers ordered categories and one meaning that makes them quantitative values. Explain why the numbers alone do not tell you the variable type.

Hints

- Think about numbers used as names for groups. - Now think about numbers that count points earned. - Does writing the same digits make the underlying quantities the same kind of variable?

Solution

1. If the numbers name school grade levels, they are ordered categories. 2. If they are points earned out of \(12\) on a quiz, they are discrete quantitative values. 3. The variable type depends on what each number means, not just how it is written.

Answer

School grade level: ordinal categorical. Quiz score out of \(12\): discrete quantitative. The column needs a clear definition before it can be classified.

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