Free math worksheets

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

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.

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

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

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

Discrete quantitative, because it is a count.

- Identify whether the numerical summary comes from a sample or the whole population. - A parameter describes an entire defined population.

1. The mean describes all firefighters employed by the city, which is the population of interest. 2. Therefore, \(38.6\) years is a parameter.

It is a parameter because it summarizes the entire population of city firefighters.

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

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.

Petal count is discrete; flower diameter is continuous.

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

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.

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

- 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.

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.

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

- 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.

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.

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

- Identify whether the numerical summary comes from a sample or the whole population. - Trace the stated percentage back to the group from which it was calculated.

1. The sample percentage, \(58\%\), is a statistic. 2. The unknown percentage among all registered voters in the county is the population parameter.

Statistic: \(58\%\). Parameter: the true countywide percentage of registered voters who support the bond.

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

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.

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

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

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.

Locker number: categorical. Backpack mass: continuous quantitative.

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

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.

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

- 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.

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.

a) Discrete quantitative b) Continuous quantitative c) Categorical

- Match each term to the sample, the population, or the items measured. - The individual bulb is measured, while the mean summarizes a group of bulbs.

1. The statistic is the sample mean, \(1184\,\text{hours}\). 2. The parameter is the unknown mean lifetime of all bulbs of that model. 3. The observational units are the individual sampled light bulbs.

Statistic: \(1184\,\text{hours}\). Parameter: the population mean lifetime. Observational units: individual sampled bulbs.

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

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.

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

- 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.

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.

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

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

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.

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

- 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.

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

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

- Trace where the numerical value came from. - The unknown population percentage is the parameter the sample statistic is intended to estimate.

1. The value \(4.7\%\) was calculated from the sample, so it is a statistic. 2. The parameter is the unknown percentage of all the retailer’s June online orders that were returned.

\(4.7\%\) is a statistic; the parameter is the unknown return percentage for all the retailer’s June online orders.

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

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.

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

- Match one recorded value to the item or event that produced it. - Decide whether the question concerns people or separate visits. - Count observational units rather than unique identities.

1. The variable is pages viewed during one website session, so each session is an observational unit. 2. The sample contains \(900\) sessions, giving sample size \(n=900\). 3. Some visitors created more than one session, so the \(520\) distinct people do not equal the number of observed visits.

The observational units are website sessions, and the sample size is \(n=900\). The \(520\) visitors are not the sample size because a visitor may contribute more than one session.

- Trace each mean back to the group from which it was calculated. - Classify the summaries by their source rather than by their numerical size. - Consider whether equal numbers must represent the same statistical role.

1. The population mean age of all \(96\) playgrounds is the parameter. 2. The sample mean age of the \(24\) selected playgrounds is the statistic. 3. A parameter and a statistic are distinguished by the group summarized, not by whether their numerical values happen to match.

Parameter: the mean age of all \(96\) playgrounds, \(11.4\) years. Statistic: the mean age of the \(24\) sampled playgrounds, also \(11.4\) years. They differ because one summarizes the population and the other summarizes a sample.

- Determine what one row represents. - Separate the number of observational units from the number of variables recorded per unit. - Recall what the symbol \(n\) counts.

1. Sample size counts observational units, not the total number of recorded cells. 2. Each row represents one patient, so there are \(75\) observational units. 3. The six variables provide multiple data values about each patient but do not create additional patients.

The sample size is \(n=75\), not \(450\). The table contains six variables for each of \(75\) patient observational units.

- 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.

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.

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

- 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.

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.

Membership level is ordinal categorical. Annual fee is quantitative.

- 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.

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.

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

- 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.

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.

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.

- 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.

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.

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

- 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.

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.

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

- 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.

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.

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

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

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.

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

- 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.

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.

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

- 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.

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.

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

- 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.

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.

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

- 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.

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.

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

- Identify the phrase that tells how many final values the question requires. - Distinguish raw sensor readings from the summarized observations used in the analysis. - Decide what one completed row of the analysis data set should represent.

1. The question asks for one result for each day, so the observational units in the final data set are the days in July. 2. All readings within a day must be combined to find that day’s maximum temperature. 3. Each datum in the final data set is the highest recorded temperature for one day.

The observational units should be the days in July, and each datum should be that day’s maximum recorded temperature.

- Match the denominator of the requested proportion to the units in the data. - Check whether one database row can stand for more than one person in the population. - Identify who actually has the characteristic being measured.

1. The question concerns individual attendees, not ticket transactions or purchasers. 2. One purchaser can represent several attendees whose transportation choices may differ. 3. The observational units should be individual festival attendees, with transportation method recorded for each attendee.

Purchasers are not the correct units because one purchase can cover several attendees. The observational units should be individual attendees.

- Identify the observational units named in the requested proportion. - Compare those units with what one file row represents. - Determine the numerator and denominator that the desired proportion needs.

1. The requested proportion uses individual shipped items as observational units. 2. An order may contain several items, and the current indicator does not show how many items were damaged or how many items were shipped. 3. The warehouse needs the total number of shipped items and the number of damaged items, ideally recorded at the item level.

The file summarizes orders rather than items, so it cannot provide an item-level damage proportion. The total number of shipped items and the number of damaged items are required.

- Consider whether one column label can refer to more than one mathematical quantity. - Ask what information is needed to attach meaning to a numerical value. - Separate a variable’s name from its operational definition.

1. Elapsed duration and clock time are different variables with different meanings, units, and useful summaries. 2. The numerical values cannot be interpreted or graphed correctly without knowing which quantity was recorded. 3. The data dictionary should identify the measured quantity and its unit or time convention, including how the value was calculated or encoded.

Analysis must wait because “time” does not identify the variable’s meaning. The metadata should state what the values measure and the unit or clock convention used, together with the recording rule.

- 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.

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.

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.

- 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.

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.

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.

- 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.

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.

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.

- 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.

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.

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.

- 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.

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.

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.

- Ask whether arithmetic with the raw responses would have a measurement meaning. - Do not assume a categorical variable must have only a few possible labels. - Identify which field is produced by counting.

1. The written comment is a categorical text variable because each value is a verbal response rather than a measured amount. 2. A categorical variable may have many possible categories; it does not need a short fixed list. 3. Word count is discrete quantitative because it is obtained by counting words.

Written comment is categorical text data. Word count is discrete quantitative. Having many distinct text responses does not make the comments quantitative.

- 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.

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.

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

- 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.

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.

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

- 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.

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.

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

- 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.

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.

“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.

- 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.

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.

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.

- 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.

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.

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.

- 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.

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.

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

- 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.

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.

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

- 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.

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.

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

- 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.

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.

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

- 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.

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.

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.

- 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.

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.

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

- 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.

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.

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.

- 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.

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.

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.

- 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.

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.

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

- Classify the physical quantity being measured. - Interpret what the nonnumeric entry says about the unknown value. - Distinguish an actual category from a measurement that could not be resolved precisely.

1. Lead concentration is an underlying continuous quantitative variable. 2. A record of “ND” means the concentration is below a detection threshold, not that the concentration is a category or exactly zero. 3. The column mixes exact measured values with censored observations whose precise quantitative values are unknown.

Lead concentration is continuous quantitative. “ND” represents a censored value below the detection limit, so the column does not contain an exact numerical value for every observation.

- 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.

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.

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

- 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.

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.

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

- Identify what one observation should represent for the question. - Determine why some real-world units appear in several rows. - Remove repeated representations without deleting distinct loans.

1. The observational unit is a library loan, not a loan-author combination. 2. Repeated rows would give extra weight to loans associated with multiple authors. 3. The final data set should contain one checkout-duration value for each of the \(900\) unique loans.

Using all \(1260\) rows would count some loans more than once. The analysis file should have one row and one checkout-duration value per unique loan, for sample size \(n=900\).

- 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.

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.

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

- Identify what one row represents. - Separate elapsed time from the event outcome. - Use the status label to interpret what the time value means.

1. The observational units are individual machines. 2. Observed days is quantitative because it measures elapsed time. 3. Event status is binary categorical. 4. For a machine still operating when observation ends, observed days is a censored follow-up time rather than an actual failure time.

The units are machines. Observed days is quantitative, and event status is binary categorical. A Still operating machine has no observed failure time; its recorded days give only the length of follow-up.

- Invent two realistic meanings for the same numerical entries. - Ask whether the numbers rank groups or measure an amount. - Classify from the variable definition rather than the column format.

1. If the values indicate school grade level, they are ordered category labels, so the variable is ordinal categorical. 2. If the values are numerical scores earned on a \(12\)-point assessment, the variable is discrete quantitative. 3. Variable type depends on the meaning and measurement rule, not only on the symbols stored in the column.

School grade level would be ordinal categorical; a score out of \(12\) would be discrete quantitative. The data dictionary is needed because identical numbers can encode different kinds of variables.