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.

- 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.

- 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

- 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 \(\$1460\).

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.