Free math worksheets

- Decide whether each question is meant to produce one fixed answer or a collection of answers. - For a collection of answers, consider whether all responses are expected to be identical. - Variability can occur in categories as well as in numerical measurements.

1. Question a) asks for one fixed result from one identified drop, so it is non-statistical. 2. Question b) expects different flight times from several helicopters, so it is statistical; the time in the air can vary. 3. Question c) asks for one fixed measurement of one ceiling, so it is non-statistical. 4. Question d) expects different categorical choices from several team members, so it is statistical; the chosen blade design can vary.

a) Non-statistical. b) Statistical; the flight times can vary. c) Non-statistical. d) Statistical; the blade-design choices can vary.

- Identify what one value in the data set measures. - Look for the question that requires several observations of that same attribute. - Eliminate questions asking for one specified event or one fixed fact.

1. The data contain several travel-time measurements, and the values vary. 2. Question b) asks about the distribution of travel times for repeated morning trips, so the data can answer it. 3. Question a) asks for one identified trip, and question c) asks for one fixed building fact.

b) “How long do elevator trips from the lobby to the sixth floor take during the morning?” It anticipates variable times from repeated trips.

- Match the number of observations and the measurement period to the question. - Look for a clearly defined attribute that can be recorded at every location. - Eliminate questions that use only one observation or are too vague to measure.

1. The plan produces one numerical count from each of \(12\) locations under the same observation time. 2. Question a) asks for those repeated counts and anticipates variability among locations. 3. Question b) uses only one location, and question c) does not define a measurable attribute.

a) It asks for repeated numerical observations from all \(12\) locations and anticipates that the counts may vary.

- Include every stated boundary on who, what, and when. - Use wording that would produce one measurement from each member of the group. - Identify the quantity expected to differ across responses.

1. The question must name the specified group, numerical attribute, and time period. 2. A complete revision is “How many minutes did Grade \(6\) students at Lincoln Middle School exercise yesterday?” 3. The number of exercise minutes can vary from student to student, so the revised question is statistical.

“How many minutes did Grade \(6\) students at Lincoln Middle School exercise yesterday?” The number of minutes would vary among students.

- Count how many people each question concerns. - Consider whether different members of the group may give different categories of response. - Numerical answers are not required for statistical variability.

1. Question a) concerns one identified student and has one yes-or-no answer, so it is non-statistical. 2. Question b) collects answers from many students and anticipates that some responses may be yes and others no, so it is statistical. 3. Statistical variability can occur among categorical responses, including two-category answers.

a) Non-statistical b) Statistical A yes-or-no question is statistical when it is asked of a group and the responses are expected to vary.

- Distinguish one object with fixed measurements from a group of objects. - A derived value can vary when the measurements used to calculate it vary. - Identify the numerical attribute that would be recorded for each item.

1. Question a) gives fixed dimensions for one rug, so it has one determined area and is non-statistical. 2. Question b) concerns many rugs whose lengths and widths, and therefore areas, may differ, so it is statistical. 3. The area in square feet can vary from rug to rug.

a) Non-statistical b) Statistical; the rug areas can vary.

- Determine how many observations each question is designed to produce. - A proportion can be a variable numerical measurement across repeated occasions. - Name the daily quantity that may change.

1. Question a) asks for one fraction from one specified day, so it is non-statistical. 2. Question b) asks for a fraction from each of many school days and anticipates that those daily fractions may differ, so it is statistical. 3. The fraction or percent of waste that is fruit can vary from day to day.

a) Non-statistical b) Statistical; the daily fraction of waste that is fruit can vary.

- Ask whether the question has one determined mathematical answer. - Identify whether multiple people will produce separate observations. - Focus on anticipated variability rather than the quantity of numerical language.

1. Question a) asks for one fixed mathematical result, so it is non-statistical. 2. Question b) collects one count from each student and anticipates different results, so it is statistical. 3. Statistical classification depends on variability among observations, not on how many numbers appear in the wording.

a) Non-statistical b) Statistical; the number listed correctly can vary among students.

- Determine whether one item or a collection is being examined. - A digit can be treated as a category when its numerical size is not the focus. - Identify the possible responses that may differ across the collection.

1. Question a) asks for one fixed digit from one identified locker number, so it is non-statistical. 2. Question b) asks for one digit from each of many locker numbers and anticipates different digit categories, so it is statistical. 3. The last digit can vary among \(0, 1, 2, \ldots, 9\).

a) Non-statistical b) Statistical; the last-digit categories can vary among lockers.

- Decide whether the category named in the question has one fixed defining property. - Identify whether a collection can contain items with different measurements or counts. - Focus on variability rather than grammar.

1. Question a) asks for a fixed defining property of triangles, so every correct answer is \(3\); it is non-statistical. 2. Question b) asks for side counts from a collection that may include different polygon types, so the counts can vary; it is statistical. 3. The key is anticipated variability in collected observations, not whether the wording uses a plural noun.

a) Non-statistical b) Statistical; the numbers of sides can vary among the mosaic’s polygons.

- Look beneath the final summary to the observations that must be collected. - Identify the possible response categories for one locker. - Decide whether those responses are expected to be identical across the group.

1. Each locker supplies a categorical observation: jacket present or jacket not present. 2. Those observations are expected to vary from locker to locker. 3. The fraction is a summary of variable data, so having one final summary value does not make the question non-statistical.

The student is not correct. The question is statistical because the yes-or-no jacket status can vary among lockers.

- Decide whether the data are a fixed stated list or observations from a group. - Identify what could differ from one member of the group to another. - Treat the requested calculation separately from the nature of the question.

1. Question a) gives a fixed list and asks for one determined calculation, so it is non-statistical. 2. Question b) requires collecting one bottle count from each student, and those counts are expected to vary, so it is statistical. 3. A statistical question is identified by anticipated variability in observations, not by the summary operation used afterward.

a) Non-statistical b) Statistical; bottle counts can vary among students.

- Decide whether a mathematical rule fixes all values in advance. - Compare equal assignment with free choices. - Focus on whether observations can differ, not how many groups are named.

1. Question a) is a fixed equal-division calculation: \(24\div6=4\). It is non-statistical. 2. Question b) asks for six observed team sizes created by student choices, and those sizes can vary. It is statistical. 3. A collection of labels does not create statistical variability when a rule already determines every value.

a) Non-statistical; each team has \(4\) students by the stated rule. b) Statistical; chosen team sizes can vary.

- Count the number of measurements requested by each question. - Separate one event total from measurements across parts of the event. - Identify what could change from one hour to another.

1. Question a) asks for one total measurement from one storm, so it is non-statistical. 2. Question b) asks for one measurement from each hour, and hourly snowfall amounts can vary, so it is statistical. 3. Repeated time intervals create a collection of observations whose distribution can be studied.

a) Non-statistical b) Statistical; hourly snowfall amounts can vary during the storm.

- Count the number of days supplying observations. - Identify the two response categories. - Decide whether the response must stay the same over time.

1. The question requests \(30\) observations across different days. 2. Each observation is categorical: jacket worn or not worn. 3. The response can vary from day to day, so the question anticipates a distribution of two categories. 4. A statistical question can use repeated observations from one person and can have binary responses.

Yes. It is statistical because the jacket status can vary across the \(30\) days.

- Use the graph’s category labels as possible responses. - Make one response correspond to each student. - State the population size shown by the graph.

1. One valid question is, “Which library section did each of the \(40\) students visit first?” 2. Each student supplies one categorical observation. 3. The observation can vary among Fiction, Nonfiction, Graphic Novel, and Poetry. 4. The bar heights summarize the frequencies of those variable categories.

One valid question is: “Which library section did each of the \(40\) students visit first?” The section category can vary among students.

- Name the group supplying observations. - Identify the measured quantity and its unit. - Decide whether all members must have the same measurement.

1. The population is all sixth-grade students included in the project. 2. The attribute is distance from home to school, measured in miles. 3. Distances can vary among students. 4. The question anticipates a distribution of numerical measurements, so it is statistical.

The question is statistical. It studies home-to-school distance, in miles, across sixth-grade students, and those distances can vary.

- Do not confuse uncertainty about one future event with variability in a data set. - Change the question so it refers to repeated observations. - Name the measurement that could differ across those observations.

1. The original question asks about one specified bus trip on one specified day, so it has one eventual answer and is non-statistical. 2. A valid statistical revision must gather delay data from several trips of the route. 3. One valid revision is “How many minutes late was the \(7{:}15\) a.m. bus on each school day this month?” The number of minutes late can vary from day to day.

a) No. It concerns one bus trip and will have one answer. b) One valid revision is: “How many minutes late was the \(7{:}15\) a.m. bus on each school day this month?” The delay in minutes would vary by day.

- Classify the question by what variability was reasonably expected before data were collected. - Separate the design of the question from the particular outcome of one data collection. - Ask whether multiple observations were needed to answer the question.

1. The question concerns a group of \(18\) lockers and asks for a measurement from each one. 2. Before collecting the data, the manager reasonably anticipated that the widths might vary. 3. Therefore, the question is statistical. The fact that all observed values turned out equal does not change what the question anticipated before measurement.

Yes. The question is statistical because it was asked about a group and anticipated possible variability in locker widths. Equal observed results do not make the original question non-statistical.

- Look for wording that asks about a group rather than one identified item. - Identify the numerical attribute whose values would form the data set. - Think about what information is needed to describe a typical value across the group.

1. The question is statistical because it asks for a typical value across a group of graphic novels and anticipates different page counts. 2. The needed data are the page counts of the graphic novels in the library, or of a clearly defined set of them. 3. One book gives only one page count and does not show the variability or center of the collection.

a) Yes. Page counts are expected to vary among graphic novels. b) Collect the page count for each graphic novel in the defined library collection. c) One book cannot show the distribution or support a statement about a typical page count.

- Focus on anticipated variability rather than the form of the answers. - Categorical responses can form a data set. - A numerical answer can still be one fixed fact.

1. Question a) asks several orchestra members for categorical answers that can vary, so it is statistical even though the answers are words or category names. 2. Question b) asks for one fixed numerical fact about one identified violin, so it is non-statistical even though its answer is a number. 3. Therefore, whether answers are numerical does not by itself determine whether a question is statistical.

The statement is false. Question a) is statistical because the instrument categories can vary among members. Question b) is non-statistical because it asks for one fixed count about one violin.

- Count how many observations each question is intended to produce. - Consider whether repeated measurements under similar conditions must be identical. - Focus on variability in the results, not on the number of physical objects.

1. Question a) asks for the result of one identified trial, so it is non-statistical. 2. Question b) asks for \(15\) trial times and anticipates that the results may vary from trial to trial, so it is statistical. 3. A statistical question can concern repeated measurements of the same object when those measurements may vary.

a) Non-statistical b) Statistical Repeated trial times can vary even when the same toy is tested, so they form a data set with anticipated variability.

- Separate the nature of the question from the name of the calculation used in its answer. - Ask whether the data are already fixed or must be collected from varying observations. - Focus on anticipated variability.

1. Question a) gives one fixed data set, so its range is one determined value. It is non-statistical as a question about collecting variable data. 2. Question b) requires collecting several wing-length measurements that are expected to vary, so it is statistical. 3. A question is classified by whether it anticipates variability in data to be collected, not by whether its answer uses a statistic.

a) Non-statistical b) Statistical A calculation such as range can be applied to fixed numbers, but the question is statistical only when it anticipates variable data.

- Distinguish the number of data values collected from the number of values in the final summary. - Consider whether several people can provide different category responses. - A one-word final answer can still summarize a variable data set.

1. Question a) concerns one identified pair of shoes and has one fixed categorical answer, so it is non-statistical. 2. Question b) requires collecting shoe-color categories from many students and anticipates variability among those responses, so it is statistical. 3. The final summary may be one category, but it is based on a variable data set.

a) Non-statistical b) Statistical Question b) requires variable categorical data from a group, even though the final mode is one color.

- Replace a judgment word with an observable response. - Name the group whose data will be collected. - Make the possible answers clear enough to sort into categories.

1. The word “best” does not define a measurable attribute or a common rule for judging responses. 2. A clear statistical version identifies a population and a categorical observation that can vary. 3. One valid revision is, “How does each sixth-grade student usually travel to school: walking, biking, by car, or by bus?” 4. The travel categories can vary among students.

One valid revision is: “How does each sixth-grade student usually travel to school: walking, biking, by car, or by bus?” The revised question is statistical because the way students travel can vary among students.

- Make one version point to exactly one identified item. - Make the other version name a collection of items. - State what measurement could vary across that collection.

1. One non-statistical version is, “How long is the song ‘Morning Bell’?” It asks for one fixed duration. 2. One statistical version is, “How long is each song on the class playlist?” It asks for a set of durations that can vary. 3. The specific population and number of observations determine whether variability is anticipated.

Non-statistical: “How long is the song ‘Morning Bell’?” Statistical: “How long is each song on the class playlist?” The first has one fixed duration; the second collects varying durations from many songs.

- Identify how many observations the question requests. - Compare the possible day counts across the months. - Separate variability from randomness or uncertainty.

1. The question asks for \(12\) observations, one day count for each month. 2. The counts are not all the same: months can have \(28, 29, 30\), or \(31\) days. 3. The values are fixed and known, but they vary across the collection. 4. Statistical classification depends on variability in the observations, not on randomness or lack of knowledge.

Yes. It is statistical because the numbers of days vary among the months, even though the calendar values are fixed and known.

- Replace a one-day snapshot with repeated observations. - Keep the measurement times consistent across days. - State what numerical value is recorded each time.

1. Count the bicycles in the racks at the same chosen time before school and the same chosen time after school on each of several school days. 2. Each observation is a bicycle count paired with a time period and day. 3. Counts can vary from day to day and between the two time periods. 4. Repeated observations allow the school to compare the two distributions rather than relying on one day.

Count bicycles before and after school at consistent times over several days. The counts are expected to vary across days and between the two time periods.

- Count the number of measurements that will be recorded. - Consider whether physical trials can differ slightly despite a common setup. - Distinguish reducing variability from eliminating the need to study it.

1. The question requests \(15\) melting-time measurements. 2. Small differences in cube size, tray contact, air movement, or measurement can cause the times to vary. 3. The question anticipates a distribution of melting times, so it is statistical. 4. Controlling conditions reduces unwanted differences but does not change the question into a single fixed calculation.

The question is statistical because the \(15\) measured melting times can vary even under controlled conditions.

- Decide whether the answer comes from a fixed model or repeated observed data. - Identify what is recorded for each set of trials. - Separate a constant theoretical value from variable experimental results.

1. Question a) asks for a fixed probability based on six equally likely outcomes, so it is non-statistical. 2. Question b) asks for a count from each repeated set of rolls, and those counts can vary from set to set, so it is statistical. 3. A fixed probability describes the model, while experimental counts form a variable distribution.

a) Non-statistical b) Statistical; the count of \(6\)s can vary among sets of \(30\) rolls.

- Identify whether the question refers to a schedule or repeated events. - Do not classify a question from one vocabulary word. - State the observations and time period explicitly in the revision.

1. If the question asks for the school’s scheduled opening time, it has one fixed answer and is non-statistical. 2. The word “usually” is vague and does not identify observations to collect. 3. A clear statistical revision is, “At what time did the school doors actually open on each of the last \(20\) school days?” 4. Actual opening times can vary from day to day.

The scheduled opening-time interpretation is non-statistical. A statistical revision is: “At what time did the school doors actually open on each of the last \(20\) school days?”

- Choose an attribute that can reasonably take the listed whole-number values. - Name the group that supplies the eight observations. - Make one value correspond to each person.

1. One valid question is, “How many pets does each of these eight classmates have?” 2. The measured attribute is a count of pets for each classmate. 3. The data show several different counts, and the question anticipates that classmates may have different numbers of pets. 4. Other contexts using a reasonable whole-number count per classmate are also possible if the population and attribute are stated clearly.

One possible question is: “How many pets does each of these eight classmates have?” The attribute is number of pets, which can vary among classmates.

- Identify what the responses mean before looking at their codes. - Ask whether differences between code numbers represent measured differences. - Separate statistical variability from the type of variable collected.

1. The question is statistical because travel categories can vary among students. 2. The numbers are labels for categories, not measurements of amount. 3. Arithmetic such as averaging the codes would not have a meaningful travel-unit interpretation. 4. Coding changes how responses are stored, not the categorical nature of the data.

The question is statistical, but the variable remains categorical. The codes \(1,2,3,4\) are labels, not numerical measurements.

- Keep the measured attribute, unit, and timing consistent. - Change only the group supplying the observations. - Consider how group characteristics could affect the collected values.

1. One valid student question is, “How many pounds does each sixth-grade student’s backpack weigh at the start of school today?” 2. One valid teacher question is, “How many pounds does each teacher’s backpack weigh at the start of school today?” 3. Both collect a numerical weight using the same unit and time condition. 4. The groups may carry different materials and amounts, so their centers, variability, and shapes can differ.

Student question: “How many pounds does each sixth-grade student’s backpack weigh at the start of school today?” Teacher question: “How many pounds does each teacher’s backpack weigh at the start of school today?” The distributions may differ because the populations may carry different items.

- Identify one observation produced by each question for one student. - Decide whether responses can differ across students. - Distinguish category identities from counts of categories.

1. Question a) collects club-name categories, which can vary among students, so it is statistical. 2. Question b) collects a numerical count for each student, which can also vary, so it is statistical. 3. The first studies which categories occur, while the second studies the number of selected categories per student.

Both questions are statistical. a) studies variation in club categories. b) studies variation in numerical club counts.

- Look at the observations needed before the final conclusion can be stated. - Separate the form of the final answer from the variability of the underlying data. - Identify the measured attribute for each object.

1. Each pencil must be measured, producing one length observation per pencil. 2. The lengths may vary, even though the final conclusion is a single yes-or-no statement. 3. The question anticipates a distribution of measurements and asks whether all observations meet one condition. 4. Therefore, it is statistical.

The student is incorrect. The question is statistical because the pencil lengths can vary and must be measured across the whole box.

- Identify the observations that must be collected before calculating the sum. - Decide whether those observations are expected to match. - Separate the number of final answers from the number of data values.

1. The height of each chorus student must be measured. 2. Those height observations can vary among students. 3. The final sum is a summary calculated from a variable data set. 4. A question can be statistical even when it requests one summary value.

The statement is incorrect. The question is statistical because it requires variable height measurements from all chorus students before the total can be found.

- Identify when the measured attribute might change. - Decide how many observations are needed to study variability. - Keep measurement conditions consistent enough for meaningful comparison.

1. The question concerns sound level, which can vary by time and day. 2. One reading produces no distribution and cannot show typical level or variability. 3. A stronger plan records sound level in the same unit at several fixed times during lunch on multiple school days. 4. The observations should use the same measurement method and clearly defined cafeteria location.

One reading is insufficient. Record repeated sound-level measurements at consistent times and locations across several lunch periods.

- Identify the observation recorded for each student. - Decide whether those category responses can differ. - Separate the final summary from the data used to produce it.

1. Each student supplies a home-language category. 2. Those categories can vary among students. 3. The number of distinct categories is a summary found after collecting the variable categorical data. 4. Requesting one summary count does not make the underlying question non-statistical.

Yes. It is statistical because home-language categories vary among students and must be collected before the number of distinct categories can be counted.

- Name both groups being compared. - Replace “consistent” with a measurable attribute whose spread can be studied. - Use matching sample conditions for the two groups.

1. One valid revision is, “What are the weights, in ounces, of \(30\) snack packets selected from Machine A and \(30\) selected from Machine B during the same shift?” 2. The populations are packets produced by the two machines. 3. The attribute is packet weight in ounces. 4. Equal sample sizes and the same shift create a reasonable basis for comparing variability.

One valid revision is: “What are the weights, in ounces, of \(30\) snack packets selected from Machine A and \(30\) selected from Machine B during the same shift?”

- Name a group of observations rather than one particular person. - Make the quantity measurable with a stated unit. - Keep the observation period consistent so the data answer one clear question.

1. One valid question is, “How many seconds do pedestrians wait to cross at this intersection during the morning arrival period?” 2. One observation is the waiting time, in seconds, for one pedestrian. 3. A consistent plan is to record every selected pedestrian during the same defined morning time window on several school days. 4. The individual waiting times are expected to vary, so the question is statistical.

One valid response is: “How many seconds do pedestrians wait to cross at this intersection during the morning arrival period?” Record one waiting time in seconds for each selected pedestrian during the same defined time window on several days.