Classifying variables practice problems

Run step 1 on all eight: would adding or averaging the values give a number that measures something? Run step 2 only on the ones that survive step 1.

(a) Zip code. An average zip code points to no location, and 90210 is not more of anything than 10001. The digits are an address label, so this is categorical and step 2 does not apply.

(b) Number of siblings. An average number of siblings is a real quantity, so it is quantitative. The possible values are 0, 1, 2, 3, and onward, listable with nothing possible between 2 and 3, so it is discrete.

(c) Height. An average height is meaningful, so it is quantitative. Between any two heights another height is possible, so it is continuous. A ruler reading to the nearest centimeter does not change that.

(d) Blood type. O, A, B, and AB are names with no order and no average, so this is categorical.

(e) Time to run 400 meters. An average time is meaningful, so quantitative, and elapsed time fills an interval, so continuous.

(f) Number of items in a basket. A count you can average, so quantitative, and the possible values are 0, 1, 2, and onward, so discrete.

(g) Letter grade. A, B, C, D, and F are category labels. They carry an order, which is more than blood type has, but there is no measured distance between a B and a C, so this is categorical.

(h) Daily rainfall. Depth of water is measured and averaged, so quantitative, and it fills an interval, so continuous. Many days record 0.0 mm, and a pile of zeros is a fact about the weather, not evidence that the variable is discrete.

(a) Jersey number is a name written with digits: it identifies a player and measures no quantity, so it is categorical. Goals scored counts a quantity you can add, so it is quantitative, and its possible values are 0, 1, 2, and onward, so it is discrete.

(b) Jersey numbers: the sum is $4 + 7 + 9 + 12 + 18 + 22 = 72$, so the mean is $\frac{72}{6} = 12$. Goals: the sum is $0 + 1 + 1 + 2 + 3 + 5 = 12$, so the mean is $\frac{12}{6} = 2$.

(b) Only the second is a summary of anything. "This team averages 2 goals per player" describes the season. "This team averages jersey number 12" describes nothing, and the fact that one player does wear 12 is a coincidence of these six labels: move that jersey to 13 and the mean becomes $\frac{73}{6} \approx 12.17$, a number nobody wears, without changing anything about the team.

(c) After renumbering, the jersey sum is $1 + 2 + 3 + 4 + 5 + 6 = 21$, so the mean jersey number is $\frac{21}{6} = 3.5$. The goals did not change, so the mean goals is still $\frac{12}{6} = 2$.

(d) The mean jersey number moved from 12 to 3.5 while not one fact about the six players changed, and the mean goals stayed at exactly 2. That is the operational meaning of "arithmetic on the values is meaningless": a summary that moves when someone relabels the categories was never measuring the players.

One more distinction worth naming. Jersey number here is functioning as an identifier, the way the Player row does, so it labels the rows rather than describing them. Identifiers are categorical when they are treated as variables at all, and they are usually not graphed or summarized.

The test is not what the recorded numbers look like. Ask what set of values the quantity could take, then ask whether that set can be listed one value at a time.

(a) Age. The quantity behind the record is elapsed time, which fills an interval: between 15 years and 16 years every age in between is possible and real. The form rounds it, and rounding is the instrument. Age is continuous.

(b) Proportion of 20 free throws made. The player makes some whole number $k$ of shots from 0 to 20, so the proportion can only be $\frac{k}{20}$: the 21 values $0, 0.05, 0.10, \ldots, 1.00$. Nothing at 0.07 is possible. Decimals throughout and still discrete, because those 21 values can be listed one at a time with nothing possible in between. Shortness is not the test, listability is: a list of 21 values and a list that never stops are both lists.

(c) Volume of juice. Volume fills an interval, so between any two fill levels another is possible. Continuous.

(d) Number of defective bulbs. The possible values are the 25 whole numbers 0 through 24, a finite list. Finite counts as listable, so this is discrete. Discrete does not require the list to be endless.

(e) Price. A price is a whole number of cents, so its possible values are countable and strictly it is discrete. It is routinely modeled as continuous, because a cent is tiny next to typical prices, and either answer is defensible as long as you say which one you are using and why. Note that (b) and (e) are not the same situation for that modeling choice, though both are discrete under the listability test: 21 values spaced 0.05 apart are nowhere near a continuum, while the possible prices under 20 dollars run into the thousands, spaced a cent apart, which is close enough to one that treating them as continuous costs almost nothing.

(f) Leaf length. Length fills an interval, and recording to the nearest centimeter is a choice about the ruler. Continuous.

Read (a) and (b) together, because they break the two halves of the same false rule. "Whole numbers mean discrete" fails on age, which is written as an integer every time and is continuous. "Decimals mean continuous" fails on the free throw proportion, which is decimal every time and is discrete.

(a) The five ratings are category labels with a real order and no measured distance between neighbors: nothing in the data says the step from 3 to 4 is the same size as the step from 4 to 5. So the rating is a categorical variable with ordered categories. Wider statistics calls this an ordinal variable; the AP course sorts every variable into categorical or quantitative, and this one lands on categorical.

(b) Check both counts first: $6 + 4 + 5 + 10 + 15 = 40$ this year and $2 + 8 + 12 + 11 + 7 = 40$ last year.

(b) This year, using codes 1 to 5: $6(1) + 4(2) + 5(3) + 10(4) + 15(5) = 6 + 8 + 15 + 40 + 75 = 144$, so the mean code is $\frac{144}{40} = 3.60$. Last year: $2(1) + 8(2) + 12(3) + 11(4) + 7(5) = 2 + 16 + 36 + 44 + 35 = 133$, so the mean code is $\frac{133}{40} = 3.325$. The difference is $3.60 - 3.325 = 0.275$. The arithmetic is right.

(c) Now use the codes 1, 2, 3, 4, 10, which keep the categories in exactly the same order. This year: $6(1) + 4(2) + 5(3) + 10(4) + 15(10) = 6 + 8 + 15 + 40 + 150 = 219$, so the mean is $\frac{219}{40} = 5.475$. Last year: $2(1) + 8(2) + 12(3) + 11(4) + 7(10) = 2 + 16 + 36 + 44 + 70 = 168$, so the mean is $\frac{168}{40} = 4.20$. The improvement is now $5.475 - 4.20 = 1.275$.

(c) Nobody's rating changed. The improvement is 0.275 under one order-preserving coding and 1.275 under another, so "0.275 points" is a fact about the numbers someone assigned to the labels, not about the service. The direction happened to survive here; it is not guaranteed to, and a coding change can reverse a comparison outright.

(d) The median category. With 40 ratings the median sits between the 20th and 21st values in order. This year the cumulative counts are 6, 10, 15, 25, 40, so positions 16 through 25 are all rating 4 and the median category is 4. Last year the cumulative counts are 2, 10, 22, 33, 40, so positions 11 through 22 are all rating 3 and the median category is 3. Both are read off the counts and the ordering alone, so no order-preserving recoding can move either one.

(d) The share rating 4 or 5. This year $\frac{10 + 15}{40} = \frac{25}{40} = 0.625$, and last year $\frac{11 + 7}{40} = \frac{18}{40} = 0.45$. That is a rise of $0.625 - 0.45 = 0.175$, or 17.5 percentage points. This uses only counts, so it too is immune to recoding.

(e) Report the median category rising from 3 to 4 and the share rating 4 or 5 rising from 45% to 62.5%. Both are true of the customers rather than of the code sheet. Averaging ordinal codes is common practice, and the honest version of it states the assumption out loud: that the five steps are being treated as equal in size, which nothing in the survey measured. Display the counts with a bar graph, keeping the bars in rating order, not a histogram.

(e) One caution on scope: these are two samples of 40 customers, and nothing here says how they were selected, so neither number is being offered as an estimate for all of the cafe's customers.

(a) One row of this data set is one dog, so the observational unit is a dog seen at this clinic last year, and $n = 800$. It is not a visit: a dog that came four times still contributes one row, and its four visits are a value inside that row.

(b) Breed is a group label with no order and no average, so categorical. Weight in kilograms is a measured quantity that fills an interval, so quantitative and continuous. Number of visits is a count you can average with possible values 1, 2, 3, and onward, so quantitative and discrete. Rabies vaccination is a yes-or-no label, so categorical.

(c) The 73% is a summary statistic computed from the vaccination column: $\frac{584}{800} = 0.73$. A variable is a characteristic that varies from dog to dog, and 73% varies from nothing, because there is only one of it for this data set. The variable is "vaccinated for rabies, yes or no", and 73% is the proportion of one of its two categories.

(c) The same number could be a variable in a different data set. Record the percent vaccinated at each of 60 clinics and the observational unit becomes a clinic, $n = 60$, and percent vaccinated is then a quantitative variable measured on each one. What a number is depends on the unit it was collected from.

(d) The 18.4 kg is the mean of the weight column, and the weight column is not "a dog's weight" in general. It is the weight recorded at that dog's first visit of the year, one value per dog. It says nothing about how a dog's weight changed across the year, and it describes only the 800 dogs this clinic saw, not dogs in the city or dogs of any particular breed.

(e) The student has used the wrong test. "Could be any number" is about how many values there are, and discrete variables can have endlessly many values: 1, 2, 3, and onward never stops. The test is whether the possible values can be listed one at a time with nothing possible between neighbors. A dog cannot make 2.6 visits, so the values are listable and the variable is discrete. Continuous would require that between any two possible values another one is possible, which is true of the weight column and false of the visit count.

(a) Question A records minutes, an amount you can average, so it produces a quantitative variable, and elapsed time fills an interval, so it is continuous. Question B records one of two labels, so it produces a categorical variable, with two categories.

(b) For A, report a center and a spread, such as a mean and a standard deviation or a median and an interquartile range, and graph it with a dotplot, stemplot, histogram, or boxplot, because those place values on a number line. For B, report counts and a proportion, and graph it with a bar graph or a pie chart. Putting B's two categories on a histogram would claim its axis was a number line; see histograms vs bar graphs.

(c) A converts to B. Two hours is 120 minutes, so go down the minutes column, mark every student above 120 as a yes, and you have a categorical column: $\frac{189}{300} = 0.63$. The conversion works because the categorical variable is defined by a cut point on the quantitative one.

(c) B does not convert to A. A yes tells you a student was above 120 minutes and nothing more, so no mean, median, or standard deviation of minutes can be recovered from it. Collapsing a quantitative variable to a category throws away information permanently, which is the practical reason to collect the finer measurement when you can.

(d) The gap is $216 - 189 = 27$ students, or $\frac{27}{300} = 0.09$ of the group. Since the same 300 students answered both questions, this is not sampling variability and not two different groups. At least 27 students gave a pair of answers that contradict each other, and possibly more, since a student who said yes while reporting 90 minutes and a student who said no while reporting 150 minutes partly cancel in that difference of 27. The likely reason is that the two questions do not mean the same thing to a student: "more than 2 hours" invites a rough judgment, while a minute count invites an estimate the student has to construct.

(d) State what it does not tell you. Nothing here says which of the two numbers is closer to the truth. It says the wording of the question is part of what got measured, so the variable is defined by the instrument that produced it and not only by the behavior it is aimed at.

Check the totals before judging anything: $48 + 24 + 36 + 12 = 120$, so all 120 students are accounted for, and the proportions are $\frac{48}{120} = 0.40$, $\frac{24}{120} = 0.20$, $\frac{36}{120} = 0.30$, and $\frac{12}{120} = 0.10$, which add to 1.

(i) Wrong word and wrong idea. Favorite lunch is categorical, so it has no mean at all: you cannot add pizza to salad. What the student means is the mode, the most common category, and pizza is indeed the mode at 48 students, or 40%. Corrected: "the most common favorite lunch is pizza, chosen by 48 of the 120 students."

(ii) The reasoning confuses the tally with the thing tallied. The variable is favorite lunch, and its values are pizza, salad, sandwich, and other, which are labels. The numbers 48, 24, 36, and 12 are frequencies, and a frequency is always a number no matter what is being counted, so its being numeric proves nothing about the variable. The type belongs to what is counted; see count data. Favorite lunch stays categorical.

(iii) Wrong graph. A histogram puts values into intervals on a number line, and its bars touch because the intervals touch. Pizza, salad, sandwich, and other are not intervals of anything and have no width, so the correct display is a bar graph with separated bars, or a pie chart of the four proportions.

(iv) The conclusion is right and the reason is wrong, which on a free-response question earns nothing. Commute time is continuous because elapsed time fills an interval, not because a mean came out with a decimal in it.

(iv) Counterexample, which is how to kill this reasoning outright: five students report 0, 1, 1, 2, and 3 siblings, so the mean is $\frac{0 + 1 + 1 + 2 + 3}{5} = \frac{7}{5} = 1.4$ siblings. Number of siblings is plainly discrete, and no student has 1.4 siblings. A mean does not have to be one of the possible values, so the shape of a mean says nothing about the type of the variable.

(a) One row is one swimmer entering on that Saturday, so the observational unit is a swimmer entry and $n = 1800$. It is not a pool: there are only 5 pools, and a pool is a value of one of the columns, not a row.

(b) Age in whole years: quantitative and continuous. It is written as an integer because the form asks for whole years, but the quantity behind it is elapsed time, which fills an interval. Party size: quantitative and discrete, since parties come in 1, 2, 3, and onward with nothing between. Membership status: categorical, two labels. Pool entered: categorical, five labels. Minutes stayed: quantitative and continuous, since time fills an interval.

(c) The mean stay of 74.6 minutes is a measure of center for the minutes column, a quantitative variable. The mean party size of 2.3 people is a measure of center for the party size column, also quantitative. The 62% is a proportion, which along with counts and the mode is the kind of summary a categorical variable allows, and it comes from the membership column: $\frac{1116}{1800} = 0.62$.

(c) Note that 2.3 is not a possible party size, and that is fine. A mean of a discrete variable need not be one of its values, exactly as in problem 7.

(d) The staffer has classified the labels, not the variable. Numbering the pools 1 through 5 assigns names written with digits, the same move as a jersey number in problem 2. A mean pool number of 3.1 identifies no pool and would change the moment the pools were renumbered, while not one swimmer's day would be different. Pool entered is categorical, and the summaries available are counts and proportions per pool, displayed with a bar graph.

(e) Name the population before answering. If the population of interest is the swimmers who entered these 5 pools on that Saturday, then all 1,800 of them were measured, this is a census of that group, and 62% is a parameter. If the department means all swimmers across the season, those 1,800 are a sample from it, and 62% is a statistic.

(e) Under the second reading, add the warning the design earns: the Saturday was picked for convenience rather than at random, and weekend swimmers plausibly buy day passes more often than weekday swimmers do, so 62% may sit above the season-long share. Which population you name decides both the word and whether the number can be trusted for it.