Name:
Date:
Instructor:
How to use this test. This test covers
describing data — reading frequency tables and histograms, and computing and choosing between the mean, median, mode, and range. Choose the
best answer and write its letter on the answer line.
- Every question has a Nudge — a hint about how to think it through, never the answer itself. Try the question first; read the nudge only if you get stuck.
- For table and histogram questions, point to the exact row or interval you are using before you calculate.
- When computing a median, always sort (or confirm the sort of) the data first.
- The big idea of this test: the mean and the median can tell very different stories about the same data. Watch for when and why.
- After each answer, jot your reasoning on the Show your work / why lines. A calculator is allowed.
STIMULUS 1 · FREQUENCY TABLEReading & Relative Frequency
Weekly Study Hours — Survey of 40 College Students
A survey asked 40 students how many hours they studied per week. Use the table to answer Questions 1–6.
Key concepts: Frequency = how many are in each group | Relative frequency = group count ÷ total | Mode of grouped data = the interval with the highest count
Table 1. Weekly study hours reported by 40 students
| Study Hours per Week | Number of Students |
|---|
| 0–4 hours | 6 |
| 5–9 hours | 12 |
| 10–14 hours | 14 |
| 15–19 hours | 5 |
| 20–24 hours | 3 |
Total: 40 students.
1.How many students studied 10 or more hours per week?
- A. 18 students
- B. 14 students
- C. 22 students
- D. 26 students
Nudge — Decide which rows count: every interval that starts at 10 or higher. Add the student counts from just those rows.
My answer:
2.What percent of the students studied 5–9 hours per week?
- A. 25%
- B. 30%
- C. 33%
- D. 12%
Nudge — Percent = (count in that interval ÷ total students) × 100. Remember the total is 40, not the interval count.
My answer:
3.Which interval is the mode (the most frequent study range)?
- A. 0–4 hours
- B. 5–9 hours
- C. 10–14 hours
- D. 15–19 hours
Nudge — The mode of grouped data is about frequency, not size. Scan the count column for the largest number and read across to its interval.
My answer:
4.How many students studied fewer than 10 hours per week?
- A. 18 students
- B. 12 students
- C. 22 students
- D. 6 students
Nudge — “Fewer than 10” covers only the intervals below 10. Identify which rows those are, then add their counts.
My answer:
5.What is the relative frequency of the 20–24 hour interval, expressed as a percent?
- A. 3%
- B. 0.3%
- C. 12%
- D. 7.5%
Nudge — Relative frequency = interval count ÷ total. Work out the decimal first, then move the decimal to convert it to a percent.
My answer:
6.Which statement is best supported by the table?
- A. Most students studied at least 15 hours per week.
- B. Exactly half of the students studied between 5 and 14 hours.
- C. Four out of five students studied fewer than 15 hours per week.
- D. The most common study range was 5–9 hours.
Nudge — Test each statement against the actual counts before choosing. “Most” means more than half — check the arithmetic for every claim rather than trusting a first impression.
My answer:
STIMULUS 2 · DISTRIBUTIONShape, Center & Position
Daily Customers at a Café — 30-Day Month
A café recorded how many customers it served each day over a 30-day month. The frequency table below groups the days by customer count. Use it for Questions 7–12.
Key concepts: The count in each row = number of days in that range | Shape is named for the direction of the tail, not the peak | The median’s position: with 30 values it lies between the 15th and 16th
Table 2. Daily customer counts over 30 days
| Customers per Day | Number of Days |
|---|
| 40–49 customers | 9 |
| 50–59 customers | 8 |
| 60–69 customers | 6 |
| 70–79 customers | 4 |
| 80–89 customers | 3 |
Total: 30 days. (These are the exact bar heights from the original histogram.)
7.On how many days did the café serve between 60 and 69 customers?
- A. 4 days
- B. 6 days
- C. 8 days
- D. 9 days
Nudge — Read the count straight from the 60–69 row of the table. Make sure you are on the right interval before writing anything down.
My answer:
8.On how many days did the café serve 70 or more customers?
- A. 3 days
- B. 4 days
- C. 7 days
- D. 13 days
Nudge — “70 or more” spans two intervals. Identify both of them, then add their day counts together.
My answer:
9.Which best describes the shape of this distribution?
- A. Skewed right — most days cluster at lower customer counts, with a tail stretching toward higher values.
- B. Skewed left — most days cluster at higher counts, with a tail toward lower values.
- C. Symmetric — the left and right sides mirror each other.
- D. Uniform — every interval occurred about equally often.
Nudge — Find where the large counts cluster and which direction the “tail” of smaller counts trails off. Remember the skew is named after the direction of the tail, not the peak.
My answer:
10.Which interval contains the median of the 30 days?
- A. 40–49
- B. 50–59
- C. 60–69
- D. 70–79
Nudge — With 30 days the median sits between the 15th and 16th values in order. Build a running (cumulative) total down the rows and see which interval the 15th and 16th days fall into.
My answer:
11.Approximately what percent of days had fewer than 60 customers?
- A. 43%
- B. 50%
- C. 57%
- D. 63%
Nudge — “Fewer than 60” means the two lowest intervals. Add their counts, divide by 30 days, then convert to a percent.
My answer:
12.In a right-skewed distribution like this one, how does the mean typically compare to the median?
- A. The mean is less than the median.
- B. The mean and median are always exactly equal.
- C. The mean and median have no consistent relationship.
- D. The mean is greater than the median, because the high-value tail pulls the mean upward.
Nudge — The mean feels every value, including the few large ones out in the tail; the median only cares about the middle position. Ask which measure the tail is able to drag.
My answer:
STIMULUS 3 · MEAN vs. MEDIANThe Effect of an Outlier
Salaries at Brightline Design Co.
The eight annual salaries at a small design company, in thousands of dollars, already sorted:
42 45 48 50 52 55 58 260
The $260K salary belongs to the owner. Use this data for Questions 13–18.
Key concepts: Median of 8 values = average of the 4th and 5th | An outlier pulls the mean toward it but barely moves the median | “Typical” is a judgment call — choose the measure that honestly represents most of the data
13.What is the median salary at the company (in thousands)?
- A. $51
- B. $50
- C. $52
- D. $76.25
Nudge — With 8 salaries the median is the average of the two middle values (the 4th and 5th). The list is already sorted, so locate those two positions.
My answer:
14.What is the mean salary at the company (in thousands)?
- A. $51
- B. $61
- C. $76.25
- D. $87.1
Nudge — Add all eight salaries, then divide by 8. Be sure the owner’s $260K is included in your total and that you divide by every employee.
My answer:
15.Which measure better represents what a typical employee at this company earns, and why?
- A. The median, because the owner's $260K salary pulls the mean far above what most employees actually earn.
- B. The mean, because it uses every salary in the calculation.
- C. The mean, because it is always the more accurate measure of center.
- D. Neither — the mode should always be used for salary data.
Nudge — Seven of the eight salaries sit between $42K and $58K. Ask which measure of center lands inside that everyday range and which gets dragged far outside it by one value.
My answer:
16.If the owner's $260K salary is removed, what is the mean of the remaining 7 salaries (in thousands)?
- A. $44
- B. $50
- C. $51
- D. $76.25
Nudge — Take the sum of all eight salaries, subtract the $260K owner’s salary, then divide by the new count of 7.
My answer:
17.When the owner's salary is removed, which statement correctly describes what happens to the two measures of center?
- A. The median changes more than the mean.
- B. Both measures change by about the same amount.
- C. Neither measure changes.
- D. The mean drops dramatically while the median barely moves.
Nudge — Recompute the mean and the median without the $260K value, then compare how far each one shifted from its original value. Which measure is far more sensitive to that one extreme salary?
My answer:
18.The company hires a new employee at $49K. Including the owner, what is the new median of all 9 salaries (in thousands)?
- A. $49
- B. $49.5
- C. $50
- D. $51
Nudge — Insert $49K into the sorted list so you have 9 values. With an odd count, the median is the single middle value — find its position after re-sorting.
My answer:
STIMULUS 4 · CORE SKILLSMean, Median, Mode & Range
Computing Every Measure from One Data Set
Six scores from a class quiz, already sorted:
4 6 6 7 9 16
This one small data set has a different mean, median, and mode — Questions 19–24 ask you to find each and explain why they differ.
Key formulas: Mean = sum ÷ count | Median = middle value (average the two middles if the count is even) | Mode = most frequent value | Range = largest − smallest
19.What is the mean of the data set?
Nudge — Add all six values to get the total, then divide that sum by 6.
My answer:
20.What is the median of the data set?
Nudge — The data is already sorted. With 6 values there is no single middle, so average the two middle values (the 3rd and 4th).
My answer:
21.What is the mode of the data set?
Nudge — The mode is the value that appears most often, not the largest value. Count how many times each number repeats.
My answer:
22.What is the range of the data set?
Nudge — Range measures spread: subtract the smallest value from the largest. The maximum by itself is not the range.
My answer:
23.The mean of this data set (8) is noticeably larger than its median (6.5). What best explains why?
- A. The unusually large value (16) pulls the mean upward but has little effect on the median.
- B. The mean is always larger than the median in every data set.
- C. The median must have been calculated incorrectly.
- D. The mode of 6 forces the median to be lower than the mean.
Nudge — Picture removing the 16 for a moment: would the mean and median still be far apart? Use that to decide which value is doing the pulling, and on which measure.
My answer:
24.If the value 16 is replaced with 10, what is the new mean?
Nudge — Swapping 16 for 10 lowers the total. Recompute the new sum, then divide by the same count of 6 — every value participates in the mean, so it must change.
My answer:
Constructed Response — Which Rent Number Should You Trust?
Choosing the honest measure · show all work.
Scenario. A housing counselor is comparing monthly rents (in dollars) for five available apartments in each of two neighborhoods.
Riverdale: 1,200 1,250 1,300 1,350 1,400
Oakwood: 1,100 1,150 1,200 1,250 3,800
Show all calculations. Label which number is the mean and which is the median at every step.
Nudge — Plan before you write. (1) For each neighborhood, compute both the mean and the median — sort first, then average the middle value for the median and add-and-divide for the mean. (2) Notice where the mean and median agree and where they disagree, and look for the value causing the gap. (3) In Part C, let that comparison decide which measure is the honest one for each neighborhood. (4) In Part D, back your explanation with the specific numbers you calculated, not just the word “misleading.”
Part A — Compute. Calculate the mean and the median rent for Riverdale. Show your work.
Part B — Compute. Calculate the mean and the median rent for Oakwood. Show your work.
Part C — Decide. The counselor wants one number per neighborhood to describe a “typical” rent. Which measure should be reported for each neighborhood? Justify your choice using the data.
Part D — Interpret. A rental website advertises: “Average rent in Oakwood: $1,700.” In 1–2 sentences, explain to an apartment hunter why this number is misleading, using what you calculated.
Reflection
Answer honestly — this helps identify what to work on next.
1. In your own words: when should someone report the median instead of the mean? Give one real-life example.
2. Which was harder for you — reading values off the table, or deciding which measure of center to use? Why?
3. Pick one question you got wrong or guessed on. Explain what the correct approach should have been.
— End of Student-Facing Test —
FOR INSTRUCTOR USE — NOT FOR DISTRIBUTION
Mentor Guide
MA 321 · Test 2 — Describing Data | how to review this test and score the constructed response
1. Reviewing this test without an answer key
This test ships without a printed answer key on purpose. The point of review is not a score — it is to understand how the student is thinking. Work each missed item with the student, reconstructing the method rather than reading off a letter.
- Ask first, tell second. Before confirming anything: “Walk me through how you got this.” Their reasoning is your diagnostic.
- Return to the data. For every miss, go back to the table or data set and locate the evidence together: “Which row? Which number?”
- Correct answers still need explanation. An unexplained right answer is a finding, not a pass — a right answer for the wrong reason is a future wrong answer.
- Name the specific misconception — “you picked the interval with the biggest values instead of the biggest count” — not just “review mode.”
- One central concept: the mean absorbs the size of every value; the median only counts positions. If a student misses Q15, Q17, and Q23, treat it as one gap and teach it once, deeply.
2. What the nudges are (and how to use them)
Each question carries a Nudge — a printed hint about how to attack the item (which rows to add, sort before you find the median, name the skew by the tail). A nudge never states or implies the answer; a stuck student gets unstuck by reasoning, not by being told.
- Open (first exposure): student reads the nudge freely whenever stuck — best for a first pass.
- Delayed (default): student attempts first, reads the nudge only if they cannot start. Have them tick which questions needed a nudge — that tick list is your diagnostic.
- Closed (exam simulation): cover the nudges, then review them afterward while going over reasoning. Use once the skills are solid.
3. What each section tests & the common misconceptions
Q1–6 — Frequency table
- Q1/Q4 are complements (22 + 18 = 40). One right, one wrong usually means a mis-grouped interval — have them bracket the rows in pencil before adding.
- Q2/Q5 percent errors are almost always “forgot the total is 40.” If they answer 12% on Q2, they read a raw count as a percent — ask “12 out of how many?”
- Q3 classic mode error: picking the largest hours (20–24) instead of the largest count. Mode = most popular, not biggest.
- Q6 require arithmetic on all four claims, not intuition — this is claim-verification practice.
Q7–12 — Distribution (table converted from a histogram)
- Q7/Q8 pure row-reading; Q8 spans two intervals — have them circle both before adding.
- Q9 most common error is naming the skew after the peak instead of the tail: peak is on the left, tail stretches right, so it is skewed right.
- Q10 median-by-cumulative-count is new to most. Build the running total together: 9, 17, 23, 27, 30 — then find where days 15 and 16 land.
- Q11 the “complement” trap (days at 60+) gives the right arithmetic on the wrong group — a reading error, not a math error.
- Q12 connect forward to Stimulus 3: the long right tail does to the mean exactly what the owner’s salary does in the next section.
Q13–18 — Salaries (outlier effect, the heart of Chapter 3)
- Q13/Q20 even-count median: average the two middle values, do not take just one. If both are missed, that is a confirmed pattern — drill the even-count rule.
- Q14 dividing by 7 instead of 8, or reporting the median, are the two usual slips.
- Q15 Option B (“the mean uses every value”) is true as a fact but wrong as a conclusion — acknowledge the true part, then ask whether using every value helps when one is $260K.
- Q16/Q17 a before/after experiment. Have the student build a two-row table (mean and median, with-owner vs. without) to make the sensitivity difference visceral.
- Q18 tests re-sorting after an insert and the switch from even to odd count.
Q19–24 — Core skills
- Q19–22 four measures from one data set. Swapping two (e.g. mode reported as median) means the procedures are fine but the vocabulary is not — have them write one-line definitions from memory, then retry.
- Q21 choosing 16 confuses “most frequent” with “largest” — same misconception as Q3.
- Q23 the interpretation item: a student who computed Q19/Q20 but misses Q23 can calculate but not explain — the more important gap for MA 321.
- Q24 a what-if; assuming the mean does not change is the error to name.
4. Constructed Response — how to score
Grading principles. Parts A–B are computation; C–D are judgment and communication — weight them equally, because MA 321 grades the writing. If A or B has an arithmetic slip but the right method, trace it through C and D and credit consistent reasoning. Score holistically.
Expected results: Riverdale mean $1,300 / median $1,300 (they agree, so either works); Oakwood mean $1,700 / median $1,200 (they disagree because of the $3,800 outlier, so the median is the honest choice). Part D should note the $1,700 ad is technically correct but unrepresentative — four of five apartments cost $1,250 or less, so a hunter should expect about $1,200.
| Score | Level | What this looks like |
|---|
| 4 | Advanced | All four means/medians correct. Part C chooses the median for Oakwood with the outlier named explicitly and recognizes either measure works for Riverdale. Part D explains the $1,700 ad is inflated by the one $3,800 unit and states what a hunter should actually expect (~$1,200), using computed numbers. |
| 3 | Proficient | Computations correct or nearly so (one slip). Part C picks the median for Oakwood with a general justification (“there’s an outlier”) but may not address Riverdale. Part D says the ad is misleading and gestures at the outlier, but without specific numbers. |
| 2 | Developing | At least one neighborhood’s mean and median correct. Part C states a choice without data-based justification, or applies “always use median” without recognizing Riverdale’s agreement. Part D attempted but vague. |
| 1 | Emerging | Computations attempted with method errors (median unsorted, mean divided by wrong count). Parts C/D blank or restate the question. Aware of the task but cannot carry it through. |
Quick checklist: ☐ Riverdale mean $1,300 ☐ Riverdale median $1,300 ☐ Oakwood mean $1,700 ☐ Oakwood median $1,200 ☐ Part C names the $3,800 outlier ☐ Part D uses computed numbers
5. Running the session & general principles
- The single sentence this whole test teaches: “The mean absorbs the size of every value; the median only counts positions.” If the student can say that and give an example by the end of review, the session succeeded.
- Suggested pacing: Stimulus 1 + Q1–6, break, Stimulus 2 + Q7–12, then Stimuli 3–4 + Q13–24, then the constructed response (which can be homework).
- Do not re-explain the same concept the same way twice — switch representation: a number line, physical coins, or an Excel column they can edit and watch the mean move.
- Excel reinforcement: enter the salary data, use AVERAGE and MEDIAN, then change 260 to 60 and watch which cell moves — Chapter 3’s Excel lab does exactly this.
- Ask “what does this number mean in words?” after every computation. A student who computes a median of $51K but cannot say “half the employees earn less than this” has only half the skill.
- Connect forward: Test 3 (media literacy) shows news graphics that exploit the mean/median gap — everything here is preparation for spotting that.