# Twelve data-reasoning practice exercises

Version 1.0.0 · September 14, 2026 · Skills Path / Editorial Desk Media

Original synthetic practice material. No classroom, learner-study or educator-review results are claimed. Arithmetic has automated checks; teaching quality still requires educator review. All examples and answers are included here: no account, paid software or external dataset is required.

## Learning objectives

By the end of a practice session, attempt to:

- Identify the unit and denominator before computing a summary.
- Weight combined rates by the population described in the question.
- Distinguish repeated groups, excess rows and exact duplicates.
- State what is known and unknown when data is missing.
- Separate percentage points, relative growth and conditional proportions.
- Explain how subgroup weights and overlapping memberships change a result.

These are proposed learning objectives, not demonstrated learning gains.

## Audience and prerequisites

Designed for adult self-study, introductory undergraduate study or professional data-literacy practice. Prior knowledge: basic arithmetic, fractions and percentages, and reading rows and columns. A four-function calculator is optional. No spreadsheet or programming experience is needed. Facilitators should judge fit for their learners before use.

## How to use the three variants

1. **Worked:** read the answer-key steps, then explain why the common wrong answer fails.
2. **Partially supported:** use the question and two hints; cover the answer key until an attempt is complete.
3. **Independent:** use only the question, table and assumptions. Show a calculation and write one sentence naming the denominator or counting policy.

The three variants share the same underlying dataset. Moving from worked to independent is supported practice, not evidence of transfer to an unseen task. For an actual teaching pilot, an educator should add a reviewed new-number problem and assess delayed reasoning independently.

A suggested first session uses exercises 1, 3 and 5, followed by a discussion of what changes if the denominator or missing-value policy changes. Timing is left to the facilitator; no completion-time claim has been measured. Keep learner identities and responses out of public feedback records.

## Accessibility and adaptation

All information is text or small tables, with labeled headers. No answer depends on color, images, audio, animation, dragging or a time limit. Read “null” as “not observed” where the assumptions specify missing data. Percent signs can be read as “percent”; a slash in a formula means division. A facilitator may provide large print, read tables aloud in header order, or transpose tables to a list. Check the final teaching platform with its actual assistive technology; this text alone does not certify an accessibility standard.

When adapting, preserve the stated counting policies and recompute the key after changing numbers. Keep explanations alongside numerical answers. Translation requires language review; no translated teaching version is represented here.

## Learner pages

### 1. Combine unequal class sizes (weighted-completion)

Objective: Use group size to combine completion rates.

| Group | Completed | Enrolled |
| --- | --- | --- |
| Small | 18 | 20 |
| Large | 54 | 80 |

Assumptions:

- Each learner occurs in exactly one group.
- Completion status is known for all 100 learners.

**Independent prompt:** What percentage of all enrolled learners completed? State the numerator and denominator.

**Partial-support hints:**

- Combine the counts before dividing.
- The two classes have different numbers of learners; an equal average of their percentages changes the weighting.

Worked version: use answer key 1 after your attempt.

### 2. Average cost per notebook (weighted-cost)

Objective: Weight unit costs by quantities.

| Batch | Notebooks | Cost per notebook (dollars) |
| --- | --- | --- |
| First | 10 | 2 |
| Second | 30 | 4 |

Assumptions:

- All notebooks are interchangeable for this calculation.
- Costs include all charges; quantities are fully received.

**Independent prompt:** What is the average cost per notebook across both batches?

**Partial-support hints:**

- Compute total dollars spent on each batch.
- Divide total dollars by total notebooks, not by the number of batches.

Worked version: use answer key 2 after your attempt.

### 3. Count removable exact duplicates (duplicate-rows)

Objective: Distinguish excess rows from every row in a duplicate group.

| Learner ID | Workshop |
| --- | --- |
| A | Mapping |
| A | Mapping |
| B | Charts |
| C | Tables |
| C | Tables |
| C | Tables |

Assumptions:

- A duplicate means identical values in both columns.
- Keep one copy of each distinct row; no empty values occur.

**Independent prompt:** How many rows are removed when one copy of each distinct row is retained?

**Partial-support hints:**

- Count how many copies to retain, not how many rows belong to a repeated group.
- For a group containing n identical rows, remove n − 1.

Worked version: use answer key 3 after your attempt.

### 4. Count repeated identifiers (duplicate-key-groups)

Objective: Keep repeated keys distinct from identical rows.

| Record key | State |
| --- | --- |
| K1 | draft |
| K1 | final |
| K2 | draft |
| K3 | draft |
| K3 | final |
| K3 | archived |

Assumptions:

- A repeated key is any key appearing more than once.
- Different states are meaningful; do not remove rows based on the key alone.

**Independent prompt:** How many distinct keys occur more than once?

**Partial-support hints:**

- Group by the first column only.
- Count groups whose size exceeds one; do not sum their row counts.

Worked version: use answer key 4 after your attempt.

### 5. Average only observed readings (missing-observed-mean)

Objective: State the missing-value policy and keep genuine zeros.

| Record | Reading |
| --- | --- |
| R1 | 0 |
| R2 | 10 |
| R3 | null (not observed) |
| R4 | 5 |
| R5 | null (not observed) |

Assumptions:

- null means not observed, never zero.
- Compute the mean of observed readings only.
- No claim is made about the mean of all five underlying true values.

**Independent prompt:** What is the mean of the observed readings, retaining the real zero?

**Partial-support hints:**

- A real zero is a measured value and belongs in the denominator.
- Write both the included values and the number of included values.

Worked version: use answer key 5 after your attempt.

### 6. Report a bound with unknown statuses (missing-completion-bound)

Objective: Distinguish a lower bound from a completed-data estimate.

| Learner | Status |
| --- | --- |
| L1 | complete |
| L2 | complete |
| L3 | complete |
| L4 | complete |
| L5 | complete |
| L6 | incomplete |
| L7 | incomplete |
| L8 | null (not observed) |
| L9 | null (not observed) |
| L10 | null (not observed) |

Assumptions:

- Every row is a distinct enrolled learner.
- null means completion status unknown.
- Known statuses are correct; unknown learners may be complete or incomplete.

**Independent prompt:** What is the smallest possible completion percentage among all ten enrolled learners?

**Partial-support hints:**

- Use every enrolled learner in the denominator.
- A lower bound is a possible minimum, not a claim that unknown statuses are actually incomplete.

Worked version: use answer key 6 after your attempt.

### 7. Separate points from relative growth (percentage-points)

Objective: Subtract rates when the requested unit is percentage points.

| Period | Completed | Enrolled |
| --- | --- | --- |
| Before | 20 | 100 |
| After | 25 | 100 |

Assumptions:

- The periods contain comparable groups for this descriptive calculation.
- No causal effect is inferred.

**Independent prompt:** How many percentage points did the completion rate increase?

**Partial-support hints:**

- Express both rates as percentages first.
- Use subtraction without dividing by the baseline percentage.

Worked version: use answer key 7 after your attempt.

### 8. Find relative growth from a baseline (relative-change)

Objective: Use the original count as the denominator of relative change.

| Period | Completed projects |
| --- | --- |
| Before | 20 |
| After | 25 |

Assumptions:

- These are counts across equally long periods.
- The original count is nonzero.

**Independent prompt:** What is the percentage increase in completed projects relative to the original count?

**Partial-support hints:**

- Subtract to find the increase in projects.
- The baseline is the original period, not the final period.

Worked version: use answer key 8 after your attempt.

### 9. Interpret a flagged document (flag-denominator)

Objective: Use the requested conditional denominator.

| Actual document state | Flagged | Not flagged |
| --- | --- | --- |
| Has formatting error | 8 | 2 |
| No formatting error | 32 | 58 |

Assumptions:

- All 100 documents were checked and the actual state was verified.
- Each document appears in exactly one cell.
- This is a synthetic document example, not a model accuracy claim.

**Independent prompt:** Among flagged documents, what percentage actually has a formatting error?

**Partial-support hints:**

- The words 'among flagged' determine which column is the denominator.
- Count both correct and incorrect flags.

Worked version: use answer key 9 after your attempt.

### 10. Choose the requested middle value (median-outlier)

Objective: Compute a median without substituting a mean.

| Task | Minutes |
| --- | --- |
| A | 4 |
| B | 5 |
| C | 100 |
| D | 5 |
| E | 6 |

Assumptions:

- All five observations are genuine and equally weighted.
- The question asks for a median, not a prediction of a future task.

**Independent prompt:** What is the median task duration?

**Partial-support hints:**

- Sort the values numerically.
- Choose the middle position; do not average all values.

Worked version: use answer key 10 after your attempt.

### 11. Compare with explicit subgroup weights (subgroup-weights)

Objective: Separate an aggregate comparison from an equally weighted subgroup comparison.

| Method | Difficulty | Successful | Attempted |
| --- | --- | --- | --- |
| A | Easy | 9 | 10 |
| A | Hard | 30 | 100 |
| B | Easy | 80 | 100 |
| B | Hard | 2 | 10 |

Assumptions:

- Success means the same outcome in every row.
- Compute a descriptive standardized comparison giving Easy and Hard equal 50% weights for each method.
- These data do not establish a causal benefit or justify a universal population weighting.

**Independent prompt:** Under equal Easy/Hard weights, how many percentage points higher is method A's success rate than method B's?

**Partial-support hints:**

- Compute one rate per method and difficulty.
- Use the specified 50/50 weights, not the observed numbers of attempts.

Worked version: use answer key 11 after your attempt.

### 12. Count a union once (overlapping-groups)

Objective: Remove the overlap when combining two memberships.

| Membership | Learners |
| --- | --- |
| Spreadsheet workshop | 30 |
| Chart workshop | 22 |
| Both workshops | 12 |

Assumptions:

- Counts refer to distinct learners within each workshop.
- The twelve learners in both are already included in each of the first two counts.
- No other workshops are part of this question.

**Independent prompt:** How many distinct learners attended at least one of the two workshops?

**Partial-support hints:**

- A learner attending both contributes one to each workshop count.
- For 'at least one', keep the overlapping learners once.

Worked version: use answer key 12 after your attempt.

## Worked answer key

### Answer 1: Combine unequal class sizes

**72 percent.**

1. Count all completions: 18 + 54 = 72.
2. Count all enrolled learners: 20 + 80 = 100.
3. Compute 72 / 100 × 100 = 72%.

The population contains 100 learners, so each learner gets equal weight. The group rates, 90% and 67.5%, should not receive equal group weight.

Common wrong answers:

- 78.75: This is (90 + 67.5) / 2: it gives a group of 20 the same weight as a group of 80.

### Answer 2: Average cost per notebook

**3.5 dollars per notebook.**

1. First batch cost: 10 × 2 = 20 dollars; second batch cost: 30 × 4 = 120 dollars.
2. Total cost is 140 dollars for 40 notebooks.
3. 140 / 40 = 3.50 dollars per notebook.

Each notebook, rather than each purchase event, is the unit being averaged.

Common wrong answers:

- 3: Averaging the two listed unit prices ignores that three times as many notebooks cost 4 dollars.

### Answer 3: Count removable exact duplicates

**3 rows.**

1. There are six input rows and three distinct rows: A/Mapping, B/Charts and C/Tables.
2. The A group removes one excess row; the C group removes two.
3. Total removed: 1 + 2 = 3 rows.

Five rows belong to duplicate groups, but retaining one from each group means only three are removable.

Common wrong answers:

- 5: This counts all rows in repeated groups, including the copies that must remain.
- 2: This counts repeated groups, not excess rows.

### Answer 4: Count repeated identifiers

**2 keys.**

1. K1 appears twice, K2 once and K3 three times.
2. Only K1 and K3 meet the more-than-once condition.
3. There are two repeated keys; all six full rows are different.

Repeated identifiers can signal multiple versions, not accidental copies. This question counts groups and does not authorize deletion.

Common wrong answers:

- 3: This is the number of excess occurrences if one row per key were retained, a different question.
- 0: There are no identical full rows, but there are repeated keys.

### Answer 5: Average only observed readings

**5 reading units.**

1. Observed readings are 0, 10 and 5; the two null entries are excluded.
2. Their total is 15 and their count is three.
3. 15 / 3 = 5 reading units.

The observed mean uses three readings. Treating unknown values as zero invents observations; deleting the genuine zero also changes the question.

Common wrong answers:

- 3: Dividing 15 by all five records silently treats the missing readings as zero.
- 7.5: Dropping the real zero leaves only 10 and 5 and overstates the observed mean.

### Answer 6: Report a bound with unknown statuses

**50 percent.**

1. Five of ten learners are definitely complete.
2. The minimum occurs if none of the three unknown learners is complete.
3. 5 / 10 × 100 = 50%. The maximum would be 8 / 10 × 100 = 80%.

Completion lies between 50% and 80% under these assumptions. The observed-status rate of 5/7 does not identify the whole group's rate.

Common wrong answers:

- 71.428571: This approximately describes only the seven known statuses, not the minimum among all ten learners.
- 80: This is the upper bound if all three unknown statuses are complete.

### Answer 7: Separate points from relative growth

**5 percentage points.**

1. The rates are 20 / 100 = 20% and 25 / 100 = 25%.
2. Subtract the percentages: 25 − 20 = 5.
3. The increase is 5 percentage points; the relative increase is a different quantity.

A change from 20% to 25% is 5 percentage points. Its 25% relative increase uses another denominator and unit.

Common wrong answers:

- 25: This is relative percentage growth, not the percentage-point difference.

### Answer 8: Find relative growth from a baseline

**25 percent.**

1. The absolute increase is 25 − 20 = 5 projects.
2. Divide the increase by the original 20 projects: 5 / 20 = 0.25.
3. Multiply by 100 to express 25% relative growth.

Relative growth compares the change with the original amount. Counts are not rates, so an absolute increase of five projects is not five percentage points.

Common wrong answers:

- 20: Dividing the increase by the final 25 uses the wrong baseline.
- 5: This is the increase in projects, not its percentage relative to the original count.

### Answer 9: Interpret a flagged document

**20 percent.**

1. There are 8 + 32 = 40 flagged documents.
2. Eight of those 40 actually have an error.
3. 8 / 40 × 100 = 20%.

The 80% rate of flagging erroneous documents (8/10) answers a different conditional question. Both quantities can be true at once.

Common wrong answers:

- 80: This is the percentage of erroneous documents that were flagged, reversing the condition.
- 8: This divides by all 100 documents instead of the 40 flagged documents.

### Answer 10: Choose the requested middle value

**5 minutes.**

1. Order the observations: 4, 5, 5, 6, 100.
2. With five observations, the third is the middle value.
3. The median is 5 minutes; the mean is 120 / 5 = 24 minutes.

The median is defined by position. The unusually long task affects the mean much more, but it is not automatically an error to delete.

Common wrong answers:

- 24: This is the arithmetic mean, not the median.
- 100: This is the middle row of the unsorted display; a median requires ordering the values.

### Answer 11: Compare with explicit subgroup weights

**10 percentage points.**

1. Method A subgroup rates: 9/10 = 90% and 30/100 = 30%; their equal-weight average is 60%.
2. Method B subgroup rates: 80/100 = 80% and 2/10 = 20%; their equal-weight average is 50%.
3. The standardized difference is 60 − 50 = 10 percentage points. Raw aggregate rates would rank B above A because the subgroup mixes differ.

Equal subgroup weights are a chosen comparison policy. The raw aggregates, 39/110 and 82/110, answer a different question; neither automatically estimates a causal effect.

Common wrong answers:

- -39.090909: This approximately subtracts raw aggregate rates, ignoring the explicitly requested equal subgroup weights.
- 20: This adds the two ten-point subgroup advantages instead of averaging them.

### Answer 12: Count a union once

**40 learners.**

1. Adding the workshop counts gives 30 + 22 = 52 memberships.
2. The twelve learners in both were counted twice.
3. Subtract them once: 52 − 12 = 40 distinct learners.

The union includes single-workshop and both-workshop learners. Subtracting the intersection twice would count only people who attended exactly one workshop.

Common wrong answers:

- 52: This counts memberships rather than distinct learners.
- 28: Subtracting the overlap twice counts exactly one workshop, excluding learners who attended both.

## Reuse rights

The original exercise text, synthetic tables and explanatory answer keys in this package and the companion skills-reasoning manifest are made available under [Creative Commons Attribution 4.0 International (CC BY 4.0)](https://creativecommons.org/licenses/by/4.0/). This notice does not relicense existing site code, other publications, unrelated downloads, trademarks or linked third-party material.

Suggested attribution: “Twelve data-reasoning practice exercises, Skills Path / Editorial Desk Media, version 1.0.0 (September 14, 2026), CC BY 4.0.” Link to the material and license when practical, and identify adaptations. The [license text](https://creativecommons.org/licenses/by/4.0/legalcode) governs reuse. References are linked rather than copied. Original text and numbers were created for this package; there are no stock assets, customer records or proprietary test items.

## Review and adoption record

- Arithmetic: automated recomputation available in the project test suite; not a teaching-quality evaluation.
- Educator review: not obtained.
- Teaching uses observed: zero.
- Directory submission: not submitted.
- Directory acceptance: not accepted.
- Adoption: no evidence available.

Before describing this as educator-reviewed, obtain an actual educator's review of clarity, prerequisite fit, distractors and scaffolding. Record the review date, the permitted reviewer description and changes made. Review two actual teaching uses within 30 days if they occur; otherwise record adoption evidence as unavailable. Directory acceptance, if later obtained, is separate from educator review and classroom adoption.