Follow a question through data, analysis, a figure and an abstract.
These are teaching analyses of existing data or a mathematical construction. They are not claims of new experiments or publication-ready discoveries. Reproduce the result, then develop and review your own question.
The examples below are teaching reproductions, not new discoveries. These extensions are exercises to perform, not additional results we have measured. Stage labels are suggestions; choose by readiness with a teacher or mentor.
Middle school: begin with the carpet counting exercise, then try reading a data table and figure. Use the “Middle school” task in each example.
High school: reproduce a calculation, define a comparison and examine a limitation using the “High school” tasks.
College / expert: audit assumptions and reproducibility using the “College / expert review” tasks. Explain one improvement without supplying the learner’s conclusion.
Does water-to-binder ratio track concrete strength at 28 days?
Question → measurement plan
Within the UCI Concrete Compressive Strength dataset, describe the association between water-to-binder mass ratio and compressive strength for specimens tested at 28 days. Binder means cement + blast-furnace slag + fly ash. Holding age constant removes one source of variation; other mix differences remain.
Use all 1,030 original rows. Select age = 28 days, calculate water ÷ binder from kg/m³ values, and keep strength in MPa. Preserve the original zero-based row index for traceability. The source has no empty cells in this conversion and 25 exact duplicate rows across the full dataset; they are retained, not treated as proven independent repeats.
Measurements → analysis
The age filter produces 425 rows. Compute Pearson’s correlation as Σ[(x − mean(x))(y − mean(y))] divided by √[Σ(x − mean(x))² × Σ(y − mean(y))²]. The result is r = -0.769. Plot every selected row; do not replace unusual values without a stated reason.
Download the 425-row analysis CSV. Recompute the correlation from its ratio and strength columns. Check duplicate sensitivity and other mixture variables before proposing a predictive model.
Figure 1. Each point is a source row; overlaps are possible. The negative association does not show that changing only water caused the observed strength differences.
Analysis → abstract
Question. How does water-to-binder ratio relate to compressive strength in concrete tested at 28 days?
Methods. We selected 425 rows from a published 1,030-row dataset, retained duplicates, calculated the water-to-binder ratio and summarized its association with strength using Pearson correlation.
Result. Ratio and strength were negatively associated (r = −0.769).
Interpretation. This descriptive result is consistent with a mixture relationship in the selected data. Uncontrolled composition differences and uncertain independence prevent a causal conclusion. Further work should examine duplicate sensitivity and additional mixture variables.
Abstract → appropriate venue
A developed project with a clear contribution and mentor review could fit an undergraduate conference such as NCUR. This replication alone does not establish journal-level novelty. Ask an editor before targeting AJUR.
Source: I-Cheng Yeh (1998), Concrete Compressive Strength, UCI, DOI 10.24432/C5PK67, CC BY 4.0. Analysis and figure are derived from that dataset; age filtering and ratio calculation are the stated changes.
Adapt this example
Middle school: Read the first ten rows of the supplied 28-day CSV. Find the units, calculate the minimum and maximum strength, and describe the scatterplot in two sentences. Do not mix or test concrete for this exercise.
High school: Recreate the scatterplot and correlation for all 425 rows. Explain why holding age at 28 days does not control all mixture differences. State a competing explanation before interpreting the association.
College / expert review: Audit duplicate handling and the ratio definition. Propose a sensitivity analysis or adjusted model, justify its assumptions and inspect whether conclusions depend on those choices.
Make a labeled figure or a one-minute explanation of what you checked. Keep your own calculations and credit the source example.
02 · Mathematics · Computational reproduction
When does box counting recover a known fractal dimension?
Question → measurement plan
Construct a depth-five Sierpiński carpet on a 243 × 243 grid. Retain a cell only if none of its five base-three coordinate digit pairs equals (1, 1). This leaves 8⁵ = 32,768 occupied cells. Count boxes intersecting an occupied cell at side lengths 243, 81, 27, 9, 3 and 1 cell.
For each length s, the aligned grid uses the integer pair (floor(x/s), floor(y/s)). Repeat with shift floor(s/2) added to both coordinates before dividing. Boxes extending beyond the original boundary are retained. At s = 1 the shift is zero. This is a finite-grid sensitivity check, not a new definition of dimension.
Measurements → analysis
Aligned counts are 1, 8, 64, 512, 4,096 and 32,768. The slope of log N against log(243/s) is log(8)/log(3) = 1.892789. Shifted counts are 4, 16, 96, 688, 5,280 and 32,768. Their finite-scale ratios differ: grid alignment matters.
Download box counts and grid settings. Reproduce the digit rule with a small script or spreadsheet; compare slopes over different subsets of scales instead of choosing only a favorable fit.
Figure 2. A method check against a construction with known scaling. For the ideal carpet, each stage retains eight copies at one-third scale, giving 8 × (1/3)ᴰ = 1. A finite image or short straight-looking segment does not establish an unknown object’s fractal dimension.
Analysis → abstract
Question. Does box counting reproduce known scaling, and how sensitive is the finite result to grid alignment?
Methods. We generated a depth-five Sierpiński carpet and counted occupied boxes at six powers-of-three scales using aligned and shifted grids.
Result. Aligned counts recovered the construction’s dimension of 1.892789. Shifted-grid counts differed at five scales.
Interpretation. The calculation validates the aligned implementation for this construction and demonstrates finite-grid sensitivity. It does not establish new mathematics or justify applying the same fit to arbitrary images without validation.
Abstract → appropriate venue
Use this first as a methods exercise. A substantive extension comparing estimators and uncertainty could be discussed with a mentor for NCUR or a future undergraduate regional meeting. The construction and numbers here are generated from the stated rule, with no external dataset.
Adapt this example
Middle school: Draw the first two carpet stages on squared paper. Count occupied squares, label the grid size and make a small table. Explain the pattern without treating a drawing as experimental evidence.
High school: Reproduce the aligned counts and log–log slope. Compare with the supplied shifted-grid counts and explain why grid alignment matters at finite resolution.
College / expert review: Review scale selection and offsets. Design a convergence check across depths and report residuals; distinguish a numerical estimate from the exact mathematical dimension.
Make a labeled figure or a one-minute explanation of what you checked. Keep your own calculations and credit the source example.
03 · Molecular data · Representation fidelity
How much can coordinate rounding change a measured distance?
Question → measurement plan
Use the deposited PDB-format coordinates for ubiquitin 1UBQ, chain A. Select the Cα atoms at residue pairs 1–2, 1–20, 1–40, 1–76, 20–40 and 40–60. Measure Euclidean distances in ångströms (Å) from the three-decimal deposited coordinates, then round each coordinate to two, one and zero decimals and remeasure.
The reference is the downloaded coordinate file, not exact molecular truth. The entry reports a 1.8 Å X-ray resolution; coordinate decimal places and experimental resolution are different quantities. The file is pinned by SHA-256 in the source record. Residue pairing is preselected, not exhaustive.
Measurements → analysis
For each pair, calculate √[(x₁ − x₂)² + (y₁ − y₂)² + (z₁ − z₂)²], then take the absolute difference from the deposited-coordinate result. Across these six pairs the largest changes are 0.006476 Å at two decimals, 0.099554 Å at one decimal and 0.697506 Å at zero decimals. Python’s round-to-nearest, ties-to-even operation is used on parsed floating-point coordinates.
Download all 24 distance comparisons. Match the chain, residue numbers and atom names before recalculating. The three-decimal row is a zero-change reference, not an independent validation experiment.
Figure 3. Distance changes reflect this representation change and these selected pairs. They do not demonstrate a biological effect or a change in protein function.
Analysis → abstract
Question. How does coordinate rounding affect selected distances in a deposited protein structure?
Methods. We measured six chain-A Cα pairs in 1UBQ before and after rounding coordinates to fewer decimal places.
Result. Maximum absolute changes were 0.006476 Å, 0.099554 Å and 0.697506 Å at two, one and zero decimal places, respectively.
Interpretation. Coarse rounding altered the selected geometric measurements. This tests representation fidelity only; it does not measure experimental uncertainty, molecular dynamics or biological function.
Abstract → appropriate venue
A broader, justified comparison of data representations could become an undergraduate presentation after mentor review. Check NCUR’s current abstract rules; a journal submission would need a substantive contribution beyond this tutorial.
Middle school: Round a few ordinary lengths to whole units, compare the before/after differences, then read the protein figure’s axes. Explain why rounding is a change in representation, not proof that an object moved.
High school: Recalculate one selected C-alpha pair with the three-dimensional distance formula. Repeat after rounding both points; compare the absolute difference with the CSV and state the units.
College / expert review: Audit atom/chain selection and floating-point rounding. Design a broader sensitivity analysis with justified pair selection, and distinguish coordinate precision from experimental resolution.
Make a labeled figure or a one-minute explanation of what you checked. Keep your own calculations and credit the source example.
Use this pattern for your own project
Write the question before selecting measurements. Keep raw values, exclusions and calculation steps. Label missing evidence as missing; a blank measurement is not zero. Draft a conclusion that the evidence can actually support, then choose a venue.
