Carry out a statistical investigation from question to interpretation

Learning goal: Use the GAISE investigation cycle to formulate a statistical question, collect appropriate quantitative data, analyze variability and interpret results in relation to the original question. Complete an actual investigation rather than filling a conclusion with invented measurements.

A statistical question anticipates variation among observations and can be answered with data. “How thick is this one book?” asks for one measurement. “How do the thicknesses of books on Shelf A compare with those on Shelf B today?” anticipates varied numerical measurements within and between two defined groups. The second question provides a reason to summarize distributions rather than report only one value.

The investigation follows four connected stages: formulate questions, collect data, analyze data and interpret results. These stages may require returning to an earlier decision. If the proposed measurement cannot be made consistently, revise the collection plan. If the intended population is too broad for the available sample, narrow the question instead of pretending the data cover it.

Stage 1 — Formulate. Select two accessible designated shelves and record the date. Define each population as the books actually on that shelf at the time of selection. The question concerns those shelf populations, not all published books. Predict that thickness will vary, but do not predict exact results and then choose books to match the prediction. Explain why thickness is quantitative and identify millimeters as the measurement unit.

Stage 2 — Collect. List or number every eligible book on each shelf before selecting. If a shelf has more than ten eligible books, use a fair random-number or number-slip method to select ten different books from it, without replacement. If a shelf has ten or fewer eligible books, measure every one of them and identify that group as a census, not a sample. Decide how to handle an unavailable book before sampling; record any replacement and its rule rather than quietly substituting a convenient volume.

Measure each closed book's thickness including its covers, with a ruler perpendicular to the two cover surfaces near a straight part of the spine. Use the same method and round to the nearest millimeter. Record the shelf, book identifier, measured value and any difficulty such as a curved or unusually compressible cover. Do not squeeze one book hard and leave another loose, because inconsistent technique can create variation unrelated to the comparison.

Keep a data table with one row per observed book. Preserve all values, including an unusually thick volume, unless a documented measurement or recording error justifies correction. Check a surprising value by remeasurement and keep a note of any change. A value is not an error merely because it complicates the pattern.

Stage 3 — Analyze. Make dot plots on the same millimeter scale so that the groups can be compared honestly. Show every observation, stacking repeated values. Describe each group's center, spread and any visible clusters or unusual values. Calculate a mean and, with the companion lesson's method, mean absolute deviation when useful. A median and interquartile range may also help describe a group with an extreme value, but explain the choice rather than reporting every measure without purpose.

Compare an individual book with other individuals and with its group's distribution. A 30-millimeter book may be thick relative to a group centered near twelve but ordinary in a group centered near thirty. The same measurement can have a different contextual position. Avoid treating the group mean as a thickness that every book must have.

Use roughly equal variability when making the specific center-difference-in-MAD-units comparison taught in the companion lesson. If the groups have very different spreads, report that fact instead of presenting one convenient shared variability as if the condition were met. Graphical overlap and numerical summaries should be interpreted together.

Stage 4 — Interpret. Answer the original shelf-comparison question with evidence from the plots and summaries. State which group tends to be thicker in your sample, how much overlap exists and how the selection method limits a population conclusion. If samples are small or the difference is unclear relative to variation, say so. Do not convert a shelf comparison into a causal claim that a category of book makes people read better; no reading outcomes were measured.

An investigation report should allow another person to understand and evaluate the process. Include the question, population boundaries, selection method, measurement procedure, raw data, graph, calculations and a qualified conclusion. A polished graph without the collection record cannot show whether the sample or measurement supports the claim. Each stage contributes evidence for the next.

Worked question revision: “Are books on Shelf A better?” lacks a defined numerical measure of better. “How do book thicknesses on Shelf A and Shelf B compare today?” identifies a measurable variable and expects variation. Thickness does not become a measure of quality merely because it is easy to count.

Worked measurement check: A record lists one book as 240 millimeters when neighboring observations are around twenty. Remeasure before deleting it. If the original entry accidentally recorded height instead of thickness, correct the variable with a note. If a genuinely thick object is eligible under the plan, retain it and discuss its effect.

Worked conclusion form: “In our selected books, Shelf A has a larger sample mean, but the dot plots overlap and the samples contain only ten books each. The random selection supports a cautious comparison of these shelf populations, not a statement about all library books.” Fill in actual values and direction from the completed investigation; the model sentence is not a prewritten result.

Required investigation: Carry out all four stages using the two shelves or a comparable adult-approved measurable object collection. If changing the setting, redefine the populations, numerical variable and method before collecting. Record actual measurements and preserve the raw table. Make a common-scale graph and show calculations that another learner can reproduce.

Independent performance: Present a report with a one-paragraph account of each stage and an appendix containing the data, plots and computations. Include two individual-to-group comparisons, one between-group comparison and a justified limit on generalization. A partner audits the procedure, recreates one summary and asks whether the conclusion answers the stated question. Revise any mismatch between the population named in the question and the group described in the conclusion.