Ask, collect, analyze and interpret a statistical investigation

Learning goal: Formulate a statistical question that anticipates variability, plan and collect quantitative data, analyze the distribution and draw conclusions tied to the original question. Carry out all four stages rather than treating a graph alone as a completed investigation. The four stages (formulate a question, collect data, analyze data, interpret results) are the GAISE model of statistical reasoning that Ohio's standard names. This lesson reports the number of observations, describes the attribute and how it was measured, and describes the shape of the distribution in words. It uses only the range as a measure of spread; calculating the mean, median and interquartile range, and choosing among measures, is taught in the companion lesson on center and variability.

“How long did this one attempt take?” asks for a single recorded duration. “How do completion times vary across attempts at our classroom sorting task?” anticipates different numerical answers and can be investigated statistically. A statistical question specifies an attribute, a group or process to observe and an expectation that values may vary.

Make the measurement procedure precise. For a sorting investigation, use the same twelve cards and the same three category labels in each attempt. Start timing at the participant’s first touch of the cards, not when the participant is allowed to begin; stop when all cards are placed and the participant says “finished.” Record duration in whole seconds using a stated rounding rule. If accuracy matters, record sorting errors separately rather than silently treating an inaccurate fast attempt as equivalent to an accurate one.

Choose which attempts to observe before seeing the results. A classroom plan might include two attempts from each of six volunteers in an agreed order, giving twelve observations. These are twelve attempts from six participants, not twelve independent people. Repetition and practice may influence times, so retain participant codes and attempt order in the working notes without publishing personal names.

For the worked example only, use this invented sorted data set of twelve durations in seconds: 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 11. These values are deliberately simple for learning calculations. They are not actual results of the proposed twelve-card task and should not be presented as collected classroom evidence.

A distribution describes how values are arranged and how often they occur. The example has a concentration from three through six seconds, a peak at four seconds with three observations, no observations at eight, nine or ten seconds, and a relatively isolated eleven-second value. The higher end extends farther from the main cluster than the lower end, suggesting a right-side tail.

Compare observations at several levels. The eleven-second attempt exceeds the two-second attempt by nine seconds. Compared with the group, eleven is higher than the other eleven observations. That statement describes its position; it does not explain why it occurred. A pause, a task difference or a measurement error would require additional evidence.

Analyze with a suitable display and numerical summaries, selected for the purpose. A dot plot shows every observation, so it displays the variability within the group and lets you place one individual against the group. To compare one individual with another, subtract their values. A measure of spread, such as the range, summarizes the variability in one number. The companion lessons show a dot plot, histogram and box plot for exactly this data set, along with mean, median, range and interquartile range. Keep the count twelve visible. A graph that accidentally omits the eleven-second observation would change both the distribution and its summary measures.

Interpret in relation to the original question: in this invented example, recorded attempts vary from two to eleven seconds, with most values between three and six. That conclusion is about the example's attempts. It does not establish how all students, all classrooms or every possible sorting task would perform.

For your real investigation, inspect unexpected observations before deciding how to treat them. If notes document a stopwatch malfunction, explain the exclusion and retain a record. If an unusually long attempt was correctly observed, deleting it merely to make the graph look regular would misrepresent the data. Report what was measured and any limitations in the collection plan.

The four-stage cycle can lead to another question. If second attempts appear faster, a follow-up could examine practice under a consistent repeated-attempt design. The initial summary alone does not prove that practice caused the difference. Use the next plan to gather relevant evidence rather than turning a possible explanation into a conclusion.

Worked question revision: Vague: “Are our cards good?” Investigable: “How many seconds do attempts take to sort these twelve cards into the stated categories under our timing procedure?” The revision identifies a numerical attribute and process. A separate question would be needed to study sorting accuracy or opinions.

Worked count correction: “Twelve students took part” does not follow from twelve attempts. If six volunteers each made two attempts, report six participants and twelve observations. The distinction affects how broadly the results can be interpreted.

Worked analysis choice: For the twelve invented durations, choose a dot plot because it shows all twelve values, spread from 2 to 11 seconds: that is the variability within the group. Individual to individual: the 7-second attempt took 7 − 4 = 3 seconds longer than a 4-second attempt. Individual to group: 7 seconds is longer than ten of the other eleven observations and shorter than one (11 seconds). A numerical measure of spread, the range, is 11 − 2 = 9 seconds; the companion lesson calculates the others.

Actual investigation: In four stages, write a statistical question, agree on a measurement and selection plan, collect a feasible set of classroom observations and keep raw records. Use nonpersonal tasks such as sorting or measuring paper constructions. Record units, rounding, repeated attempts and deviations from the procedure.

Independent report: Display the actual data, describe center, spread and shape (using the measures from the companion lesson), compare individual values with the group and state a conclusion answering the original question. Include the observation count and measurement method. Explain one limitation and one follow-up question. Submit raw data and calculations so a reviewer can trace the graph. Keep invented worked-example data separate from your collected results.