ACARA v9 CONTENT DESCRIPTION “plan and conduct statistical investigations of situations that involve bivariate data; evaluate and report findings with consideration of limitations of any inferences”
Builds on: Two-Way Tables. The displays and summaries built across this strand now serve a larger purpose: a full statistical investigation. This unit ties them together into the cycle of asking a question, collecting data, analysing it, and interpreting and reporting the answer.
Statistics as an investigation
The individual tools of statistics, displays, averages, quartiles, scatterplots, tables, all serve a larger purpose: answering real questions about the world through data. A statistical investigation is the whole process of doing this, and like mathematical modelling it follows a cycle rather than a single step. You pose a question, collect data to address it, analyse that data to find patterns, and interpret the results to answer the question, often looping back as one answer suggests the next question. Learning to run an investigation, not just to calculate, is the capstone of statistics: it is where the separate techniques come together into a way of reasoning from evidence to a conclusion you can defend.
The statistical investigation cycle
An investigation moves through question, collect, analyse and interpret, often looping back.
A statistical investigation follows a cycle: pose a question, collect data to address it, analyse that data with displays and summaries, then interpret the results to answer the question. Like modelling, it often loops, as findings raise new questions. Stepping through the stages shows how an investigation is planned, not improvised.
Posing a good question
Every investigation begins with a question, and its quality decides everything that follows. A vague question like is study good cannot be answered with data, because nothing in it can be measured. A good statistical question instead names clear, measurable variables and asks about the relationship between them, such as is there an association between the hours a student studies and their test score. Such a question is answerable: you can imagine exactly what data would settle it. Posing a sharp, measurable question is a genuine skill, and time spent getting it right is never wasted, since a woolly question leads only to a woolly conclusion no amount of clever analysis can rescue.
Asking an answerable question
A good statistical question identifies measurable variables and asks about their relationship.
An investigation begins with a question, but a vague one like "is study good?" cannot be answered with data. Reveal how to sharpen it into a proper statistical question.
Collecting and displaying data
With a clear question, the next stage is to collect data that genuinely bears on it, from a sample that fairly represents the group you want to draw conclusions about. Once gathered, the data must be displayed in a way suited to its type, and matching the display to the data is itself part of analysing well. Two numerical variables, like study hours and score, call for a scatterplot, which shows how they vary together. Two categorical variables would call for a two-way table instead, and a single variable for a boxplot or histogram. Choosing the right display turns a column of raw numbers into a picture in which a pattern, if there is one, can actually be seen.
Collect, then display appropriately
Choose a display that suits the data: a scatterplot for two numerical variables.
With data gathered on hours studied and test score, the next step is to display it. Plot the points to reveal the appropriate display for two numerical variables.
Analysing the data
Analysis is where the display is turned into a precise description. For bivariate numerical data, this means identifying the direction of any association, whether positive or negative, its strength, whether strong or weak, and often a line of best fit to summarise the trend and estimate its rate. In the study example, the points might show a strong positive association, with scores rising by about four marks for each extra hour of study. For categorical data, analysis means comparing conditional proportions across groups. In every case, analysis replaces a vague impression that the variables seem related with specific, defensible statements about how they are related, which is exactly what the question demands.
Analyse: describe the relationship
Analysis describes the direction, strength and rate of the association, often with a line of best fit.
The points trend upward, but analysis needs more than a glance. Reveal the line and the summary that describe this relationship precisely.
Interpreting and reporting
The final stage closes the loop: interpreting the analysis to answer the original question, and reporting the work honestly. Interpretation states the finding in plain terms, that in this sample more study is associated with higher scores, and crucially states its limits. The data shows an association, not a proven cause; it comes from one sample that may not represent everyone; and any prediction is reliable only within the range of the data collected. A good report sets out the question, the data and how it was gathered, the analysis, the finding, and the assumptions made, so that others can scrutinise and trust the conclusion. This honesty about what the data does and does not show is the mark of sound statistical reasoning, and it completes the investigation that the opening question began.
Interpret and report
Interpret by answering the question, state the limits, and report method and assumptions honestly.
Interpreting means answering the original question in plain terms: in this sample, more study is associated with higher scores. The interpretation always refers back to the question that started the investigation.
Quick self-check
1. A statistical investigation is best described as:
2. Which is a good statistical question for an investigation?
3. For an investigation into two numerical variables, an appropriate display is:
4. After finding a strong positive association between study and scores in one sample, a careful conclusion:
5. The final stage of a statistical investigation is to: