108i - Reporting on Data Distributions

Lesson 108 — Guided Practice: Reporting on Data Distributions

Strand: Statistics | Descriptor: AC9M8ST02 | Duration: 45 minutes

Learning Intentions

• To structure a clear statistical report describing a data distribution.
• To select and justify appropriate statistics when reporting on shape, centre and spread.

Success Criteria

I can:

1. Identify the key components of a statistical report: context and source, shape, centre, spread, outliers, conclusion.
2. Select and justify the most appropriate measure of centre and spread for a distribution’s shape.
3. Write a clear, accurate paragraph describing a distribution’s shape, centre and spread using data.
4. Critique a weak statistical report and identify what is missing.

Warmup

(6 minutes — spot what’s missing, pairs)

A student wrote this “report” about a data set: “The mean was . It was skewed.”

1. List everything you’d want to know that this report doesn’t tell you.
2. Rewrite it as a single, more complete sentence (you may invent plausible details).

Expect students to raise: what was measured and on whom (context); where the data came from (source); the sample size; which way it’s skewed and what that means for choosing mean or median; a measure of spread; any outliers; units.

Today’s focus: every one of those missing pieces has a name, and a good statistical report includes all of them, in order.

Activities

Activity 1 — Explicit Instruction: Anatomy of a Statistical Report (13 min)

I do — a model report. “Commute time to school (minutes) for a sample of Year 8 students, collected via a class survey (primary data).”

Sorted data:

The model report, labelled by component:

[Context & source] This report describes the distribution of commute times to school for a sample of Year 8 students, collected first-hand through a class survey (primary data).

[Shape] The distribution is right-skewed: most students travel for a relatively short time, but a small number travel much further.

[Centre] Because of this skew, the median ( minutes) is a more representative “typical” value than the mean ( minutes), which is pulled upward by the longest commute.

[Spread] The interquartile range (IQR minutes) shows the middle of students’ commute times fall between and minutes — a much narrower spread than the full range ( minutes) suggests.

[Outlier] One student’s -minute commute is a likely outlier, well beyond the rest of the sample.

[Conclusion] Overall, most students in this sample have a short-to-moderate commute, with one clear exception.

Report checklist (display permanently):

Component	What it must say
Context & source	What was measured, on whom, and whether the data is primary or secondary
Shape	Symmetric, skewed left/right, or uniform — with a reason
Centre	The chosen measure, justified by the shape
Spread	The chosen measure, justified by the shape
Outliers	Named, with their effect on the mean noted
Conclusion	A plain-language summary sentence

We do. “Weekly pocket money (10$ Year 8 students, from a published national consumer research report (secondary data).”

Sorted data:

Together, build the report sentence-by-sentence against the checklist, using sentence starters:

• “This report describes…, collected/sourced from… (primary/secondary).”
• “The distribution is… [shape], because…”
• “The [mean/median] of best represents the centre, because…”
• “The spread, measured by…, shows…”
• “A likely outlier is…, which…”
• “Overall,…”

Activity 2 — Guided Practice: Write Your Own Report (14 min)

Pairs. Choose one data set. Write a full report using the checklist and sentence starters.

Option A (primary). “Time (minutes) students spent on screens after school, collected via a class survey.”

Sorted data:

Option B (secondary). “Usual weekly hours worked by a sample of part-time employed teenagers, from a published ABS Labour Force Survey extract.”

Sorted data:

Every report must: name the source as primary or secondary; describe shape with a reason; justify the choice of centre and spread; note any outlier; end with a one-sentence conclusion.

Activity 3 — Peer Critique and Refine (7 min)

Swap your report with another pair. Audit against this checklist, then return it for one repair.

Check	✓ / ✗ / ?
Context, population and source (primary/secondary) are clearly stated
Shape is named and justified with a reason, not just asserted
Centre measure is justified by the shape, not just calculated
Spread measure is justified by the shape, not just calculated
Any outlier is identified and its effect on the mean explained
A clear, plain-language conclusion sentence is included

Expected common faults, for the debrief: stating a shape without justifying it from the mean/median comparison; reporting a statistic without saying what it means in context; omitting the source (primary/secondary); forgetting units.

Checks for Understanding

(5 minutes — exit ticket, collected)

Given this data: “Number of books read last term by a sample of students, from a class library survey (primary data)”: .

1. Calculate the mean and median.
2. Describe the shape, with a reason.
3. State the most appropriate measure of centre for this data, and justify your choice.
4. Reasoning. Write a full report (3–4 sentences) using the checklist from today’s lesson.

Answers: 1. Mean ; median ; 2. Right-skewed — the mean () is above the median (), pulled up by the value ; 3. Median, because it is not distorted by the outlier (), unlike the mean; 4. Sample answer: “This report describes the number of books read last term by students, from a class library survey (primary data). The distribution is right-skewed: most students read – books, but one student’s books is a clear outlier. Because of this skew, the median ( books) better represents a typical student than the mean ( books), which is pulled upward by the outlier. Overall, most students read a small number of books, with one notable exception.”

Common Misconceptions

Misconception	How to pre-empt it
Reporting a statistic (e.g. “the mean is ”) without any context or units.	Checklist requires context and source as the first sentence of every report.
Naming a shape without justifying it from the data (mean vs median, or the graph).	Model report explicitly shows the reasoning: “because the mean is pulled upward…”
Treating “report” as just listing all five statistics with no interpretation.	Distinguish calculating (Lesson 107’s skill) from reporting (today’s skill) — a report explains what the numbers mean.
Choosing mean by default, regardless of shape.	Every report must justify its choice of centre using the shape.
Omitting outliers, or removing them silently rather than reporting their effect.	Report checklist has a dedicated “outliers” row.
Ending a report with a number instead of a sentence.	Require an explicit plain-language conclusion, as in the model report.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A report states: “The mean of the values is .” Six of the values are . Find the seventh value.

Answer

E2 (AMC Junior style). A report claims: “The median student score was out of , so nearly everyone scored well.” The full data set is: . Critique this conclusion.

Answer

The median () is technically correct, but the conclusion is misleading — three students scored very low (), a real spread the median alone hides. A complete report would include the range or IQR and mention these low scores, rather than implying the whole group did well.

E3 (Challenge). A data set has values. Its mean is and its median is , and every value lies between and . Does this guarantee the distribution is symmetric? Explain, referring to what a report can and cannot claim.

Answer

No — equal mean and median, and a bounded range, still do not guarantee symmetry (see Lesson 107, E3). A responsible report should say “the mean and median are close, which is consistent with a roughly symmetric distribution,” rather than claiming symmetry as a certainty, unless the actual shape has been checked against a graph.

E4 (Investigation). Two reports both describe “typical commute time: minutes” using the median. Report A has IQR minutes; Report B has IQR minutes. Explain why a complete report needs the IQR, not just the median, to be genuinely useful.

Answer

Both groups have the “same” typical commute, but Report A’s commuters are far more consistent (most within a few minutes of ), while Report B’s vary hugely — some very short, some very long. A reader relying only on the median would wrongly assume the two groups are similar; the spread reveals they are not.

Homework

1. List, in order, the six components a complete statistical report should include.
2. Given the data set (screen breaks taken by students in a day, primary data, class survey), calculate the mean, median, mode, range and IQR.
3. Describe the shape of the Q2 data set, with a reason.
4. Write a complete report (using the checklist) for the Q2 data set.
5. A report says: “The data was skewed, so we used the median.” Explain why this sentence, while true, is not a complete justification. Rewrite it more fully.
6. Reasoning. Explain why a report should state whether data is primary or secondary, even if the statistics themselves would be calculated the same way either way.
7. Challenge. A poorly written report says: “Most people earn about 60{,}00035, 38, 40, 42, 45, 48, 300$. Rewrite this as a complete, accurate report.

Answers: Q2 — mean ; median ; mode and (bimodal); range ; , , IQR . Q3 — strongly right-skewed: mean () well above median (), pulled up by the outlier . Q5 — it names the shape and the choice, but doesn’t explain why skew makes the median more appropriate (that the mean is distorted by extreme values); fuller version: “The data was right-skewed, so the median was used because it is not affected by the extreme high value, unlike the mean.” Q7 — sample report: “This report describes weekly earnings (in thousands of dollars) for a sample of people. The distribution is strongly right-skewed: six people earn between 35{,}000$48{,}000$300{,}000$42{,}000$78{,}000$), which is heavily inflated by the outlier. Overall, most people in this sample earn a moderate income, with one very high exception.”