Mathematics & Statistics 1.1 — Explore data using a statistical enquiry process
NCEA Level 1 · Achievement Standard 91944 · Study guide + self-marking quiz
What this standard actually asks you to do
You run one complete statistical investigation from start to finish: pose a question, get some data, graph it, describe what you see, and conclude. It is internal, so your teacher assesses it during the year — there is no exam for 1.1.
The three grade levels — word for word
| Grade | What the standard requires |
|---|---|
| ACHIEVED |
Explore data using a statistical enquiry process involves:
|
| MERIT |
…with statistical justification:
|
| EXCELLENCE |
…with statistical insight:
|
A Describe what you see. → M Back it up with numbers, and finish the whole cycle. → E Bring in real-world knowledge and honestly critique your own process.
How much data you need
| Investigation style | Minimum recommended |
|---|---|
| Relationship | 30 pairs of data |
| Comparison | 30 values in each group (100 or 1000 per group if sourcing from a big dataset) |
| Time series | 5 complete cycles |
| Experimental probability | 30 trials |
Rules of the assessment
- You choose your own investigative question / statement, with teacher approval. Your teacher may correct the wording.
- You may work in a group for planning and collecting data, but every other stage must be your own individual work.
- You can use technology (spreadsheets, NZGrapher, CODAP, iNZight).
- Your teacher can give a milestone check and guidance on your plan and sample size — but not corrections of specific detail.
- Roughly 1,000–1,500 words plus 2–3 visualisations is typical. Depth beats length.
How 1.1 fits with the rest of Level 1 Maths
| Std | Title | Assessment |
|---|---|---|
| 91944 (1.1) | Explore data using a statistical enquiry process | Internal · 5 cr |
| 91945 (1.2) | Use mathematical methods to explore problems that relate to life in Aotearoa NZ or the Pacific | Internal · 5 cr |
| 91946 (1.3) | Interpret and apply mathematical and statistical information in context | External · 5 cr |
| 91947 (1.4) | Demonstrate mathematical reasoning | External · 5 cr |
PPDAC — the statistical enquiry cycle
Almost everyone uses PPDAC. It is the skeleton your report hangs on. The single biggest thing that separates Achieved from Merit is connecting the stages to each other instead of treating them as five separate boxes to fill in.
P — Problem
Write an investigative question you can actually answer with data. A good one names four things:
- the variable you'll measure (and its units)
- the group(s) being compared or related
- the population it's about
- a word showing you mean typical / tend to, not every individual
P — Plan
Say exactly how the data will be obtained. Cover: who (population and how you'll sample from it), what (variables and units), how (the measuring or sourcing method, step by step), how many (and why that's enough), and what could go wrong (sources of variation).
D — Data
Primary data = you collect it (measure, survey, experiment). Secondary data = someone else collected it (CensusAtSchool, Stats NZ, NIWA, sports records).
If secondary, you still must explain the original collection: who collected it, when, from whom, and how. Note the metadata too — for example, in a set of test scores the scores are the data, while who sat the test, when, and in what subject is the metadata.
Show your cleaning: values you removed and why, units you converted, obvious typos you fixed.
A — Analysis
Graph it appropriately, then describe features in context. Every observation gets the same three-part treatment (see tab 5).
C — Conclusion
Answer your original question, in a sentence that echoes its wording. Then, for Excellence, reflect: how well did the process work? What limited it? What would you change and why?
The four investigation styles
You must pick one of these four. Choose early, because it decides your graph, your measures and the type of conclusion you're allowed to draw.
1 · Comparison — numerical comparison of two or more groups
Graph: box and whisker plot (parallel boxplots). Dot plots, stem-and-leaf and histograms are supporting evidence only — never sufficient on their own.
Features to describe: centre, spread, shape, shift and overlap of the two groups, clusters, unusual points.
Measures for Merit: medians, quartiles, IQR, difference between medians (DBM), overall visible spread (OVS), values of unusual points.
Special skill: making an informal sample → population inference ("making the call").
Making the call
You are describing a sample, but your question is about a population. "Making the call" is deciding whether the difference you can see is big enough to claim it holds back in the population.
- Samples of about 30 in each group — use the three-quarters/half guideline:
- Boxes don't overlap at all (or only just touch) → make the call.
- Boxes overlap, but each group's median sits outside the other group's box → make the call.
- Otherwise → you can't make the call. That doesn't mean the groups are the same; it means your sample isn't enough to tell.
- Samples of 100 or 1000 per group — judge the distance between the medians as a proportion of the overall visible spread. Bigger samples let you call smaller differences. Use the "How to make the call" guideline sheet your teacher gives you, since the exact fraction depends on sample size.
- Uneven group sizes? The smaller group decides which method you use.
2 · Relationship — between two numerical variables
Graph: scatter graph.
Features: direction and strength of the trend, clusters, unusual or interesting points, patterns.
Measures for Merit: actual data values inside clusters, quadrant counts, coordinates of unusual points.
Allowed: an informal prediction, either by reading off the trend line or by substituting into its equation.
3 · Time series — one variable measured over time
Graph: time series graph.
Features: long-term trend, seasonality and cycles (and their length), spikes and troughs, variation, unusual points.
Measures for Merit: values at spikes/troughs, the length of a season or cycle, size of the rise or fall.
You need 5 complete cycles (e.g. 5 years of monthly data if the season is annual).
Include a forecast, made informally by eye from the graph. It's legitimate to argue that a forecast isn't useful — but you must justify why. Formal forecasting methods are out of scope.
4 · Experimental probability — events with at least two stages
Graph: bar/frequency graph of outcomes, two-way table, or long-run relative frequency graph.
Features: centre, spread, shape, clusters, patterns, unusual results.
Measures for Merit: median, mean, experimental probabilities, values from the two-way table.
At least 30 trials, and the event must have two or more stages — a single die roll or one spinner spin is too simple, so use a digital simulation to build it up to the right level.
Choosing the graph
| Your investigation | Use |
|---|---|
| Comparing two or more groups on a numerical variable | Box and whisker plot (add a dot plot underneath as support) |
| Relationship between two numerical variables | Scatter graph |
| One variable measured over time | Time series graph |
| Outcomes of a probability experiment | Bar / frequency graph, two-way table, or long-run relative frequency graph |
Reading a box plot
The measures you need
| Measure | How to find it | What it tells you |
|---|---|---|
| Median | Order the data; take the middle value (average the middle two if n is even) | The typical / centre value. Not dragged around by outliers. |
| Mean | Add all values ÷ number of values | Centre, but pulled towards extreme values. |
| LQ & UQ | Median of the lower half; median of the upper half | Where the middle 50% starts and ends. |
| IQR | UQ − LQ | Spread of the middle 50%. Bigger IQR = more varied group. |
| Range | max − min | Total spread. Very sensitive to one weird value. |
| DBM | Difference between the two medians | How far apart the two groups are. |
| OVS | Largest value across both groups − smallest across both | The yardstick you compare DBM against. |
Worked example — quartiles by hand
Times (minutes) for 11 students to get to school:
12, 15, 15, 18, 20, 22, 24, 25, 28, 30, 35
- n = 11, so the median is the 6th value = 22 min
- Lower half (the 5 values below the median): 12, 15, 15, 18, 20 → LQ = 15 min
- Upper half: 24, 25, 28, 30, 35 → UQ = 28 min
- IQR = 28 − 15 = 13 min
- Range = 35 − 12 = 23 min
Note: software (Excel, NZGrapher, CODAP) sometimes computes quartiles slightly differently. That's fine — just be consistent and say which tool you used.
Sources of variation — the vocabulary that earns marks
| Type | Example in a reaction-time study |
|---|---|
| Natural variation | People are simply born with different reaction speeds. |
| Measurement variation | The online timer rounds to the nearest 0.01 s; a laggy laptop adds a delay. |
| Occasion variation | The same student is slower first thing in the morning than after lunch. |
| Induced variation | Some students had already practised the test; others hadn't. |
| Sampling variation | A different sample of 30 students would give a slightly different median. |
Naming these in your Plan (what you expected and how you'd manage it) and again in your Conclusion (what actually happened) is one of the cheapest routes to Merit and Excellence.
Where to get good data
- CensusAtSchool / TatauranaKiTeKura — real NZ student data, sample sizes of 100 or 1000 on tap
- Stats NZ — population, housing, income, transport
- NIWA / CliFlo — rainfall, temperature (perfect for time series)
- Sport NZ, club and school records — scores, times, attendance
- Measure it yourself — arm span, hand span, reaction time, memory tests, kicking accuracy
Writing that hits A, then M, then E
The three-part sentence — use it for every single observation
ACHIEVED level
"The Year 13 box is further to the left than the Year 9 box, so Year 13 students in my sample tended to have faster reaction times."
Feature named ✓, context ✓, but no measure — this is describing.
MERIT level
"Year 13 students in my sample tended to react faster than Year 9 students. The Year 13 median was 0.24 s compared with 0.32 s for Year 9 — a difference of 0.08 s, which is nearly a fifth of the overall visible spread of 0.44 s. Because each group's median falls outside the other group's box, I can call it back to the population: Year 13 students at our school tend to have faster reaction times than Year 9 students."
Feature ✓ + measures ✓ + context ✓ + sample→population ✓
EXCELLENCE level — adds real-world knowledge and honest reflection
"Year 13 students tended to react faster (median 0.24 s vs 0.32 s, a difference of 0.08 s against an overall visible spread of 0.44 s), and I can make the call because both medians sit outside the other group's box. Reaction time is known to improve through adolescence as the nervous system finishes developing, which fits the direction of this difference. However, the Year 13 group sat the test in the computer lab during class, while the Year 9 group did it at home on their own devices — so some of the 0.08 s gap could be induced variation from different devices and different levels of distraction, not age. The Year 9 group also had the wider box (IQR 0.14 s vs 0.09 s), which is consistent with less controlled conditions. If I ran this again I would test both groups on the same machines in the same room, and select students randomly from the whole year level rather than only from the two classes I could access."
Sentence starters
The four mistakes that cost the most marks
- Graph-speak with no context. "The bar is higher." Higher what, for whom, and so what? Every sentence must land back on your investigative question.
- Not explaining the data source. Jumping from question straight to graph. Where did it come from, who was in it, how many, and why is that suitable?
- Overclaiming from a sample. "This proves all Year 13s are faster." You have a sample; you can only say the population tends to, and only if you can make the call.
- A conclusion that doesn't answer the question. Re-read your Problem statement and answer it in its own words.
Pre-submission checklist
| Level | Check |
|---|---|
| A | My investigative question names the variable, units, group(s) and population. |
| A | I've explained where the data came from, who's in it, how many, and how it was collected. |
| A | I have at least one appropriate, fully labelled visualisation. |
| A | I describe at least 3 features (centre, spread, shape/trend, unusual points) in context. |
| M | Every feature is backed by a specific measure from my data. |
| M | All five PPDAC stages are present and refer back to each other. |
| M | I've distinguished my sample from the population and made (or declined) a call, with reasons. |
| E | I've brought in outside knowledge about the context that explains what I found. |
| E | I've reflected on the enquiry process itself and named specific limitations. |
| E | My improvements are specific and justified, not "I'd get more data". |
Quiz — 22 questions across Achieved, Merit and Excellence
ACHIEVED Knowing the standard
ACHIEVED Reading and calculating
Shared context for Q11, Q12, Q13 and Q16. A student sourced reaction times (seconds, from an online tap-test) for a sample of 35 Year 9 students and 35 Year 13 students at their school.
Five-number summaries. Year 9: min 0.18, LQ 0.26, median 0.32, UQ 0.40, max 0.58. Year 13: min 0.14, LQ 0.20, median 0.24, UQ 0.29, max 0.44.
MERIT Justifying with measures
EXCELLENCE Insight and reflection
Written tasks — compare your answer to the model
These are the parts that actually decide your grade. Write your own answer first, then open the model.
Mini investigation — put it all together
Go to CensusAtSchool NZ and pull a random sample of 100 Year 11 students. Choose one numerical variable (e.g. arm span, travel time to school, hours of sleep) and one grouping variable (e.g. gender, travel method, region). Then produce a one-page investigation:
- An investigative question with variable, units, groups and population.
- A paragraph explaining the source of the data and two sources of variation you'd expect.
- Parallel box plots, fully labelled, with a dot plot underneath as support.
- Three described-and-justified features, each with a measure and context.
- A conclusion that makes (or declines) the call, plus one paragraph reflecting on the process.
Mark it against the checklist in tab 5. If every row has a tick, you're at Excellence.