# Collecting, Displaying and Misusing Data

> Types of data, census and samples, sampling methods and bias, stem-and-leaf plots, histograms, misleading graphs and time-series data.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/collecting-and-displaying-data/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The canteen survey
The school canteen wants to know if students would buy a new healthy wrap. The manager asks the first 20 students in the lunch line on Monday. Eighteen say yes. Great news?

Not really. The first students in line might be the hungriest, Monday might be different from other days, and 20 students is a small group. How data is **collected** decides whether it can be trusted. And how it is **displayed** decides whether people read it correctly.

**In short:** Data is either categorical (groups, like colours) or numerical (counts or measurements). A sample must be chosen fairly to represent the whole population, and graphs like histograms, dot plots and box plots show data clearly when their scales are not misleading.

## Starting point: Years 9 and 10

### Types of data

- **Categorical** data is in groups: favourite sport, eye colour. **Nominal** has no order, **ordinal** has an order (small, medium, large).
- **Numerical** data is numbers: **discrete** is counted (number of pets), **continuous** is measured (height).

### Census or sample?

A **census** collects data from every member of a population. A **sample** uses part of it. Samples are cheaper and faster, but only useful if they **represent** the population.

### Sampling methods

- **Simple random:** everyone has an equal chance, like drawing names from a hat.
- **Systematic:** choose every $k$th person from a list, such as every 10th student.
- **Stratified:** split the population into groups (strata), then sample each group in proportion to its size.

### Worked example: Stratified sample
A school has 300 Year 9, 250 Year 10 and 450 Year 11 students. For a sample of 50 from 1000 students, take $\dfrac{300}{1000} \times 50 = 15$ Year 9 students. The same working gives 12.5 Year 10 and 22.5 Year 11 students, which round to 13 and 22 so the total stays 50.

### Bias

A sample is **biased** if some people are more likely to be chosen, or if the questions push people towards an answer. The canteen survey was biased: only hungry students on one day were asked. Leading questions like "Don't you agree the wrap is delicious?" also cause bias.

### Displays

- **Dot plot** and **column graph** for small data sets or categories.
- **Stem-and-leaf plot**: the stem is the tens digit, the leaf the units. It keeps every value and shows the shape. A **back-to-back** stem-and-leaf plot compares two groups.
- **Histogram**: bars touching, for grouped numerical data.

| Stem | Leaf |
|---|---|
| 1 | 2 5 8 |
| 2 | 0 3 3 7 9 |
| 3 | 1 4 |

This shows 12, 15, 18, 20, 23, 23, 27, 29, 31, 34. The mode is 23 and the median is the average of 23 and 23, so 23.

### Misleading graphs

Watch for an axis that does not start at zero, uneven scales, pictures where both width and height grow, and missing labels. They can make a small change look huge.

### Time-series data

A **time series** records the same thing over time, like monthly rainfall or daily temperatures. Plot it as a line graph with time across the bottom.

*Diagram: Time-series graph of ice cream sales over two years, rising each summer and with an upward trend*
*Sales peak every summer (seasonal) and slowly rise overall (trend).*

### Key idea: Reading a time series
- **Trend:** the long-term direction (up, down or steady).
- **Seasonal:** a pattern that repeats at the same time each cycle.
- **Irregular:** random ups and downs with no pattern.

### Watch out: A big sample is not always a good sample
1000 biased responses are worse than 100 random ones. How the sample is chosen matters more than its size.

**Mind map**

```text
Data
  Types
    Categorical: nominal, ordinal
    Numerical: discrete, continuous
  Collecting
    Census or sample
    Random, systematic, stratified
    Bias
  Displays
    Dot plot, column graph
    Stem-and-leaf
    Histogram
  Misleading graphs
    Axis not from 0
    Uneven scales
  Time series
    Trend
    Seasonal
    Irregular
```

### Year 11: Comparing two data sets
To compare two groups, use the same kind of display for both: a back-to-back stem-and-leaf plot, or two box plots on the same scale. Compare the **centre** (median or mean), the **spread** (IQR or standard deviation) and the **shape** (symmetric or skewed), and mention any outliers.

**Example.** Class A's median test score is 72 with IQR 10. Class B's median is 68 with IQR 25. Class A did better on average and its results were much more consistent.

**Reading a box plot.** The four parts of a box plot (lower whisker, two halves of the box, upper whisker) each hold about 25% of the data. So about 75% of the values lie between $Q_1$ and the maximum, and about a quarter of 680 students, 170, scored above $Q_3$.

**Cumulative frequency** is a running total: daily counts 2, 6, 3, 5 have cumulative frequencies 2, 8, 11, 16.

**Pareto charts.** List the categories from most to least frequent as columns, and draw a line showing the cumulative percentage. For faults caused 50, 30, 10, 6 and 4 times (out of 100), the line reads 50%, 80%, 90%, 96%, 100%. The **Pareto principle** says a few causes (often about 20%) explain most of the effect (often about 80%), so fixing the biggest causes first gives the most improvement.

### Year 12: Smoothing a time series
A **moving average** smooths out seasonal and irregular changes so the trend is clearer. A 3-point moving average replaces each value with the mean of it and its two neighbours: for 10, 16, 13, 19, the moving averages are $\dfrac{10 + 16 + 13}{3} = 13$ and $\dfrac{16 + 13 + 19}{3} = 16$.

A least-squares line through the smoothed values gives a model of the trend, which can be used to forecast, with the usual warning that extrapolation is risky.

## Quick summary

- Data is categorical (nominal or ordinal) or numerical (discrete or continuous).
- A census asks everyone. A sample asks some, and must be representative.
- Sampling: simple random, systematic, stratified. Bias makes results untrustworthy.
- Displays: dot plots, column graphs, stem-and-leaf plots, histograms. Watch for misleading scales.
- Time series show trend, seasonal and irregular patterns.

## Quiz

Try these questions from the chapter quiz.

1. "Favourite colour" is what type of data?
   - A. categorical
   - B. discrete numerical
   - C. continuous numerical
   - D. ordinal numerical
   - Answer: A. categorical. It is a category, not a number.

2. "Time to run 400 m" is what type of data?
   - A. categorical
   - B. discrete
   - C. continuous
   - D. nominal
   - Answer: C. continuous. Time is measured.

3. T-shirt sizes small, medium and large are:
   - A. nominal
   - B. ordinal
   - C. continuous
   - D. discrete
   - Answer: B. ordinal. Categories with a natural order.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/collecting-and-displaying-data/#quiz