# Data, Graphs and Errors

> Results tables, graphs, trends, errors and uncertainty, and how to judge and improve an experiment.

From HSC Chemistry on WhizzNotes, free study notes for NSW students: https://whizznotes.com/chemistry/data-graphs-and-errors/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The rusting nails
Ms Okafor's class put iron nails into four test tubes: dry air, tap water, salty water and oil. Each day for a week they measured how much rust had formed by weighing the nails.

Group A wrote numbers on scraps of paper and lost two of them. Group B kept a neat table, weighed each nail three times, and drew a column graph. At the end, Group B could say clearly: "Salty water rusted the nails fastest. Oil stopped rust almost completely."

Group A had the same experiment but no clear answer. Good data handling is what turns measurements into a conclusion.

**In short:** Results go in a table with units in the headings, then onto a line graph with a line of best fit for continuous data. Random errors scatter results and are reduced by repeating, while systematic errors shift every result and are fixed by changing the method or equipment.

## Results tables

A results table keeps your data organised.

- The **independent variable** goes in the **first column**.
- Each heading has the **quantity and the unit**: "Mass of rust (g)". Units go in the heading, not after every number.
- Leave columns for each **trial** and the **average**.
- Use the same number of decimal places down a column.

| Temperature (°C) | Trial 1 (s) | Trial 2 (s) | Trial 3 (s) | Average (s) |
|---|---|---|---|---|
| 20 | 62 | 58 | 60 | 60 |
| 40 | 41 | 39 | 40 | 40 |
| 60 | 26 | 24 | 64 | 25 |
| 80 | 15 | 16 | 14 | 15 |

Look at 60 °C. The third trial, 64 s, does not fit. It is an **outlier**. Leave it out of the average and repeat that trial if you can: $(26 + 24) \div 2 = 25$.

## Graphing data

1. Independent variable on the **x-axis**, dependent variable on the **y-axis**.
2. Label both axes with the quantity and unit.
3. Use an even scale that fills most of the grid.
4. Plot neat crosses, then draw a **line of best fit** (straight or a smooth curve). Never join dot to dot.
5. Give the graph a title.

| Use a... | When the independent variable is... | Example |
|---|---|---|
| **Line graph** | A number that can take any value (continuous) | Temperature, time, mass |
| **Column graph** | A name or group (categorical) | Type of liquid, metal used |

*Diagram: Line graph of dissolving time against temperature. The curve falls steeply then levels out.*
*As temperature goes up, dissolving time goes down. The curve shows the trend.*

## Describing trends and patterns

A **trend** is the overall direction of the data. Describe it in words and use numbers.

- **Positive trend**: as x increases, y increases.
- **Negative trend**: as x increases, y decreases.
- **No trend**: no clear link.
- **Directly proportional**: a straight line through the origin. Double x and y doubles.

Good answer: "As the temperature increased from 20 °C to 80 °C, the dissolving time decreased from 60 s to 15 s. The decrease was larger at low temperatures."

## Errors

| Error | What it does | Example | How to reduce it |
|---|---|---|---|
| **Random error** | Results scatter up and down unpredictably | Starting the stopwatch a little late or early | Repeat trials and average |
| **Systematic error** | Every result shifts the same way | A balance that reads 0.2 g with nothing on it | Fix or calibrate the equipment. Repeating does not help |
| **Mistake** | Human slip, gives an outlier | Reading 64 instead of 24 | Repeat that trial |

**Parallax error** happens when you read a scale from an angle. Always read a measuring cylinder at eye level, from the bottom of the curved surface (the **meniscus**).

## Accuracy, precision, reliability, validity

| Word | Meaning | Improve it by |
|---|---|---|
| **Accurate** | Close to the true value | Calibrated equipment, removing systematic error |
| **Precise** | Repeated results close together. Also, an instrument with small divisions | Finer instruments (a 10 mL cylinder instead of 100 mL) |
| **Reliable** | Repeating gives similar results | More trials, averaging |
| **Valid** | Fair test that measures what it claims | Controlling variables |

### Watch out: Do not say "human error"
In an evaluation, "human error" earns no marks. Name the actual problem and the fix: "Starting the stopwatch by hand caused random error. Use a light sensor or repeat more times."

## Suggesting improvements

When asked to improve an experiment, pick the weakness and give a precise fix.

- Results scattered: **more repeats**.
- Big gaps between x values: **more intervals** in the range.
- Hard to see the end point: use a **better end point** (for example, a cross under the beaker that disappears).
- Readings too coarse: use **more precise equipment**.
- A variable not controlled: **control it** (for example, use a water bath to keep temperature steady).

**Mind map**

```text
Data and errors
  Tables
    Independent in first column
    Units in headings
    Trials and average
  Graphs
    Line: continuous
    Column: categories
    Line of best fit
  Trends
    Positive or negative
    Proportional: straight through origin
  Errors
    Random: repeat
    Systematic: fix equipment
    Outlier: leave out, repeat
  Quality
    Accurate, precise
    Reliable, valid
```

### Year 11: Uncertainty and significant figures
Every reading has an **uncertainty**. For a scale it is usually **half the smallest division**. A 100 mL measuring cylinder marked every 1 mL gives $\pm 0.5\text{ mL}$.

$$\text{percentage uncertainty} = \frac{\text{uncertainty}}{\text{reading}} \times 100\%$$

Reading 20.0 mL with $\pm 0.5\text{ mL}$ gives $\frac{0.5}{20.0} \times 100\% = 2.5\%$.

A bigger reading gives a smaller percentage uncertainty. That is why chemists measure larger amounts where they can.

When you calculate, give the answer to the **fewest significant figures** used in the data. $2.5 \text{ g} \div 1.25 \text{ mL} = 2.0 \text{ g/mL}$ (2 significant figures).

### Year 12: Percentage error and the true value
**Percentage error** compares your result with an accepted value:

$$\text{percentage error} = \frac{\left| \text{experimental} - \text{accepted} \right|}{\text{accepted}} \times 100\%$$

If your density of copper is 8.6 g/cm³ and the accepted value is 8.96 g/cm³:

$$\frac{8.96 - 8.6}{8.96} \times 100\% \approx 4\%$$

Now compare the percentage error with your total percentage uncertainty. If the error is **bigger** than the uncertainty, there is probably a **systematic error** you have not found. If it is smaller, the difference can be explained by random error alone.

## Quick summary

- Tables: independent variable first, units in headings, trials and an average.
- Leave outliers out of the average and repeat them.
- Line graph for continuous data, column graph for categories. Draw a line of best fit.
- Describe trends with direction **and** numbers.
- **Random** errors scatter results: repeat. **Systematic** errors shift all results: fix the equipment.
- Accurate means close to the true value. Precise means close together. Reliable means repeatable. Valid means a fair test.
- Improvements must be specific, never just "human error".

## Quiz

Try these questions from the chapter quiz.

1. In a results table, the independent variable goes in the:
   - A. First column
   - B. Last column
   - C. Title only
   - D. Average column
   - Answer: A. First column. The independent variable is listed first, then the measurements.

2. Three trials give 30 s, 32 s and 31 s. What is the average?
   - A. 30 s
   - B. 31 s
   - C. 32 s
   - D. 93 s
   - Answer: B. 31 s. (30 + 32 + 31) ÷ 3 = 31 s.

3. Trials give 22 s, 24 s and 58 s. What should you do with 58 s?
   - A. Include it in the average
   - B. Treat it as an outlier, leave it out and repeat that trial
   - C. Double the other results
   - D. Change the hypothesis
   - Answer: B. Treat it as an outlier, leave it out and repeat that trial. 58 s does not fit the other two, so it is an outlier.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/chemistry/data-graphs-and-errors/#quiz