Data analysis, error and uncertainty: HSC Investigating Science Module 5

What this dot point is asking

NESA wants you to process raw quantitative data into summary statistics, represent data with appropriate graphs and tables, account for measurement error and uncertainty, and identify outliers. Quantitative data analysis is examined in nearly every Investigating Science paper.

The answer

Processing data turns raw measurements into evidence that can be interpreted against a hypothesis. The standard steps:

1. Tabulate raw data.
2. Calculate summary statistics (mean, median, range, standard deviation).
3. Identify outliers and decide whether to exclude.
4. Estimate uncertainty.
5. Graph the result with error bars.
6. Interpret in light of the hypothesis.

Summary statistics

Mean. Sum of values divided by count. The most common measure of central tendency for normally distributed data.

Median

Middle value when data is ordered. Less affected by outliers than the mean.

Range

Difference between maximum and minimum. A simple measure of spread.

Standard deviation

A more rigorous measure of spread that quantifies how tightly values cluster around the mean.

Uncertainty

Every measurement has uncertainty arising from instrument resolution and natural variation. Standard ways to report:

• Absolute uncertainty. $24.7 \pm 0.1$ cm.
• Percentage uncertainty. $\frac{0.1}{24.7} \times 100 = 0.4 \%$.

For multiple measurements, the uncertainty is usually estimated as half the range or as the standard deviation of the mean.

Significant figures

Report data with significant figures appropriate to the precision of the instrument.

• A ruler graduated to 1 mm reads to $\pm 0.5$ mm; report measurements to 1 decimal place in cm.
• A digital thermometer reading to 0.1 degrees Celsius reports to that resolution; report no extra digits.

When calculating, the answer cannot be more precise than the least precise input. $24.7$ cm $\times 3.1$ cm rounds to $77$ cm$^2$ (two significant figures, matching the $3.1$).

Random and systematic error

Random error. Unpredictable variation between repeated measurements, caused by chance fluctuations. Magnitude varies; direction varies. Random error reduces with averaging.

Systematic error. Consistent bias in one direction caused by miscalibration, methodological flaw or biased observer. Systematic error does not reduce with averaging.

Outliers

A data point well outside the cluster of others. A common rule is values more than 2 to 3 standard deviations from the mean. Options:

1. Investigate. Check whether the value is a transcription error, a faulty instrument or a real but rare observation.
2. Repeat the measurement if possible.
3. Exclude with justification. Document the reason for exclusion. Do not silently drop outliers; that is selective reporting.

Graphing

Choice of graph.

• Line graph. Continuous independent variable (time, temperature).
• Bar graph. Categorical independent variable (treatment groups).
• Scatter plot. Investigating correlation between two continuous variables.

Error bars. Vertical lines showing the range or uncertainty around each data point. Mandatory for any quantitative graph in Investigating Science.

Axes. Labelled with quantity and unit. Independent variable on the x-axis, dependent variable on the y-axis. Origin clearly marked.

Interpreting in light of the hypothesis

A finding is meaningful when:

1. The treatment effect is larger than the uncertainty in the measurement.
2. The result is reproducible across replicates.
3. Alternative explanations (confounders, instrument bias) can be ruled out.

A treatment difference smaller than the error bars is not evidence of effect.

Examples in context

Example 1. Australian Bureau of Statistics Sydney population recount. After the 2021 Census, the ABS identified that an early estimate of Sydney's population had to be adjusted downward by roughly 11,000 people once dwelling-occupancy data was reconciled across local government areas. The original estimate sat outside the 95 per cent confidence interval implied by post-enumeration survey calibration, marking it as a potential outlier rather than a real demographic shift. The ABS published a revision with a clear uncertainty range (plus or minus 0.4 per cent for state-level totals). The case illustrates good practice: investigate the outlier (overcount in temporary accommodation), document the cause, and reissue with corrected uncertainty rather than silently dropping the figure.

Example 2. Cape Grim CO2 baseline uncertainty. Atmospheric CO2 at Cape Grim is measured by non-dispersive infrared spectroscopy with quoted uncertainty of plus or minus 0.1 parts per million for hourly averages. To distinguish real annual growth (now about 2.4 ppm per year) from random noise, scientists average tens of thousands of clean-air observations and calculate the standard error of the annual mean. Outliers in the raw record (often from local biomass smoke or instrument drift) are flagged and excluded with a documented filter. Error bars on the published time series show plus or minus 0.2 ppm at the 95 per cent level, allowing the growth trend to be clearly distinguished from year-to-year variability.

Try this

Q1. A class of 30 students measures the period of a pendulum and records times of 1.40, 1.42, 1.41, 1.40, 1.39, 2.10, 1.41 seconds for one student. Identify the outlier using a simple criterion and justify whether it should be excluded. [3 marks]

• Cue. 2.10 s is more than 2 standard deviations above the mean. Investigate (likely a mis-start of the stopwatch); document and exclude with justification.

Q2. Annual rainfall at Bourke is reported as 340 plus or minus 60 mm for the 2010s decade and 290 plus or minus 50 mm for the 2020s decade. Comment on whether this constitutes evidence of a drying trend. [3 marks]

• Cue. Difference (50 mm) is less than the combined uncertainty (about 78 mm). Difference not significant from this dataset alone; longer time series needed.

Q3. A student measures the mass of 10 fresh leaves on a balance reading to 0.01 g. (a) State how to report the uncertainty in a single measurement. (b) Explain how to estimate the uncertainty in the mean. (c) Identify one source of systematic error in this experiment. [2+2+2 marks]

• Cue. (a) plus or minus 0.005 g (half the smallest division). (b) Standard error: s divided by square root of n. (c) Balance not zeroed; leaf surface moisture; air currents in the lab.