Process, analyse and interpret quantitative and qualitative data, including identifying and accounting for sources of error and uncertainty

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.