Appendix 1 2(d)(i), 2(d)(iii) - Rates, sampling and descriptive statistics

Rates and denominators

A total often reflects the size of the population or the duration of observation. A rate relates a quantity to an appropriate denominator so comparisons become meaningful.

In original illustrative data, 12 reported water-supply interruptions among 600 households represent $12/600\times1000=20$ reports per 1,000 households. A different area has 15 among 1,500 households, or 10 per 1,000. The second has more reports but a lower rate. Check whether repeated reports from the same household are included: a report rate is not automatically the percentage of households affected. Definitions and periods must match.

If mapped coastline retreat is 3.6 m over four years, its average rate is $3.6/4=0.9\,\mathrm{m\,yr^{-1}}$. This does not establish steady annual retreat; one storm could account for much of it. Rainfall of 15 mm in 30 minutes gives an average intensity of 30 mm per hour, not 0.5 mm per hour. Convert time before calculating.

A proportion rising from 20% to 25% increases by five percentage points, but by $(25-20)/20\times100=25\%$ relative to its initial value. Distinguish these statements. Round at the end, retain units and avoid giving more precision than the observations support.

Measurement and error

A numerical observation depends on an operational definition and a measurement procedure. Decide what is measured, where and when, the instrument's units and resolution, and whether repeated observations describe the same phenomenon.

Accuracy concerns closeness to the true or accepted reference value. Precision concerns how closely repeated readings agree; a precise instrument can consistently read too high. Random error varies unpredictably between readings, while systematic error shifts readings consistently, for example a miscalibrated ruler or a sensor offset. Repeated readings can reduce the effect of independent random error on a mean, but averaging does not remove a systematic bias. Correct or calibrate the instrument, not just increase the sample size.

Original illustrative channel-width readings are 4.8, 5.0 and 5.2 m. Their mean is 5.0 m. The half-range is $(5.2-4.8)/2=0.2$ m, which gives a simple description of repeatability, not a complete confidence interval. If the tape adds 0.3 m to every reading, the mean still overestimates width by 0.3 m. Bank movement between readings is genuine variation rather than necessarily instrument error.

Keep instrument resolution and uncertainty separate. A display resolving 0.01 m does not prove accuracy to ±0.01 m. Record method limitations, calibration, repeats and conditions. A positional error can assign a reading to the wrong land-use zone even when the reading itself is accurate.

Sampling design and representativeness

A sample studies part of a defined population, such as all eligible households or all possible quadrat positions in a study area. A sampling frame lists or maps the units from which selection is made. Define its boundaries and period before claiming the result represents the population.

Random sampling gives each eligible unit a known chance of selection, commonly by numbering a frame and selecting random numbers. It reduces selection by convenience but may leave spatial gaps. Systematic sampling selects regular intervals after a random starting point; it gives coverage along a transect but may coincide with a repeating pattern, such as every sixth unit being a corner plot. Stratified sampling divides a population into relevant groups and samples within each. In proportional stratification, each group's sample share matches its population share; select randomly within groups where possible.

Consider an original illustrative frame of 120 equal-area plots: 72 residential, 36 industrial and 12 park plots. A proportional sample of 20 selects $72/120\times20=12$ residential, six industrial and two park plots. For a systematic sample across the whole numbered frame, the interval is $120/20=6$; a randomly selected start of 4 gives 4, 10, 16, …, 118. Before using it, check whether frame ordering repeats every six units. Proportional allocation may give too few park plots for a secure park-specific conclusion. Oversampling parks can improve that comparison, but a pooled estimate then needs appropriate weighting.