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Distinguish trace rainfall, measured zero and missing data. Compare cleaning decisions with an original synthetic table and explain their effects on means and sample size.
Before you startWeather data context, means and sampling frames.
01 / Three records, three meanings
0: a recorded zero. tr: trace rainfall below 0.05 mm. n/a: unavailable.
Do not turn missing information into a measured zero.
A trace entry records a tiny amount, not an absent observation. It usually needs a stated numerical approximation for arithmetic. A missing entry provides no rainfall amount for that day.
Pearson’s 2023 examiner report, question 3 explains why discarding trace days changes calculations. Follow the convention specified by the question; otherwise state and justify your approximation.
Invented teaching data, not Met Office records. Daily rainfall (mm):
| Day | Record |
|---|---|
| 1 | 0 |
| 2 | tr |
| 3 | 0.6 |
| 4 | n/a |
| 5 | 1.4 |
| 6 | tr |
| 7 | 0.2 |
| 8 | 0 |
Total 2.2 mm ÷ 7 observed days ≈ 0.314286 mm/day.
Trace is approximated as zero; both trace days remain in the count. The missing day is excluded.
Numerical displays are rounded to six decimal places. A mean over observed days is not a known mean for all eight days.
02 / Replace trace transparently
Known numerical total: 0 + 0.6 + 1.4 + 0.2 + 0 = 2.2 mm
The two trace entries contribute zero under this approximation.
There are seven observed days
Count the two trace days; exclude only the n/a day.
Mean ≈ 2.2/7 = 0.314286 mm/day
This is the observed-day mean under the stated convention.
Using 0.025 mm for each trace is a midpoint-style approximation to a value below 0.05; it is not a claim to know the actual amounts. Here it gives 2.25/7 ≈ 0.321429 mm/day. State which convention you use.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
A sample has 12 dates: eight numerical rainfall values, three traces and one unavailable entry. With trace treated as zero, how many days enter the mean?
Trace days have an observation; the unavailable day does not.
Eleven: eight numerical records plus three trace records.
The eight numerical values total 5.5 mm. Find the eleven-day mean if each of the three traces is approximated as zero.
Divide by the observed-day count, not the numerical-cell count.
5.5/11 = 0.5 mm/day.
03 / Missing is not zero
If you calculate a mean from the observed entries, report the number used and explain which entries were unavailable. Replacing n/a with zero silently assumes no rain fell, which the record does not say.
Missingness may be related to the measurement process or weather. Excluding missing days can therefore affect representativeness; it is a transparent calculation rule, not a guarantee of an unbiased estimate of all days.
Observed rainfall values on four days are 0, 0.4, 0.8 and 2.0 mm; a fifth is unavailable. Find the observed-day mean and explain why dividing by five is unjustified.
The known total is 3.2 mm.
3.2/4 = 0.8 mm/day. Dividing by five gives 0.64, implicitly treating the unknown rainfall as zero.
Can the exact five-day mean be recovered from those four values and n/a?
The missing amount contributes to the total.
No. If the missing amount is x mm, the five-day mean is (3.2 + x)/5. Its value is unknown without more information.
04 / How much can traces change a mean?
With r traces and n observed days, replacing every trace by zero changes the mean by less than 0.05r/n mm.
This bound concerns trace approximation, not unknown missing values.
In the model, r = 2 and n = 7, so the change is less than 0.1/7 ≈ 0.014286 mm/day. This explains why a declared zero approximation can be reasonable for some purposes. It does not justify discarding those two observed days.
There are four trace entries among 20 observed days. Bound the difference between the exact observed-day mean and the mean obtained by replacing all traces with zero.
Each omitted trace amount is below 0.05 mm.
The total difference is less than 0.2 mm, so the mean difference is less than 0.2/20 = 0.01 mm/day.
In the model, discarding both trace days gives 2.2/5 = 0.44. Explain why this is not the same as approximating trace as zero.
The denominator changed.
Approximating retains seven observed days, giving 2.2/7. Discarding removes two low-rainfall observations and changes the population of days being averaged.
05 / Requested versus usable samples
Requesting 30 randomly selected calendar dates does not guarantee 30 usable readings for a variable. You could sample from all eligible dates and report missingness, or deliberately define a frame of dates with that variable available. The latter changes the population directly represented.
If comparing two variables, use records where both needed values are available and retain the pairing. Different variables can have different usable sample sizes.
Twenty-five dates are chosen. Temperature is available for 24 and rainfall for 23, but two different dates are missing rainfall and a third is missing temperature. How many complete temperature–rainfall pairs remain?
Exclude the union of missing dates.
Three distinct dates lack at least one variable, leaving 25 − 3 = 22 complete pairs.
A student removes all missing-wind-speed dates before selecting a random sample. What population is directly sampled?
The eligible frame now contains only available readings.
The dates with wind-speed readings available in the chosen station and period. The sample is not automatically representative of dates with unavailable readings.
06 / Cleaning is a documented decision
Preserve the original table. Create a working copy or an additional cleaned column, record trace substitutions and identify missing values explicitly. Inspect unusual values against units and context. An extreme but valid observation should not be removed merely because it changes the mean.
A rainfall value is much larger than nearby entries. Should it automatically be deleted?
Unusual is not the same as erroneous.
No. Check the source, units and plausibility. Retain a valid extreme value; if an error is established, document the correction or exclusion.
07 / State value rules and counts
Write a defensible one-sentence report for the model using trace as zero.
Include the approximation, count and missingness.
For the seven observed days in this invented eight-day table, the mean rainfall is approximately 0.314 mm/day when both traces are treated as zero; one unavailable day is excluded, so this is not the known mean for all eight days.
Section 1 of 7 · Three records, three meanings