Worked examples
Ten situations, all fictional, each with the arithmetic done both ways. If you only read one thing on this site before a reporting period, this is the useful one.
Every number below is invented. They are built to make one error visible at a time. You can reproduce most of them in the calculator.
1. The observed zero
Sessions: 2 of 5, 9 of 10, 4 of 5, 0 of 5. The student was present for all four.
| Treatment | Result | What it claims |
|---|---|---|
| Keep the zero | 60% | 15 of 25 opportunities. What happened. |
| Drop the zero | 75% | 15 of 20. A day the student was present has been deleted. |
Fifteen percentage points, from one judgement call about a row that looked like no data and was not. This is the single most common way a progress percentage drifts upward.
2. Pooled against the mean of sessions
The same four sessions, summarised two legitimate ways.
| Method | Result | Weights | Fits a goal written as |
|---|---|---|---|
| Pooled totals | 60% | every opportunity equally | “on 80% of opportunities” |
| Mean of session percentages | 52.5% | every session equally | “on 4 of 5 consecutive sessions” |
Seven and a half points apart, both correct. They diverge exactly when sessions have different numbers of opportunities, which is most of the time in a real classroom.
3. The correction counted twice
Friday was written as 3 of 5 from memory, then rescored to 4 of 5 against the tally sheet.
| Treatment | Total | Problem |
|---|---|---|
| Corrected row replaces the original | 15 / 25 = 60% | None. |
| Both rows kept | 18 / 30 = 60% | The percentage happens to match, but the denominator claims thirty opportunities where twenty-five were administered. Any sample-size or coverage statement built on it is wrong. |
The percentage can survive the error while the evidence behind it does not. That is why the denominator belongs next to the number.
4. Missed intervals
Twelve planned intervals: 6 scored yes, 3 scored no, 3 missed because you were pulled away.
| Treatment | Result | What it claims |
|---|---|---|
| Occurrence out of observed | 6 / 9 = 66.7% | Correct. Report coverage of 9 of 12, or 75%, beside it. |
| Missed counted as no | 6 / 12 = 50% | Three intervals nobody watched have been scored as non-occurrence. |
If all twelve are missed, occurrence is not calculable and coverage is 0%. It is not a 0% occurrence rate, and the two mean different things to a team deciding what to do next.
5. Rate without a window
Monday: 6 events in 12 observed minutes. Tuesday: 3 events in 3 observed minutes.
| Reading | Monday | Tuesday | Conclusion |
|---|---|---|---|
| Raw count | 6 | 3 | Monday looks twice as bad. |
| Rate per minute | 0.5 | 1.0 | Tuesday is twice the rate. |
Exactly backwards, from a sheet with no window column on it. This is why the frequency layout in the sheet builder makes the window mandatory.
6. Overlapping durations
Two episodes, 90 seconds and 150 seconds, inside 600 seconds of observation.
| Case | Total | % of observed |
|---|---|---|
| Non-overlapping | 240 s | 40% |
| The 90 s sits inside the 150 s, both counted | 240 s | 40% claimed; 25% actual |
Overlap is easy to create when two people record the same session, or when one episode is logged at its start and again when it is noticed. Record the combined span once.
7. The prompted response
Ten opportunities: 4 independent correct, 2 correct after a verbal prompt, 2 incorrect, 2 not administered.
| Criterion | Result | Denominator |
|---|---|---|
| Independent only | 50% | 4 of the 8 administered |
| Independent or prompted | 75% | 6 of the 8 administered |
| Either, over all ten | 40% or 60% | Two opportunities that were never administered are in the denominator. |
Both of the first two are defensible. Switching between them between reports, without saying so, is what makes a year of data unreadable. Label the criterion every time.
8. Averaged prompt codes
A session coded I, V, G, I, P. Somebody assigns 4, 3, 2, 4, 1 and reports a mean prompt level of 2.8.
That number has no meaning. The codes are categories, not quantities: a gestural prompt is not twice a physical one, and the distance from independent to verbal is not the same as verbal to gestural. Report how many opportunities met the stated criterion, and report the distribution of prompts separately if it is useful.
9. Pooling across scopes
Three sessions in the classroom under the January criterion, three in the resource room under the March one.
Pooled, the six sessions produce a single percentage that describes neither setting and neither criterion. It will look perfectly ordinary in a report. Summarise each scope separately, and if the team genuinely wants them combined, say so explicitly and record why. The calculator will flag it when the goal, protocol or prompt condition differs across the rows you enter.
10. Percentages with no denominator
An old file has 40%, 90%, 80%, 0% and nothing else.
You can plot those four points faithfully. You cannot produce a pooled total from them, because pooling needs the counts, and there is no way to recover whether 80% was 4 of 5 or 8 of 10. Inventing a plausible denominator to make the arithmetic work produces a number that looks checkable and is not. The graph tool plots this case and refuses to pool it.
The pattern
Nine of the ten are the same mistake wearing different clothes: something that was not observed ended up in a denominator, or something that was observed got left out of one. The habit that catches nearly all of them is writing the number with its denominator attached, every time, out loud, including to yourself.