Goal and prerequisites
Use cumulative totals for intervals and percentiles, and read a fitted line without claiming causation. Review statistics before starting.
Method and its conditions
A cumulative frequency gives the number at or below a boundary. Subtract cumulative totals to obtain an interval frequency. Locate quartiles at one quarter and three quarters of the total, then read across to the curve and down to the value axis. A best-fit line is a model, not a sequence of joined observations.
Worked example: show the decisions
If cumulative counts at 10 and 20 are 18 and 46, then 28 observations lie above 10 and at or below 20. For a sample of 80, the lower and upper quartile ranks are 20 and 60; if the curve gives values 12 and 30, the IQR is 18. Interpolation from grouped data produces estimates.
Contrast: diagnose an incorrect approach
The vertical difference 60−20 is a count difference, not the IQR in measurement units. On a scatter plot, a trend can support a prediction within the observed range without showing that one variable causes the other. Extrapolation beyond that range needs additional justification.
Transfer: try a changed situation
A plotted fitted line is y=2 x+3. Predict y at x=4 and find the residual if the observed y is 10. Explain the sign.
Transfer solution and interpretation
The prediction is 11 and observed minus predicted is−1. The observation lies one unit below the model. This numerical residual assesses that point’s fit; it does not prove the line is a causal or universally valid relationship.
A cumulative-frequency table gives 9 observations at or below 10 and 24 at or below 20. How many observations are greater than 10 and at or below 20?
- Subtract the earlier cumulative count from the later cumulative count.
- Using the stated quantities: 24-9 = 15.
- Check the domain, units and whether the answer is an exact value, bound or estimate.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
0580 syllabus for exams in 2025–2027