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Linking Normal Probability Plots to CDF Probability Scales

Linking Normal Probability Plots to CDF Probability Scales
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2021-01-16 06:32
FunctionSequenceProbability & StatisticsSlider AnimationGraphing
This applet demonstrates the visual relationship between a CDF probability plot (including an animated/transitioning vertical quantile scale) and a Normal probability plot. It highlights how theoretical Normal quantiles correspond to empirical quantiles derived from sample data, making it easier to judge whether the data follow an approximately Normal distribution. Key concepts include the Normal CDF, probability/quantile scales, empirical distribution (ogive-style cumulative curves), and parameterization of the Normal model using the sample mean and sample standard deviation. Construction-wise, the GeoGebra setup uses Sequence, Normal, Sort, and Join to generate and order theoretical quantile points, then Mean and SampleSD to estimate parameters (xbar, s). Conditional logic via If is used for data handling, while Translate and the internal coordinate/scale objects (probscale, linscale/scalept, ycoords variants) implement the mapping between CDF-style axes and probability-plot coordinates. Core graphic objects include normcdf and the collection of points for curves (ogive, ogivePoints, normpdfpoints), along with PolyLine/Polygon/Line/Segment for constructing cumulative and quantile lines, and Textfield/text labels (text1, text2) for clarifying the probabilistic scales. Suitable for high-school/undergraduate probability & statistics and practical data analysis demonstrations, especially for teaching Normality assessment and probability plot interpretation.

GeoGebra Commands & Parameters

  1. Sequence(RandomUniform[0, 10], i, 1, 19) → dataunif
  2. Sequence(10 - InverseChiSquared[3, RandomUniform[0, 1]], i, 1, 19) → dataleft
  3. Sequence(RandomNormal[5, 2], i, 1, 19) → datanorm
  4. Sequence(InverseChiSquared[3, RandomUniform[0, 1]], i, 1, 19) → dataright
  5. Normal(0.5, 1 / ((2 * InverseNormal[0, 1, scalept])), x) → normf
  6. Sort(If[skewleft, dataleft, If[skewright, dataright, If[norm, datanorm, dataunif]]]) → data
  7. Translate(Rotate[Sequence[(xx, (2 * normf(xx))), xx, -3, 3, 0.05], 1.5708], (axescross - normoffset, 0)) → normpdfpoints
  8. Mean(data) → xbar
  9. SampleSD(data) → s
  10. If(probscale, t2, 1 - t2) → t
  11. Sequence((0.05 * i), i, 0, 20) → ycoords1
  12. Zip(If[yy ≟ 0, -3, If[yy ≟ 1, 3, InverseNormal[0, 1, yy] / ((2 * InverseNormal[0, 1, scalept])) + 0.5]], yy, ycoords1) → ycoords2
  13. Zip(((1 - t) * aa) + (t * bb), aa, ycoords1, bb, ycoords2) → ycoords
  14. Sequence((Element[data, i], Element[ycoords, i + 2]), i, 1, 9) → altPoints
  15. Join(Sequence[{(Element[data, i], Element[ycoords, i + 1]), (Element[data, i + 1], Element[ycoords, i + 1])}, i, 1, 19]) → ogivePoints
  16. PolyLine(ogivePoints) → ogive
  17. Zip(((1 - t) * yy) + (t * (InverseNormal[0, 1, yy] / ((2 * InverseNormal[0, 1, scalept])) + 0.5)), yy, Sequence[(0.1 * i) + 0.05, i, 0, 9]) → altycoords
  18. Polygon({Corner[2], Corner[3], (x(Corner[3]) - (x(Corner[3]) - x(Corner[4])) / 5.5, y(Corner[3])), (x(Corner[2]) - (x(Corner[2]) - x(Corner[1])) / 5.5, y(Corner[2]))}) → filler
  19. Line(normlineP1, normlineP2) → normdragline
  20. Polygon(Join[{KeepIf[y(P) < y(normlineP1), P, normpdfpoints], {(axescross - (2 * normf(y(normlineP1))) - normoffset, y(normlineP1)), (axescross - normoffset, y(normlineP1))}}]) → normarea
  21. PolyLine(normpdfpoints) → normpdf
  22. Segment((0, -0.27), (0, -0.23)) → zerotick
  23. Sequence((Element[data, i], Element[ycoords, i + 1]), i, 1, 19) → dataPoints
  24. Sequence(Ray[(axescross, Element[ycoords, i]), (1, Element[ycoords, i])], i, 1, 21, 2) → gridlines
  25. Sequence(Text[zz, (axescross - normoffset + 0.06, zz / ((2 * InverseNormal[0, 1, scalept])) + 0.49)], zz, -3, 3) → zlabels
  26. Segment((axescross - normoffset, y(normlineP1)), (axescross, y(normlineP1))) → normsegment
  27. Sequence(Segment[(axescross - normoffset - 0.04, zz / ((2 * InverseNormal[0, 1, scalept])) + 0.5), (axescross - normoffset + 0.04, zz / ((2 * InverseNormal[0, 1, scalept])) + 0.5)], zz, -3, 3) → zticks
  28. Sequence(Segment[(axescross - 0.08, Element[ycoords, i]), (axescross + 0.08, Element[ycoords, i])], i, 1, 21, 2) → tickmarks
  29. Sequence(Text[Element[ycoords1, i], (axescross - 0.36, Element[ycoords, i] - 0.015)], i, 1, 21, 2) → ticklabels
  30. Zip(Segment[(axescross - 0.04, yy), (axescross + 0.04, yy)], yy, altycoords) → alttickmarks
  31. Textfield() → inputBox1

AI Drawing Prompt Reference

Create a visual representation showing the relationship between a Normal probability plot and a CDF probability plot. Include an animated vertical quantile scale transitioning between the two plots. The graph should feature sequence-generated theoretical quantile points and parameterization using sample mean and standard deviation for the Normal model. Key elements are the Normal distribution curve, empirical cumulative distribution curve (ogive-style), quantile lines, and probability scale text labels, all interconnected through coordinate mapping.
The prompt above was compiled from the original drawing commands and is for reference only.