By R. W. Farebrother
Examines vintage algorithms, geometric diagrams, and mechanical rules for complements visualization of statistical estimation techniques and mathematical ideas in physics, engineering, and laptop programming.
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Additional info for Visualizing Statistical Models and Concepts
As the upper end of the string is pulled away from the hole along this line the weight will rise and thus gain potential energy at a steady rate. On the other hand, if the upper end of the string is allowed to move towards the hole, then the weight will fall and lose potential energy at the same steady rate. More precisely, if the hole is at the point y ¼ yi on the y-axis and the upper end of the string is at the point y ¼ a, then the distance between these two points is jyi À aj. Further, since the upper end of the string may be regarded as having moved this distance from the hole, the unit weight will be at the same distance above its lowest point, and hence the potential energy level associated with this simple model will be proportional to the function Pi ¼ jyi À aj Now, our discussion of this model is in the context of a given data set, so that the ith observation Y ¼ yi may be regarded as ﬁxed and the potential energy function Pi regarded as a function of the variable parameter a.
1991a), The conceptual model behind the picture, ASA Statistical Computing and Statistical Graphics Newsletter 2: 5–8. Sall, J. (1991b), The conceptual model for categorical responses, ASA Statistical Computing and Statistical Graphics Newsletter 3: 33–36. Skemp, R. R. (1971), The Psychology of Learning Mathematics, Penguin Books, Harmondsworth, England. Stigler, S. M. (1986), The History of Statistics: The Measurement of Uncertainty Before 1900, Harvard University Press, Cambridge, Massachusetts.
If n is an odd number satisfying n ¼ 2m þ 1 then the ring will be at rest if it is located at the ðm þ 1Þth observation. Similar, if n is even and n ¼ 2m then the ring will be at rest at any point on the interval between the mth and the ðm þ 1Þth observations. It is well known that these two properties characterise the median of a set of univariate observations. 1 is the median of the given set of n observations. In passing, we note that these results give some indication of possible values of the force function si when a ¼ yi , for smþ1 takes the value 0 in the ﬁrst case when a ¼ ymþ1 and sm takes the value À1 and smþ1 takes the value þ1 in the second case even though a may take the values ym or ymþ1 as well as any value between the two.
Visualizing Statistical Models and Concepts by R. W. Farebrother