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Since antiquity, the relationships between 2-tuples and their Pythagorean means have been represented in geometric forms. In this paper, we extend the practice to generalized power means through new representations, and also to 3-tuples. These geometric forms give rise to new algebraic expressions for summary statistics of 2- and 3-tuples.

The study of means dates to antiquity. Pappus of Alexandria, who lived at the end of the third century AD, wrote on the relationships between the “three means” and gave reference to the work of previous geometers [

Because of their practical importance, we consider these four means as special named cases of what [

We find this definition convenient to work with, although we note that the cases given for

Power means may be regarded as statistics which give more weight to large values of

In this paper, we explore geometric representations of power means as a method of investigating the properties of power means. We first describe past geometric representations of the means of 2-tuples, and then introduce our alternative method. Second, we describe how our method can be expanded to represent 3-tuples.

Reference [

“Pappus first gives a construction by which another geometer (αλλoςτις) [lit. the other] claimed to have solved this problem, but he does not seem to have understood it, and returns to the same problem later”.

It is not Pappus’s solution, but that of the unnamed geometer, with the addition of the quadratic mean by [_{i} and BC = min x_{i}, then M_{1} = FO, M_{0} = BD, M_{−}_{1} = DE and M_{2} = BF.

We firstly propose a simple extension of the traditional representation in _{i} and BC = min x_{i}, but in this new representation

Setting AB = a; BC = b, and solving _{p} gives:

While it may be debated whether Equation (2) is simpler to work with for general purposes than Equation (1), it has the clear advantage of being invertible. While we must solve Equation (1) for p numerically, solving Equation (2) for

A sampling of values of p, M_{p} and θ when a = 3 and b = 1 are shown in

fixed points in the mapping between M_{p} and BD(θ) which are

To construct a geometric representation of the power means of a 3-tuple x as a natural extension of our N = 2 representation, we establish a number of desirable criteria. An N = 3 representation should have: a) an arrangement of line segments with lengths equal to each element x in the series; b) a curved path, on which any position can be described using only an angle

In

Example Values | ||
---|---|---|

p | M_{p} | θ |

−¥ | 1.000 | 0.000 |

−1 | 1.500 | 1.318 |

0 | 1.732 | π/2 |

1 | 2.000 | 1.823 |

2 | 2.236 | 2.034 |

¥ | 3.000 | π |

However, solving the system in

A sampling of values of p, M_{p} and θ when a = 1 and c = 3 is shown in

point at

One consequence of this relationship is that, if we set

Like the traditional form in Equation (1), Equation (4) is differentiable (Equation (5a)). It is also invertible, (Equation (5b)) a property which the traditional form lacks.

Finally, a comparison of

Example Values | ||
---|---|---|

p | M_{p} | θ |

−¥ | 1.000 | 0.000 |

−1 | 1.500 | 0.813 |

0 | 1.732 | 1.047 |

1 | 2.000 | 1.318 |

2 | 2.236 | π/2 |

¥ | 3.000 | π |

then, that provided both

The authors would like to thank Rice University Professor Frank Jones, as well as William Longley and Reid Atcheson, for helpful discussions.

Sarah M.Tooth,John A.Dobelman, (2016) A New Look at Generalized Means. Applied Mathematics,07,468-472. doi: 10.4236/am.2016.76042