Confidence intervals (CIs) give information about replication, but many researchers have misconceptions about this information. One problem is that the percentage of future replication means captured by a particular CI varies markedly, depending on where in relation to the population mean that CI falls. The authors investigated the distribution of this percentage for varsigma known and unknown, for various sample sizes, and for robust CIs. The distribution has strong negative skew: Most 95% CIs will capture around 90% or more of replication means, but some will capture a much lower proportion. On average, a 95% CI will include just 83.4% of future replication means. The authors present figures designed to assist understanding of what CIs say about replication, and they also extend the discussion to explain how p values give information about replication.

For each particle in an aggregate of point particles in the plane, the set of points having it as closest particle is a convex polygon, and the aggregate V of such Voronoi polygons tessellates the plane. The geometric and stochastic structure of a random Voronoi polygon relative to a homogeneous Poisson process is specified. Similarly, those points of the plane possessing the same n nearest particles constitute a convex polygon cell in the generalized Voronoi tessellation 𝒱 (n = 2, 3, ·· ·). In fact, 𝒱 = 𝒱1, but to ease exposition n always takes the values 2, 3, ···. A key geometrical lemma elucidates the geometric structure of members of 𝒱 n , showing it to be simpler in one important respect than that of members of 𝒱; in that, for each such N-gon of given ‘type', there is a uniquely determined set of N generating particles. The corresponding jacobian is given, and used to derive the basic ergodic structure of 𝒱 n relative to a homogeneous Poisson process. Unlike 𝒱 no 𝒱 n contains any triangles. As n →∞, the vertices of the quadrangles of 𝒱 n tend to circularity, so that the sums of their opposite interior angles tend to π.

No abstract

For each particle in an aggregate of point particles in the plane, the set of points having it as closest particle is a convex polygon, and the aggregate V of such Voronoi polygons tessellates the plane. The geometric and stochastic structure of a random Voronoi polygon relative to a homogeneous Poisson process is specified.Similarly, those points of the plane possessing the same n nearest particles constitute a convex polygon cell in the generalized Voronoi tessellation 𝒱 (n = 2, 3, ·· ·). In fact, 𝒱 = 𝒱1, but to ease exposition n always takes the values 2, 3, ···. A key geometrical lemma elucidates the geometric structure of members of 𝒱n, showing it to be simpler in one important respect than that of members of 𝒱; in that, for each such N-gon of given ‘type', there is a uniquely determined set of N generating particles. The corresponding jacobian is given, and used to derive the basic ergodic structure of 𝒱n relative to a homogeneous Poisson process.Unlike 𝒱 no 𝒱n contains any triangles. As n →∞, the vertices of the quadrangles of 𝒱n tend to circularity, so that the sums of their opposite interior angles tend to π.

No abstract