Granularity and Balance in Experimental Design Fred J. Hickernell Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China, fred@hkbu.edu.hk Summary. Granularity and perfect balance are defined and discussed for multiple factor designs. The granularity of a design is related to its discrepancy, an important concept in uniform experimental design. It indicates how fine a structure in the dependence of the response on the factors can be resolved. The balance of a design is similar to the resolution of fractional factorial designs, but it is defined for a much broader class of designs. The granularities and balance of various designs, including simple random designs, orthogonal arrays, digital nets, and integration lattices are compared. Two applications, the simple pendulum and blood glucose monitoring, are used to illustrate how granularity and balance can identify good designs. Key words: blood glucose, discrepancy, grid, integration lattice, orthogonal array, pendulum, positive semi-definite kernel, simple random 2000 Mathematics Subject Classification: 11K38, 62K15, 62K20, 62K99 1 Introduction My first recollection of performing a serious experiment was as a college fresh- man taking general physics. We were asked to determine the period of a simple pendulum as a function of its mass and length, and the amplitude of the mo- tion. Before beginning the experiment we were taught how to make careful measurements and to record what we observed in a laboratory notebook. The instructor demonstrated the experimental apparatus, and guidance was given on how to write our lab reports. However, no instruction was given on how to design the experiment. The three parameters or factors to be varied were length of the pendulum, mass of the bob and amplitude of the motion. The single response was the Contemporary Multivariate Analysis and Experimental Designs—In Celebration of Professor Kai- tai Fangs 65th birthday. Edited by Jianqing Fan and Gang Li. The World Scientific Publisher, 2005.