Simple Mean, Weighted Mean, or Geometric Mean?
There are three commonly used methods to calculate an average (i.e., mean): the simple average, weighted average, and geometric average. Analysts often use averages as estimating guidance when predicting nonrecurring (NR) costs using ratios of NR costs to first unit (T1) production costs. For example, when the “average” of the NR to T1 ratios is determined, the NR cost can be estimated as a “factor” of its T1 cost. Here, the simple average is simply the arithmetic mean of these ratios while the weighted average is derived by weighting each ratio by its T1 cost. Consequently, deciding which average (i.e., factor) is the most appropriate metric for estimating the nonrecurring cost is frequently debated.
There are some academic concerns about the relevance of using simple and weighted averages. Some analysts insist that simple averages are derived by the “wrong math,” as averaging the averages breaks the fundamental rules of math. These analysts believe weighted averages are based upon the “correct math” and should be applied accordingly. Other analysts argue that both simple and weighted averages have their shortcomings and the geometric average is the appropriate approach.
This paper discusses these concerns and examines the properties of these methods. In addition, this study analyzes all three different methods of calculating an average using weighted least squares (WLS) in regression analysis. As a result, analysts will have a better understanding of the math behind each method, and this will help them select an appropriate and suitable method to meet their requirements for calculating an average of the NR to T1 ratios (or any ratios of interest). Realistic examples will be discussed using all three methods.
Note that there are many kinds of factor relationships, such as cost-to-cost, cost-to-weight, and weight-to-power factors. The analysis in this paper is applicable to any factor relationship and we simply use the NR to T1 factor as an illustrative example.
Dr. Shu-Ping Hu
Tecolote Research, Inc.
Dr. Shu-Ping Hu: Chief Statistician at Tecolote Research, Inc. Dr. Hu joined Tecolote in 1984 and serves as a company expert in all statistical matters. She has over 15 years of experience supporting Unmanned Space Vehicle Cost Model (USCM) CER development and the related database. She also has 22 years of experience designing, developing, and validating statistical, learning and regression algorithms in CO$TAT. In addition, Dr. Hu developed many of the distribution and correlation algorithms implemented in the ACE RI$K simulation tool. For over 20 years, she has been a regular presenter at many major cost conferences, advocating the most advanced cost analysis techniques, and earning several best paper awards. She earned her Ph.D. in Mathematics, with an emphasis in Statistics, at the University of California, Santa Barbara.