Summarising explained variation

The concepts of explained and unexplained variation in the response measurement are the basis of analysis of experimental data. In experiments with a single factor, explained variation is summarised by differences between the response means at the different factor levels (treatment means).

A table of treatment means is a good summary of the effect of the factor.

Are the differences real?

Although a table of treatment means provides a good summary, it is incomplete without taking into account the unexplained variation in the data.

Could differences between the means have arisen by chance?

We address this question in later pages of the chapter.

The following experiments were described more fully at the start of this section. In each case, the mean reponses for the treatments summarise their differences.

Abrasion resistance of fabrics

In the manufacture of clothing, a wear-testing machine is used to measure the resistance to abrasion of different fabrics by measuring the weight loss after a fixed number of abrasion cycles. An experiment was conducted to compare four competing fabrics.

The variation between the four treatment means is explained by differences between the fabrics. The variation in abrasion loss within each fabric type (e.g. between 1.93 and 2.38 for fabric A) is unexplained by differences between the fabrics.

Quality of silk

A measure of the quality of reeled silk is its "denier" — a number expressing the weight of a fixed length. Among many factors that affect denier, the temperature of the bath in which the cocoons are cooked is likely to be important. A completely randomised experiment was conducted with two different bath temperatures, 105°F and 110°F.

In this experiment, there is considerable unexplained variation — with the temperature constant, the range of denier value from cocoon to cocoon is about 50. From these data, it is unclear whether the difference between the two means (explained variation) could have arisen by chance — a formal test is required.

Hardwood and paper strength

An experiment was conducted to investigate how the hardwood concentration in paper affected its tensile strength. Four concentrations were used (5%, 10%, 15% and 20%) and six specimens were made up at each level in a pilot plant. The 24 specimens were tested on a laboratory tensile tester in random order.

The overall variation in tensile strengths (from the smallest value, 7, to the largest value, 25) is partly caused by differences in hardwood concentration. The differences between the four treatment means (10.00 to 21.17) describe the variation that is explained by the treatment. However the six values in each hardwood group vary and this variation is unexplained by differences between the hardwood concentrations.