Explained variation
As explained more fully in the introductory chapter, 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, the 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 in a previous page. In each case, the mean reponses for the treatments summarise their differences.
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.
Effect of copper on aquatic animals
An experiment was conducted to determine how copper concentrations in water affect the lifetime of Daphnia magna, a small aquatic animal. The lifetimes (days) were recorded from fourteen daphnia that were kept in separate containers containing water with different copper levels.
The variation between the three treatment means is explained by changes to the copper concentration. The variation of lifetimes within each treatment group (e.g. between 60 and 90 days when there is no copper) is unexplained by copper concentration.