# The Concepts of Mean and Variance

A corn breeder determines by measurement the phenotypic value (P) of a trait of interest for each plant in a group of plants. The arithmetic average of all these values is known as the phenotypic mean of that group; that is where P1= the phenotypic value of the first plant in the group,

Pn= the phenotypic value of the last plant in the group, and

n = the number of plants in the group.

The mean is a characteristic of the group.

Another characteristic of the group that is important to understanding how plant breeders develop improved varieties is known as the variance. The 10 ears in Figure 5 were harvested from 10 adjacent plants of the open-pollinated variety, Reid Yellow Dent, grown at Lincoln, Nebraska, in 2004. The length of each ear was measured with a ruled (Figure 6; the numbers 1 to 10 were randomly assigned to the ears). Figure 5:  Ten ears from 10 different plants of Reid Yellow Dent that were grown in Lincoln, Nebraska. (UNL, 2004) Figure 6:  A scatter diagram of the ear lengths of 10 ears.  (UNL, 2004)

In Figure 6, these ear lengths are plotted. The longest ear is ear #2 at 23.5 centimeters, the shortest ear is #1 at 7.5 centimeters, and the mean is 17.5 centimeters. In the graph, the dashed line represents the mean ear length. The variance is a measure of the scattering of the 10 points about their mean. The precise mathematical definition of phenotypic variance is shown in the equation below. The symbol for variance is σ2. The numerator in this formula is the sum of the squared deviation of the length of each ear from the mean. This formula is used to calculate the variance with the ear lengths below. ## Quiz

Question

In Figure 7, which of the two sets of ears would you guess has the greater variance?

Looks Good! Correct: There is more variation amount the ears from test plat 7b than in 7a.

A relatively small phenotypic variance occurs when each individual in the group has a similar value (Figure 7a; σ2p = 8.0), whereas a larger variance occurs when the differences in values among the individuals are considerable.  (Figure 7b; σ2= 23.6)

Earlier, we learned that phenotypic value is the sum of genetic value and environmental effects; that is, Likewise, the phenotypic variance can be expressed as the sum of the variance of genotypic effects (i.e., genetic variance) and of the variance of environmental effects (environmental variance). The following equation illustrates this concept. ## Quiz

Question

A corn breeder measures the phenotypic variance to be 24 and the environmental variance to be 8 for one group of corn plants they are testing.  What is the genetic variance among this group of corn plants?

Looks Good! Correct: Genotypic variation is phenotypic variance (24)- environmental variance (8).

Thus, when one observes differences in some phenotype among plants in a field, some of that phenotypic variance may be because those plants possess different genotypes for that trait (genetic variance) and some may be the result of the plants having different micro-environments (environmental variance). In the example illustrated in  Figure 7a, the 10 ears were harvested from plants that are the same single-cross hybrid. All plants of a single cross are genetically identical. Therefore, in this collection of 10 ears there is no genetic variance for ear length. That means all the observed phenotypic variance was caused by environmental differences among the plants from which the ears were harvested.  For the ears in Figure 7a The 10 ears in Figure 7b are from an open-pollinated variety that was planted in the same field as the single-cross hybrid from which the 10 ears in Figure 7a were harvested. Clearly, the phenotypic variance for ear length is much greater in the group of 10 ears in Figure 7b.

## Quiz

Question

What do you suppose is the reason for the greater phenotypic variance in Figure 7b?

Looks Good! Correct: These plants were in the same field and cultivated corn in an annual plant growing from seed each year.

Because the ears in Figure 7b where harvested from the same field as those in Figure 7a, the environmental variance associated with these two sets of ears was similar.  The primary reason for the great variance in Figure 7b is because each of these ears was harvested from a genetically unique plant.  Therefore, there is genetic variance in addition to environmental variance that contributes to the observed phenotypic variance.