# 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* P*_{1}= the phenotypic value of the first plant in the group,* *

* P _{n}*= 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).

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 σ^{2}. 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.

A relatively small phenotypic variance occurs when each individual in the group has a similar value (Figure 7a; σ^{2}_{p} = 8.0), whereas a larger variance occurs when the differences in values among the individuals are considerable. (Figure 7b; σ^{2}_{p }= 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.

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**.

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.