Will R. Getz – Fort Valley State University

Introduction

Several simple statistical tools exist for describing data that accumulate from goat herd records. These tools are most accurate when used on larger herds, or smaller herds collected over several years. The tools measure central tendency or variation (differences) in the herd, or several herds when breeders are working together.

Measures of central tendency

Among these measures, the mean carries the most importance in basic meat goat breeding. It is a population measure. For normally distributed traits, the mean determines the center of the distribution. It is the point along the horizontal axis where the bell-shaped curve is the highest.

Means change greatly with breed, management and the physical environment factors. Applying selection causes them to change over time. Means are calculated by simply adding up values from a population or from a sample taken from a population and dividing by the number of values. Means are used in many ways in meat goat breeding. The overall mean performance for a trait in a population in the genetic model for quantitative traits.

Measures of variation

Variation is the raw material for making genetic change. The mean tells us nothing about the uniformity of the population. From a breeding standpoint, variation usually refers to differences among individuals within a population. Variation exists in performance, breeding values, producing abilities and environmental effects. From a genetic standpoint, it is important that a population be variable. On the other hand there is often economic value associated with reduced variation and uniformity.

Measures of variation provide reference points in respect to knowing if an animal’s performance is extreme or just slightly above average. The most commonly used mathematical measure of variation is the standard deviation, which is defined as the square root of the variance. Variance can be defined as the average squared deviation from the mean, and is expressed in (nonsensical) squared units. This is procedure is used in order to calculate outcomes in positive terms, squaring the negative deviations from the population mean is required. For example, variation in kid harvest weight might be 225 “pounds squared”. This is difficult to conceptualize, it doesn’t make any sense. By taking the square root of that number, otherwise know as the standard deviation, the number 15 pounds is arrived at; it is easier to relate to this number in real-world terms. The “average” deviation from the mean weaning weight in this population of goats is therefore ±15 pounds.

The shape of the normal curve for a particular value indicates the amount of variation of that value in the population. A relatively flat, broad distribution indicates a high degree of variability. A tall, narrow distribution indicates a high degree of uniformity. The level of absolute variability is related to some degree to the scale of the trait in question. Variability in birth weight will be less than variability in weaning weight.

Knowing the mean and standard deviation for a value allows several generalizations to be made. For normally distributed values, approximately 68% of all observations lie within one standard deviation either side of the mean. Ninety-five percent of all observations lie at a distance less than two standard deviations from the mean. Virtually all (99%) observations are less than three standard deviations from the mean.

Covariation

There are three basic aspects of covariation. The first has to do with the direction or the sign of the relationship between variables, e.g. birth weight and weaning weight. This relationship may be positive, whereby two variables move in the same direction; negative, whereby two variable move in opposite directions to some degree at least; and when there is no pattern to the pairing, the covariation is zero or nearly zero.

The third aspect of covariation has to do with the amount of change in one variable that can be expected for a given amount of change in another variable.

Information on covariation is important to the breeder because it tells us which traits can be substituted in situations where it is highly expensive to collect data on a particular trait, or where it is rather difficult to collect data on the trait of primary interest. In these situations if there is a substantial relationship between traits and it is in the right direction, then it is possible to substitute traits knowing that the outcome will be favorable on each. Just as variation is measured by the variance and standard deviation, covariation is measured by covariance, correlation, and regression.

Of these the latter two are of most use, but the covariance must be calculated in order to derive correlation and regression coefficients. While cheap software programs are available for calculating these values, the goat breeder must appreciate the meaning of each. Correlation coefficients range from -1 to +1 where in the -1 indicates a very strong negative covariation. An example might be scrotal circumference in bucks and age at puberty in their daughters. As one increases the other decreases. Correlation coefficients are not associated with implications of cause and effect aspects. They are totally different discoveries.

The regression coefficient provides and indication of how much one variable moves when a given amount of shift occurs in another variable. The values can be any number and depend on the units being used by each variable or is some cases the measuring units may not be a constraint. We can ask the question for example, if the scrotal circumstance increases by one centimeter (cm) then how many days will the age at puberty decrease. Often issues of cause and effect are documented with regression coefficients. Regressions are used to help predict some value based on another piece of information.