






4 June 2009


BROTHER CAN YOU SPARE A SIGMA?by Tom Reed Horse International Vol 5 2009
I often hear people say (and write on the internet) that horse breeding is like playing the lottery. It is a gamble. This sentiment in some ways is correct and these individuals have identified a very important aspect of breeding: randomness. Even if one uses a proven and highly successful sire on a proven and highly successful dam there is no guarantee that the resulting foal will be exceptional. (In fact, from a statistical point of view, it is more likely that the foal will be less successful than his sire and dam.)
But those people with the lottery view of horse breeding misunderstand several important features of randomness and the breeding process. When one plays the lottery, every ticket has an equal chance of winning. However in horse breeding not every matching of a particular sire with a particular dam has an equal chance of producing a winner. The skill of the breeder in matching a particular sire to a particular dam is critically important. This is where the "art" of breeding comes into play. But equally important is that fact that breeding is a probabilistic process, not a deterministic one. Breeding is influenced heavily by the laws of probability and statistics.
For the sake of this essay let's assume that we are all breeders of showjumpers. My arguments hold for dressage, eventing, and other breeders as well but I will use showjumping breeding as the example.
The first thing to recognize is that jumping ability  and more important for us breeders, the genes in stallions and mares that transmit jumping ability to their progeny – is distributed across the population of horses. Some stallions and mares have great genetic endowments of genes associated with jumping ability and some stallions and mares have very small genetic endowments of genes associated with jumping ability. Most stallions and mares have average endowments.
Like so many things in life, "jumping genes" (as we will call it for this essay) is randomly distributed within the population of horses according to what statisticians call the Normal Distribution. The normal distribution looks like a bell curve (see Figure 1), and bell curves graphically describe a multitude of characteristics in life such as size, intelligence, weight, and, for our purposes, jumping ability. In the middle of the bell curve in Figure 1, in the middle of the two red segments, is a vertical line showing the mean (that is, the average) horse's jumping ability. Most horses can be found under the bell curve in the red area, which is made up of two segments: one red segment to the right of the mean (average and a little above average) and one red segment to the left of the mean (average and a little below average).
If you look further at Figure 1 you will see two green segments. The green segment on the right shows horses with better jumping ability than the horses in the red segments while the green segment on the left shows horses with worse jumping ability than the horses in the red segments. Further we have the two blue segments, where we find exceptionally good jumping ability on the right side of the bell curve and exceptionally bad jumping ability on the left side.
Let's look now at Figure 2. This graph shows another normal distribution (bell curve) but this time there are percentages and greek letters. This will help us better understand the bell curve shown in Figure 1.
In statistics the Greek letter µ is used to denote the mean (or average) of the population. So in this case µ represents the spot in the distribution where the average jumper can be found. Half of the horses in the population jump better than this horse, and they are found to the right of µ; half of the horses jump worse than this horse, and they are found on the left of µ.
In Figure 2 we see another Greek letter, s, called Sigma, which represents the characteristic of the distribution called the standard deviation. So on the right side of the bell curve 1s means one standard deviation above the average, 2s means two standard deviations above the average, etc. Likewise on the left side of the bell curve 1s means one standard deviation below the average, 2s means two standard deviations below the average, etc.
In Figure 2 look at the dark blue area under the bell curve. You will see two dark blue segments with 34.1% written in each. What this means is that 34.1% of the population of horses have jumping ability that is one standard deviation (1s) better than average and 34.1% of the population of horses have jumping ability that is one standard deviation (1s) worse than average. In the lighter blue segments you will see that 13.6% of the population of horses have jumping ability that is between one and two standard deviations better than average and 13.6% of the population of horses have jumping ability that is between one and two standard deviations worse than average.
Now comes the question for breeders: be totally honest with ourselves and decide where each of our mares is located in these bell curves? Are they all within the red segments in Figure 1, where we know from Figure 2 that 68.2% of mares can be found? Do we have any mares that in the green segment on the left side of the bell curve?
The next question is where do we want our breeding program to be on the bell curve? Let's look at Figure 3 and the accompanying Table 1. Figure 3 is a graphical representation of the bell curve that shows the percentage of the curve that appears to the left of wherever we are on the bell curve. I will give an example. Let's say we have determined that one of our mares is an average mare for transmitting jumping ability. In Figure 3 and Table 1 we have set the mean (that is, the average) at zero. So look in Figure 3 at the bottom axis and find zero (in the middle between –4 and 4). Now go up to the graph from zero and you'll see that 0.5 is the value on the left side axis. This means that the average mare (with a mean of zero) has 50% (0.5) of mares worse than her.
Going to Table 1, look for the entry under the x column for 1.0  this means one standard deviation above the mean. The value to the right of x=1 is .8414  which means that if our mare is one standard deviation better than the average mare for transmitting jumping ability she will be better than 84% of all mares for transmitting jumping ability. A mare that is two standard deviations (2s) better than average will be better than 97% of all mares!
So let's say that we have bred or purchased a band of mares that are all one to two standard deviations better than average. In this case we would own a worldclass band of mares. The mean (average) jumping ability within our herd would be much higher than the average for the entire population of mares and the standard deviation would be lower, meaning that we have a very good band of mares and there is less disparity between our best and worst mares than there would be between the best and worst mares in the entire population of mares. So if we drew a graph of our mare herd it would not look like a normal distribution; it would not look like a bell curve. The mean would be higher and the standard deviation would be lower so the curve would be higher and much less spread out. Can we now rest on our laurels?
No. Another important truth we can learn from statistics is called the Central Limit Theorem. This principle states that no matter what the underlying distribution is for a population if we take repeated samples from that population the samples will tend to approximate the shape of a bell curve. So our band of exceptional mares will, over time, tend to produce progeny that will form their own bell curve. Yes, the average product of our breeding program will be better than the average product of all breeding programs taken together, but over time our minipopulation of horses will be represented by their own bell curve!
As a breeder who reads Horse International magazine I assume your goal is to breed international competitors in an FEI discipline. In my next article for HI I will discuss how we as breeders can engage in continuous improvement so we can fight against the tendency toward the bell curve. Unlike in Lake Wobegone, "where all the women are strong, all the men are goodlooking, and all the children are above average", all of us cannot breed superior horses all the time. But we can engage in continuous improvement so that over time we are breeding better and better athletes. We can aspire to move one sigma to the right with dedication, attention to details, and grace bestowed by the breeding gods from above.











