The normal distribution and anthropometrics

5 minute read

Organisms within a population are variable; they are not all the same. When you measure a lot of organisms, you begin to notice that you can predict some things about their variation.

For example, stature is a very simple measurement so it may be surprising how much it can tell anthropologists a lot about individuals and populations. Stature changes as an individual grows and matures, and as she ages. The rate at which stature changes during growth reflects health and nutrition. The average stature in different populations reflects their environment and their evolutionary history. Stature can even tell us about ancient climates and migrations of peoples in the past.

To get at such interesting information, anthropologists have to understand how stature varies within a single population of people. This involves understanding some statistics.

For many traits that we measure including stature, a simple rule is at play: Most individuals will be near the average, and few individuals will be very far from the average. Here’s a plot called a histogram, showing the stature of female students from my course last year:

Female student stature histogram

A histogram of female students' stature in my Anthropology 105 course. The curving line represents the normal distribution with the same mean and standard deviation as the women in the class. </a> </div> In this plot, the x axis is stature (body height) in 5 cm increments. The height of each bar represents the number of women who have statures within that 5 cm bin. For example, 28 women had statures between 160 and 165 cm, but only 4 women had statures between 145 and 150 cm. No one was shorter than 145 cm, and no one was taller than 185 cm. The histogram shows how many individuals fall in each bin, a quick way to see the distribution of the observations. The mean stature of women in the class was 163.4 cm. The standard deviation of their statures was 6.7 cm. The standard deviation is how different women were on average from the class mean. How often are women far from the average stature? We can make another histogram to show this for the class:

Most of the women had statures between 157 and 170 cm — in other words, within one standard deviation of the mean. As we look farther and farther from the mean, we see fewer and fewer women with statures that extreme. Patterns like this one are very common in natural populations. It is in fact so common that we call it the normal distribution. Most individuals have traits near the average, and we see fewer and fewer individuals as we look far from the average. In that sense, there's something about the individuals in a human population that seems like it's not random: If you draw any individual by random chance and measure him or her, you're more likely to find a person near the average height than far from the average. You can't predict the stature, but if you guess the stature will near the average, you'll be right a good fraction of the time. It's not like flipping a coin or rolling a die. Why does this pattern emerge? Suppose we have a fair die, and roll it once. We should have an equal chance of rolling a one, two, three, four, five or six. If we roll the die many times, we should start seeing around the same proportion of ones as we see sixes, fives, and threes. This is one way to look at random chance: Each outcome has an equal probability of happening, and we can't predict which will happen in any given trial. Now suppose we take two rolls of the die and add them both together. Our trials are still random. We might roll a one and then a three, or a five and then a two, or two sixes. But when we look at the sum of the two numbers, we see a pattern begin to emerge. You will see a lot of sevens, fewer threes and nines, and very few twos or twelves. In other words, you can begin to predict that the outcome will be near the average. Why is this? There's only one way to get a sum equal to two: You have to roll two ones, "snake-eyes". But there are lots of ways to get a seven. You can get a one and a six, or a six and a one. You can get a three and a four, or a five and a two. In fact, there are twelve different ways to get a seven, and if you roll many pairs of dice, you'll get on average twelve times as many sevens as twos. In dice, if we combine two different random events, we will be more likely to see an outcome near the average than at an extreme. If we roll ten dice, or fifty, we become more and more likely to see an outcome near the average. A similar principle applies in biology. A person's body grows by a series of genetic and environmental events. Within a population, some of the factors that affect growth are random. People vary in the combination of genes they carry because of the random chance of inheriting them from their parents. And people vary in the environments they experience because of the random chance of who they are and where they live. A person's stature, in other words, is affected by many, many influences. Some genes may tend to make stature a little taller, others make it a little shorter. Some foods tend to make stature a little taller, others a little shorter. The combination of all these things determines how tall a person will be, and when we consider a population of people all together, they will tend to be clumped near the average. This is why many traits follow the normal distribution. But there are exceptions. A person's ABO blood type, for example, is a trait that has essentially no influence from the environment and is entirely determined by a combination of alleles for one particular gene. For blood type, there is no normal distribution in humans. We are A, B, AB, or O, period. Also, some biological factors may be very large in their effects. When we look at human stature, the largest single influence on stature is sex. Males average taller than females. Like most things, this is not an absolute — some tall women stand above the average man. But the difference between the sexes is enough to really change the distribution of stature in the population. When we look at females alone, we see a normal distribution, but when we mix individuals of both sexes, the pattern is much flatter. People are less clustered near the average, because half of them average taller and half average shorter.