Statistics Explained: Mean and Standard Deviation (And Why Experts Should Not Ignore Them)

The normal curve.

Without this concept, the field of modern psychology would not exist. And, when experts ignore it, they end up providing highly unreliable opinions.

Example of a normal curve

The concept behind the normal curve is simple: For any given trait, most people fall close to the average for that trait. As the trait (or lack of the trait) becomes more extreme, fewer and fewer people fall under that part of the curve. The distribution ends up looking like a bell, because most people are densely squeezed into the center and fewer and fewer people are at the extremities.

An example from psychology might be ‘level of happiness.’ Most people at any given moment will cluster in the middle around an average level of happiness. At the lowest extreme, there will be a few extremely unhappy people, who might be experiencing clinical depression. At the upper extreme, there will be a few people who might be clinically manic. This is how mental disorders are determined (how different is a “disorder” from “normal?”) and also how psychological tests are developed (how different is one person’s score on a test from the overall group of people who take that test?).

This is where it becomes important to understand what a mean is, and also what a standard deviation is.

The mean is the average for a particular trait. Using the example of height of men in the United States, the mean is 70 inches (5 ft, 10 inches). If you take all of the men in the US, add their heights together, and divide by the number of men, you should get 70 inches. That’s the mean, and it is the absolute center of the normal curve (the mean is often denoted by an x).

The line indicates x, or the mean, which is a height of 70 inches for men in the United States

Now, since every man in the United States is different in terms of height, there is what is called variance in this trait, or deviation from the mean. Some men are 75 inches, some men are 62 inches, and so on—they all vary to some degree.

And with a normal curve, you can calculate something called the standard deviation, which is the regular distance from the mean under which a standard percentage of people fall. For men’s height, the standard deviation is 3 inches.

One standard deviation above the mean (from 70 to 73 inches) contains 34.1 percent of people. One standard deviation below the mean (from 67 to 70 inches) contains a different 34.1 percent of people. For the height example, that means 68.2 percent of men fall within 67 and 73 inches—one standard deviation above and below 70 inches.

That standard deviation of 3 inches for the height of men also means that, when you push it out to two standard deviations, 95.4 percent of American men are somewhere between 64 and 76 inches tall. It might be a little unusual to see someone who is 6 feet, 4 inches tall, but it isn’t that shocking.

But, you will only find 4.6 percent of men beyond two standard deviations from the mean (2.3 percent above, and 2.3 percent below). And the further away one gets from the mean, the more unusual that person’s height will appear. On one end, you have Shaq. On the other end, you have Kevin Hart.

Shaq is 85 inches tall, or 5 standard deviations above the mean. On the other hand, Kevin Hart is 64 inches tall, or 2 standard deviations below the mean.

This is important to keep in mind: Kevin Hart is definitely shorter than average. But, he still falls within the group of 95.4 percent of people who are within two standard deviations from the mean. When you see a picture of him next to Shaq, he looks tiny. Just by comparing him to someone who is exceptionally tall, he looks like he might have had a health problem that kept him from growing. Instead, he’s just a normal level of short when compared to all men in the United States.

It looks like there is something wrong with Kevin Hart, but there isn’t. The problem is that this picture forces us to use an unreliable test: the “How Tall Does Someone Look When He Is Standing Next To Shaq Test,” or the “Shaq Test” for short. Using the Shaq Test would lead an evaluator to conclude that most men in the United States are short.

Just like height, many psychological traits can be measured with a mean and standard deviation. There is a range of severity when it comes to depression, anxiety, psychosis, and personality problems. Psychological tests utilize the concept of the normal distribution to understand average levels for different psychological traits, and then an individual’s psychological test scores for those traits can be measured based on how many standard deviations they are away from the mean. Someone who is severely depressed might be more than two standard deviations below the mean for happiness. Someone who is hearing voices and having delusional thoughts might be two standard deviations above the mean for psychosis.

It is through the scientific process of understanding exactly how a person compares to the rest of the group that allows psychologists to form reliable and relevant opinions about that person. And, the mean and standard deviation are a big part of that process.

Without this scientifically reliable process, an expert might come to conclusions that seem correct (“Look how short Kevin Hart is when he is standing next to Shaq. He must have a problem”), but in fact are not accurate at all (“Actually, Kevin Hart is below average in height, but he is within normal limits for this trait”).

This is why expert psychologists rely on scientifically normed and validated tests rather than questionnaires that seem right but are the psychological equivalent of the Shaq Test.