Normal Distributions Explained – With Real-World Examples

A thousand people walk into a bar. You measure their height as they enter. If you plot those heights on a histogram, something curious happens. Most people fall in the middle with just a few outliers on either side. Tall, short, but mostly average. This creates a smooth peak. Now, for something less human. Picture a baby elephant. Scratch that. Picture a whole herd of newborn elephants, each standing on a scale. Plot those weights. Same smooth peak. Different species, same shape. Okay, one more. Look at the daily revenue of Cafe Socratica over a year. Some days are great, some are quiet, but most hover around a typical number. Plot those, another smooth peak. This keeps happening everywhere you look. People, animals, money, all kinds of measurements, the same general shape shows up. A curve with one peak, symmetrical, tapering off to both sides. It's so common we gave it a name, the normal distribution. We picked the name normal for this distribution not because others are abnormal, but simply because this distribution is so common. It happens all the time in nature. Informally, the shape of this distribution is sometimes called the bell curve because it looks like the outline of a bell, round on top, tapering smoothly down both sides, and made of polished brass. The word distribution is a very important mathematical term. It's hard to summarize in just one sentence, so most people point to a graph of the distribution. What the graph tells you is which data are common and which data are rare. It's like a map of possibilities. The taller the curve at a particular point, the more frequently those values tend to appear in the data. On the other hand, where the curve is low, that means those values don't come up as often. Now, this isn't fortune telling. A distribution doesn't tell you what will happen. It tells you what values are typical and how much room there is for surprise. The normal distribution in particular tells us that values tend to cluster around a central point. The farther you get from the center, the less often those values occur. We don't want to give you the wrong idea. We're not saying all normal distributions are identical. The height of people, the birth weight of elephants, you can tell the difference between these distributions. The peaks of these distributions occur at different values and they taper off at different rates. So, how can you describe which normal distribution your data has? Turns out you only need two numbers, one for the peak and one for the taper. These two key numbers control the bell shape of any normal distribution curve. The first is the mean, written with the Greek letter mu. This is the average, the center of the distribution. It tells us where the peak sits. The second is the standard deviation, written with the Greek letter sigma. This tells us how spread out the values are. A small standard deviation means the values are tightly clustered. The curve is tall and narrow, dropping off sharply. A large standard deviation means the values are more spread out. The curve is lower and wider with a gradual taper. These two numbers, mu and sigma, define a normal distribution. The mean specifies the center of the bell curve. The standard deviation stretches or compresses it horizontally, shaping how concentrated or dispersed the values are around mu. Together, they give each normal distribution its unique identity. Change either one and you get a different bell curve. So, now you've got your bell curve, you know where the center is, you know how wide it is, but here's the real power of the normal distribution. Once you know mu and sigma, you can start making predictions. How likely is it for a value to land near the mean or far from it? The answer, surprisingly predictable. In any normal distribution, most of the values fall within just a few standard deviations of the mean. This is called the empirical rule and it gives us specific percentages. Roughly speaking, about 68% of the values fall within one standard deviation of the mean. About 95% fall within two standard deviations. And about 99.7% fall within three. This means that if a value is within one sigma of the mean, that is within one standard deviation, it's still pretty typical. There's another way scientists and engineers describe events that are less common and we show this graphically by highlighting the long tails. A two sigma event refers to data that lie at least two sigma away from the mean. This should make you think something a little unusual is happening. Three sigma? That means at least three sigma away from the mean and that's truly rare. In fact, the phrase three sigma event has become shorthand for something unlikely but not impossible. You may hear scientists talk about five sigma events, meaning extremely rare. So rare they might signal a real discovery or a problem with your data. Six sigma often comes up in engineering and quality control. You can see how the normal distribution gives us a common language for describing how surprising or unsurprising data is. Let's make this a little more personal. Suppose we measure the head circumference of a large group of adult humans. For men, the average head circumference is 57 cm with a standard deviation of about 2 cm. For women, the average is 55 cm with a standard deviation of roughly 1.5 cm. If head sizes are normally distributed and they mostly are, we can apply the empirical rule to both groups. For men, about 68% have head circumferences between 55 cm and 59 cm, within one standard deviation of the mean. 95% of men have a head circumference within two standard deviations, that is, between 53 cm and 61 cm. And if you want to include 99.7% of men, the three sigma range is between 51 cm and 63 cm. Now, for the women. About 68% have head circumferences between 53.5 cm and 56.5 cm. That's within one standard deviation from the mean. You'll find 95% of women fall within the two sigma range, between 52 cm and 58 cm. To cast a wider net, look within three standard deviations. About 99.7% of women have a head circumference between 50.5 cm and 59.5 cm. If you have measure handy, you can measure your own head and see where you land. Do you have a one sigma, two sigma, or three sigma head? The empirical rule lets you quickly grasp how common or rare any individual measurement is. Thanks to all the generous people supporting us on Patreon. Their contributions make Socratic a possible. YouTube ads help to some extent, but they don't go far, especially when your focus is on education. Support from our community really does make our work sustainable. Suppose you've collected some data and it looks roughly bell-shaped. That's a good sign. It might follow a normal distribution, but recognizing the shape is just the beginning. To do anything quantitative, whether you're applying the empirical rule, computing probabilities, or modeling the data, you'll need two numbers, the mean and the standard deviation. Let's walk through how to calculate them. First up, the mean or average. To calculate the mean, add up all your values and divide by the number of values. You're only halfway there. The mean tells us where the center of the data is. We also need to know how spread out the values are. That's where the standard deviation comes in. You compute the standard deviation using this formula. Where the variance is the average squared distance from the mean. To calculate the variance, we subtract the mean from each value, square it, then take the average of those numbers. Once you have the variance, you take the square root to get the standard deviation. Now, you might wonder, why do we have two different numbers that measure the spread of the normal distribution? The variance gives us an intuitive way to measure spread. By squaring the differences from the mean, we ensure that all the values are positive, which makes sense when we're talking about distances. But, there is a complication, units. If your data is in centimeters, then the variance is in square centimeters. That's harder to interpret. So, we take the square root. This gives us the standard deviation. A number that still reflects the spread, but now it's back in the same units as the original data, just like the mean. That makes it much easier to understand and compare. This is what allows you to describe a real-world data set with just two numbers, the mean and the standard deviation. Once you have those, you can sketch the bell curve, apply the empirical rule, calculate probabilities, and much more. To demonstrate these calculations, let's compute the mean and standard deviation for the heights of the 11 footballers who played for England in the 1966 World Cup final. First, we calculate the mean height. Add up all the heights, 1,948 cm, divide by 11 players. The average height is 177.1 cm. Don't forget the units. Notice that the average height is in centimeters, just like the individual heights. Next, we calculate the variance. Subtract the mean from each height, square the result, add them up, and divide by 11. This gives us 36.0826 square centimeters. We'll round this to 36.1 square centimeters. Again, pay close attention to the units. While the mean is in centimeters, the variance is in centimeters squared. And to get the standard deviation, we take the square root of the variance. That gives us 6 centimeters. Did you double-check the units? Because you should really double-check the units. The standard deviation has the same units as the mean. Centimeters. A collection of numbers describing a population are often called summary statistics, or summary stats for short. Here are the summary stats we calculated for the 1966 World Cup champions. Those sons of England, 11 lions, who brought immortal glory to their land, and carved their names into the annals of history, not merely by the strength of their legs, but by the measure of their hearts. So far, the mean and standard deviation seemed like independent ideas. Center, spread, but in fact, they work together, combining into a single formula that precisely describes the bell-shaped curve for the normal distribution. This is called a probability density function, or PDF for short. The PDF gives you the shape of the curve. It tells you how values are distributed, where they are common, and where they are rare. But how do you actually use this formula? You use the PDF to calculate the probability that a value falls within a certain range. Specifically, the probability that a value lies between two numbers A and B is the area under the curve from A to B. Finding that area usually requires calculus. You compute an integral. Luckily, you don't have to do that by hand. Software can do it instantly. And most calculators and statistics tools have built-in functions for working with the normal distribution. After seeing this highlighted region under the curve, now we can better understand why this is called a probability density function. Think about how we talk about density in physics. Density tells you how much matter is packed into a space, but you never ask how much matter is at a single point. Instead, you ask how much matter is in a region. It's the same idea here. The PDF gives you a probability density along the number line. At a single point, the probability is zero. What matters is the total amount of probability across an interval. For bonus understanding points, notice the formula has that fraction in front. It makes sure the total area under the curve adds up to exactly one, meaning 100% probability. Remember these key takeaways. The graph of the PDF is the familiar bell curve. The area defines the probabilities, and the mean and standard deviation are the control dials for this system. Let's put the PDF to work. In the United States, the average height for adult women is about 63.5 in. That's about 161.3 cm with a standard deviation of about 2.5 in or about 6.35 cm. Once again, the formula for the normal distribution looks like this, where mu, the mean, is 63.5 in and sigma, the standard deviation, is 2.5 in. As a demonstration, let's calculate what percentage of American women are 6 ft tall or taller. 6 ft is 72 in. So, we want the probability that a randomly selected woman has a height greater than or equal to 72 in. That probability is the area under the normal distribution curve from 72 in to infinity. This area is calculated using an integral. If you use software to compute this integral, you find that the probability is quite small. About 0.03% of American women are 6 ft or taller. To put that another way, out of every 1 million women, about 337 are 6 ft or taller. That's how you use the normal distribution and an integral to find probabilities, even for rare events. In practice, most people do not actually use the PDF to calculate probabilities. Instead, they use something called a Z-score. The Z-score tells you how many standard deviations a value is away from the mean. To calculate this, subtract the mean from the value and divide by the standard deviation. In our example of searching for tall American women, 6 ft or 72 in has a Z-score of 3.4. Once you have the Z-score, you can use a related function called the cumulative distribution function, or CDF. The CDF gives you the area under the bell curve to the left of a given point. In other words, it calculates the integral for you. If we calculate the CDF at a Z-score of 3.4, we get about 0.9997. This means that about 99.97% of American women are shorter than 6 ft. But we are interested in the probability of being taller than 6 ft, the area to the right. Since the total area under the curve is one, we subtract. You can find cumulative probabilities easily using spreadsheets like Google Sheets and Excel, programming languages like Python and R, or scientific calculators. All of these tools have built-in functions for working with the normal distribution. The idea remains the same. The Z-score tells you where you are on the curve, and the CDF gives you the total probability up to that point. According to us, the quality of Socratic videos is at least two standard deviations above the mean. 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