Definition
A normal distribution is a probability distribution where data spreads out in a bell shape, symmetric around the average. It describes in math what we see all over the world: most measurements pile up in the middle and get rarer toward both ends.
A Bell-Shaped Hill Rising in the Middle
Imagine you measure the height of every student in a school and stack up the results in a bar chart. What shape would you get? Most students would be packed in tightly near the average height for their age.
Students who are much shorter or much taller than average get fewer and fewer toward each end. Connect the tops of all those bars with one smooth line, and you often get a bell shape that is close to symmetric, with a peak in the middle.
You see the same picture when you pour lots of steel balls onto a pegboard. Each ball bounces randomly left and right off the pins, but in the end the most balls pile up at the average spot, forming a little mountain.
The 68-95-99.7 Rule Hidden in the Bell
If data follows a normal distribution, a certain pattern of percentages shows up again and again, no matter what the data is about. It all depends on the standard deviation, a number that tells you how widely the data spreads out from the average.
Within one standard deviation of the average, you'll find about 68% of all observations, sitting there calmly. Widen the range to two standard deviations and it covers about 95% of the data. Widen it to three, and a whopping 99.7%, nearly everything, fits inside.
Once you know this rule, you can tell where you stand without doing any complicated statistics. For example, say test scores in a nationwide exam follow a normal distribution. To land in the top 2.5%, you'd need a score at least two standard deviations above the average. You can see that right away.
To Be More Precise: Why Is the World So Full of Bell Curves?
Height, weight, test scores, even the exact thickness of bolts stamped out in a factory. Why do so many measurements come out close to a bell shape? Because most things in nature and society aren't caused by one single thing. They come from a huge number of small bits of chance piling up.
Take a person's height. Countless independent factors add together: many genes, what you ate as a kid, your sleep habits, how much you exercised. As these factors add up, extreme bits of luck that push too high or too low tend to cancel each other out. Most people end up somewhere near the ordinary middle.
Math proved this surprising pattern under the name central limit theorem. Even if the original data has a really weird shape (as long as its spread isn't infinitely large), the averages you get from drawing large enough samples will be distributed close to a normal distribution. That's the big discovery.
Common misconceptions
With enough samples, all data in the world will always follow a normal distribution.
Data that is heavily skewed with a long tail, like income distribution or city populations (power laws, for example), doesn't turn into a normal distribution no matter how many samples you take. A normal distribution shows up when many independent factors add together fairly evenly.
Where you meet it
When you collect test scores from a very large group, like students taking a nationwide exam, the result is often close to a bell shape.
The tiny differences in weight when a machine fills snack bags at a factory are often close to a normal distribution too.
In one sentence
It's the most famous distribution in statistics: when many small bits of chance add up, they form a bell shape, symmetric around the average.