One of the most important probability distributions in mathematics, statistics and science — and a beautiful example of how simple mathematical rules can describe patterns in the real world.
The Normal distribution is a continuous probability distribution with a characteristic bell-shaped curve.
Values near the centre are more common, while values increasingly far from the centre become less likely.
A surprisingly large number of natural measurements tend to cluster around an average value. The Normal distribution gives us a mathematical way to describe that pattern.
The mean, represented by the Greek letter µ (mu), determines the centre of the Normal distribution.
The standard deviation, represented by s (sigma), describes how spread out the observations are around the mean.
One of the most remarkable properties of the Normal distribution is the predictable amount of probability contained within one, two and three standard deviations of the mean.
About two-thirds of observations
Almost all observations
Virtually all observations
The rule gives us an immediate sense of how unusual an observation is. An observation more than three standard deviations from the mean is extraordinarily unusual under a Normal model.
Now let's put the ideas together. Adjust the mean, standard deviation and individual observation and see how its position within the distribution changes.
Probability of observing a value below x
Set x exactly one standard deviation above the mean. The z-score becomes +1 and approximately 84.13% of the distribution lies below that value.
Move x two standard deviations above the mean and watch the probability approach 97.73%.
So far we have looked at the Normal distribution as a mathematical curve. But what would it actually look like if we selected people at random and measured their heights?
Let's simulate exactly that.
Each bar represents the number of people whose height falls within a particular range. This is called a frequency distribution or histogram.
The smooth curve represents the theoretical Normal distribution from which the heights were generated.
Notice that the bars don't perfectly follow the curve. That's randomness. Generate another sample and the pattern changes slightly.
Increase the sample size and the histogram begins to resemble the theoretical Normal distribution more closely.
The probability density function of the Normal distribution is:
It looks complicated at first, but each part has a purpose.
The same mathematical constant e that appears in compound growth and exponential processes also appears at the heart of the Normal distribution.
The Normal distribution appears whenever many small, approximately independent effects combine to produce an outcome.
But there is an even deeper reason the Normal distribution appears so often.
It is closely connected to one of the most important ideas in probability:
When we repeatedly take samples and calculate their averages, the distribution of those averages tends towards a Normal distribution under broad conditions.
Imagine two completely different measurements.
Someone is 189 cm tall. Another person scores 89 on a test. The numbers themselves are not directly comparable.
But we can ask a more useful question:
This is the idea behind the z-score.
A z-score converts an observation into a standardised measure of its position within a distribution.