Measuring how much values vary, and putting observations onto a common scale.
The mean tells us where the centre of a dataset is. But the mean alone doesn't tell us whether the observations are tightly clustered or widely spread.
Standard deviation measures the typical distance of observations from the mean.
A small standard deviation means the observations tend to stay close to the mean. A large standard deviation means they are more spread out.
Find how far each observation is from the mean, square those distances, take their average, and then take the square root.
Standard deviation becomes much easier to understand when we think of it as a distance from the mean.
If the mean is μ and the standard deviation is σ, then the familiar landmarks are:
For a Normal distribution, these distances have a particularly useful interpretation because they correspond to predictable proportions of the population.
A z-score tells us how many standard deviations an observation is above or below the mean.
This simple calculation transforms an ordinary observation into a standardised value.
Raw values can be misleading when they come from different distributions.
Consider two students who take different tests.
Score: 80
Mean: 70
Standard deviation: 5
z = (80 − 70) / 5
Score: 85
Mean: 75
Standard deviation: 10
z = (85 − 75) / 10
Bob has the higher raw score, but Alice is two standard deviations above her mean while Bob is only one standard deviation above his.
Z-scores allow us to compare observations measured on completely different scales.
For a Normal distribution, standard deviation gives us a remarkably useful rule of thumb.
The numbers −3, −2, −1, 0, +1, +2 and +3 describe positions on the distribution regardless of the original units.
Let's return to the example from the Normal Distribution page: people's heights.
Suppose the population has an average height of 175 cm and a standard deviation of 7 cm.
Choose a height below and see where that person sits relative to the population.
Z-scores become particularly interesting when we start looking at financial returns.
Imagine that a financial asset has:
A return of −1.30% would therefore be approximately three standard deviations below the mean.
Under a Normal model, observations this far from the mean should be extremely unusual.
Financial markets regularly produce observations that appear more extreme than a simple Normal model would suggest.
That observation leads us towards one of the most important ideas in financial modelling: fat tails.
Standard deviation and z-scores work beautifully when the underlying distribution behaves approximately like a Normal distribution.
But real-world data is not always Normal.
In particular, financial returns often exhibit:
This is one reason why financial models often need distributions that are more flexible than the Normal distribution.
So far, we have considered individual observations.
But what happens when we repeatedly take samples from a population and calculate the mean of each sample?
Something remarkable happens.
This happens even when the original population is not itself perfectly Normal, provided the samples are sufficiently large and the usual conditions are met.
It is one of the foundational ideas behind statistics, estimation, confidence intervals and many forms of statistical inference.
It also gives us another important distinction: the distribution of individual observations is not necessarily the same as the distribution of their averages.