The normal distribution · RiskAtoms Academy

By the end you will be able to

1Describe the normal distribution and how its mean and standard deviation set its centre and width.
2Use the 68-95-99.7 rule for quick estimates.
3Convert a value to a z-score and read P(X ≤ x) = Φ(z) from a table or the calculator.
4Find tail probabilities and cut-offs, including the 1.645, 1.96 and 2.326 points used in risk.

Why you need it

How often should a day this bad happen? If you know the mean and the standard deviation and are willing to assume a bell curve, one number, the z-score, gives the answer.

These Core Path lessons use it:

L2.5.02: Return distributions
The normal curve is the benchmark that real return distributions are compared with, and where they differ.
L2.5.06: Value at risk: the idea
A value-at-risk under a normal assumption sits 1.645 or 2.326 standard deviations below the mean.

You need the standard deviation from M-C03: Variance and standard deviation first. Where the normal curve falls short is the subject of M-C08: Skewness and fat tails.

The idea

Where bells come from

Roll one die and every face is equally likely: a flat line. Add two dice and the total 7 is the most common, because more pairs make 7 than any other total. Keep adding dice. Each total is the sum of many small, independent pieces, and the shape that emerges is always the same smooth bell.

Add up dice

exact probabilities

Number of dice1

probability of each total

One die: every face is equally likely, so the bars are flat. Nothing like a bell yet.

That bell is the normal distribution. It appears whenever many small independent effects add up, which is why it shows up in measurement errors, heights and, as a first approximation, in the day-to-day changes of many prices. Its middle is the mean; its width is set by the standard deviation. Nothing else.

The rules

The 68-95-99.7 rule. For any normal distribution, the share of outcomes within one, two and three standard deviations of the mean is always the same.

z-scores. To get an exact probability, convert to the standard normal first.

Tails and cut-offs. Risk is mostly about the tails: how likely is something worse than x, and how bad is the worst 1%?

The rules, for reference

z = \frac{x-\mu}{\sigma} \qquad P(X \le x) = \Phi(z) \qquad P(X > x) = 1 - \Phi(z) \qquad \Phi(-z) = 1 - \Phi(z)

x = \mu + z\,\sigma \qquad \Phi^{-1}(0.95) = 1.645 \quad \Phi^{-1}(0.975) = 1.96 \quad \Phi^{-1}(0.99) = 2.326

In the calculator: normcdf(x, mu, sd) gives P(X ≤ x), norminv(p, mu, sd) gives the cut-off, and zscore(x, mu, sd) the z-score.

The standard normal table. Before calculators, everyone used a printed table of Φ(z). Here is a short one, computed by the same function as the calculator.

z	Φ(z) = P(Z ≤ z)	1 − Φ(z) = P(Z > z)
0	0.5000	0.5000
0.5	0.6915	0.3085
1	0.8413	0.1587
1.5	0.9332	0.0668
1.645	0.9500	0.0500
1.96	0.9750	0.0250
2	0.9772	0.0228
2.326	0.9900	0.0100
2.5	0.9938	0.0062
3	0.9987	0.0013

Try it

Move the mean and the standard deviation and watch the bell slide and stretch. Then shade an area and read its probability. Things to try: set a to μ + σ and check about 84% is below it; choose “Find the cut-off” with 5% and 1% and compare the z values with the table.

Try itNormal curve explorer

Open the distribution calculator →

Mean μ (%)0

Standard deviation σ (%)1

Shade

a−1.5

Think of X as a day's change in a price, in %.

Dashed lines mark 1, 2 and 3 standard deviations either side of the mean. The shaded area is the probability.

P(X \le -1.5) = \Phi\!\left(\tfrac{-1.5 - 0}{1}\right) = \Phi(-1.5) = 0.0668

Probability (shaded area)

0.0668

z for a−1.5

As a percentage6.68%

About 1 day in15

Worked examples

1. How often is a day worse than −2%?

Step 1 of 4

1
The set-up
Suppose a share’s daily change is normal with mean 0.05% and standard deviation 1.2%. On what share of days does it fall by more than 2%?

2. A quick range, then an exact one

Step 1 of 4

1
The set-up
An illustrative yearly return is normal with mean 6% and standard deviation 15%. Where does the middle 95% of outcomes lie?

3. The worst 1% of days

Step 1 of 4

1
The set-up
A daily change is normal with mean 0% and standard deviation 1.5%. Below what level do the worst 1% of days fall?

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 3

X is normal with mean 100 and standard deviation 15. What is the z-score of x = 130?

Question 2 · objective 3

What is Φ(1), the probability that a standard normal Z is at most 1? Give four decimal places.

Question 3 · objective 4

Same X (mean 100, sd 15). What is P(X > 130)? Give four decimal places.

Question 4 · objective 4

You want the cut-off with exactly 1% of a normal distribution below it. How many standard deviations below the mean is it?

A1.645B1.96C2.326D3

Question 5 · objective 1

You increase σ and keep μ the same. What happens to the bell curve?

AIt moves to the rightBIt gets wider and flatter, centred in the same placeCIt gets taller and narrowerDIt becomes lopsided

Glossary

Normal distribution: A bell-shaped, symmetric distribution set by two numbers: the mean μ (its centre) and the standard deviation σ (its width). Written N(μ, σ²).
Density: The height of the curve. Not a probability itself: probabilities are areas under the curve, and the whole area is 1.
z-score: How many standard deviations a value is from the mean: z = (x − μ) ÷ σ.
Standard normal: The normal distribution with mean 0 and standard deviation 1, written N(0, 1). z-scores follow it.
Φ (Phi): Φ(z), capital phi: the area under the standard normal curve to the left of z, so P(Z ≤ z). In the calculator: normcdf(z).
Φ⁻¹ (inverse): The z with a given area p to its left: the cut-off. In the calculator: norminv(p).
Tail: The far end of a distribution, where extreme outcomes live. A left tail holds the large negative values.