The normal distribution

What the normal distribution is

• A distribution describes how likely different values of a variable are.
• The normal distribution is a symmetric, bell-shaped curve that fits many natural measurements — heights, weights, birth weights, exam marks, the diameter of machine-made parts.
• It is described by exactly two numbers:
  • the mean $\mu$ (Greek "mu") — the centre of the bell, where it peaks
  • the standard deviation $\sigma$ (Greek "sigma") — the spread; a larger $\sigma$ gives a wider, flatter bell
• Changing $\mu$ slides the curve along; changing $\sigma$ stretches or squashes it. The shape itself never changes.

Why area means probability

• The total area under the curve is exactly 1, because the variable must take some value.
• An area under part of the curve is the probability of landing in that part. This is the single idea the whole topic rests on — every question is really asking "what fraction of the area is this?"
• Because the curve is symmetric about the mean:
  • $P(X < \mu) = 0.5$ — half the values lie below the mean
  • $P(X > \mu) = 0.5$ — and half above
• The 68–95–99.7 rule is worth knowing by heart, because it lets you check any answer at a glance:
  • about 68% of values lie within 1 standard deviation of the mean
  • about 95% within 2
  • about 99.7% within 3
• A value more than 3 standard deviations out is genuinely rare — roughly 3 in 1000 — which is why quality control and medical screening use these cut-offs.

Standardising with the z-score

• A z-score converts any value into the number of standard deviations it sits from the mean:

$$z = \frac{x - \mu}{\sigma}$$

• Each variable:
  • $x$ — the value you are interested in
  • $\mu$ — the mean of the distribution
  • $\sigma$ — the standard deviation
  • $z$ — how many standard deviations $x$ is above (positive) or below (negative) the mean
• $z = 0$ is exactly at the mean. A positive $z$ is above it, a negative $z$ below it.
• Why standardise at all? It puts every normal distribution onto one common scale, so a single set of tables — or one calculator function — handles all of them. A height in centimetres and a mark out of 100 both become plain numbers.
• z-scores also let you compare across different distributions. A student who is $z = 1.8$ in Maths and $z = 0.9$ in English did relatively better in Maths, even if the raw marks say otherwise.

The three kinds of question

• Less than a value — the area to the left. This is what most calculators give directly.
• Greater than a value — the area to the right. Because the total area is 1:

$$P(X > x) = 1 - P(X < x)$$

• Between two values — subtract one area from the other:

$$P(a < X < b) = P(X < b) - P(X < a)$$

• Subtract, never add. You are cutting the smaller left-hand area off the bigger one, leaving the strip in between.

Working backwards: the inverse problem

• An inverse question gives you the probability and asks for the value — "the tallest 10%", "the pass mark that lets 75% through", "the weight only 5% of bags fall below".
• Run the standardising process in reverse:
  • Find the z-score matching that area (the inverse normal function on your calculator, often written InvNorm).
  • Rearrange the z-score formula to get $x$ back:

$$x = \mu + z\sigma$$

• Watch which tail the question means. Calculators usually work from the left, so "the top 10%" is the value with 0.90 below it, not 0.10.
• A negative z is expected whenever the answer sits below the mean — the bottom 5% must give a negative z, and a positive one means you used the wrong tail.

Comparing a real data set with the model

• The normal distribution is a model, and real data only ever approximately follows it.
• To judge whether the model fits, compare:
  • Shape — is the data roughly symmetric and bell-shaped, or is it skewed with a long tail one side?
  • Centre and spread — do the sample mean and standard deviation match the model's $\mu$ and $\sigma$?
  • The 68–95–99.7 rule — does about two-thirds of the real data actually fall within one standard deviation?
• Say where the model fails, not just that it fits. Real measurements often break the model in predictable ways: incomes are strongly right-skewed, and anything that cannot go below zero (times, weights, counts) will be skewed if the mean sits close to zero.
• The model can still be useful when the fit is imperfect — the point is to state the limitation rather than pretend it is not there.