Welcome to the World of Continuous Distributions!

In your previous studies, you might have looked at discrete data—things you can count, like the number of students in a class or the score on a die. But what about things we measure? Think about the height of everyone in your school, the weight of apples in a bag, or the time it takes for a lightbulb to burn out. These are continuous variables because they can take any value within a range.

In this chapter, we are going to focus on the most famous continuous distribution of all: the Normal Distribution. It is often called the "Bell Curve" because of its shape, and it appears almost everywhere in nature and science!

1. What is a Continuous Distribution?

Unlike discrete distributions where we find the probability of an exact value (like \(P(X = 3)\)), in continuous distributions, the probability of an exact value is actually zero. Instead, we find the probability that a value falls within a range (like "what is the probability a student is between 160cm and 170cm tall?").

Analogy: Imagine trying to throw a dart at a number line. The chance of hitting exactly 1.500000... with infinite precision is impossible. However, the chance of hitting the "region" between 1 and 2 is very high!

Key Takeaway: For continuous distributions, we are always looking for the area under a curve to represent probability.


2. The Normal Distribution: The "Bell Curve"

The Normal Distribution is perfectly symmetrical. If you folded the graph in half at the center, both sides would match perfectly.

Key Features to Remember:

  • The curve is bell-shaped.
  • It is symmetrical about the mean (\(\mu\)).
  • The mean, median, and mode are all the same value and are located right in the middle.
  • The curve never quite touches the x-axis (it goes on to infinity in both directions).
  • The total area under the curve is always 1 (because the total probability must be 100%).

Did you know? Many things follow a Normal Distribution, such as IQ scores, shoe sizes, and even the errors made by scientists when taking measurements!


3. Understanding the Notation

When we say a variable \(X\) follows a Normal Distribution, we write it like this: \(X \sim N(\mu, \sigma^2)\)

  • \(\mu\) (mu): The mean. This tells you where the center of the bell is.
  • \(\sigma^2\) (sigma squared): The variance. This tells you how "spread out" the bell is.
  • \(\sigma\) (sigma): The standard deviation. This is the square root of the variance.

Common Mistake Alert! In the notation \(N(\mu, \sigma^2)\), the second number is the variance. When you do calculations, you usually need the standard deviation (\(\sigma\)). Always check if you need to square root that second number!


4. The Standard Normal Distribution (\(Z\))

There are infinitely many Normal Distributions (some tall and thin, some short and wide). To make life easy, mathematicians created the Standard Normal Distribution, which always has a mean of 0 and a standard deviation of 1.

We use the letter \(Z\) to represent this: \(Z \sim N(0, 1)\).

Standardizing: The Magic Formula

If you have any value \(x\) from a normal distribution, you can turn it into a \(Z\)-score using this formula: \(Z = \frac{X - \mu}{\sigma}\)

Don't worry if this seems tricky! All this formula does is tell you "how many standard deviations away from the mean" your value is.

  • A \(Z\)-score of 1 means you are 1 standard deviation above the mean.
  • A \(Z\)-score of -2 means you are 2 standard deviations below the mean.


5. Using the Tables to Find Probabilities

To find the probability \(P(X < x)\), we follow these steps:

  1. Standardize: Use the formula \(Z = \frac{x - \mu}{\sigma}\) to get a \(Z\)-value.
  2. Look it up: Use the Normal Distribution tables provided in your exam booklet to find the area to the left of that \(Z\)-value. This area is often called \(\Phi(z)\).

Quick Review of Table Rules:

  • To find \(P(Z < z)\): Just look up \(z\) in the table.
  • To find \(P(Z > z)\): Since the total area is 1, use \(1 - P(Z < z)\).
  • To find \(P(a < Z < b)\): Find the area for the big one and subtract the area for the small one: \(P(Z < b) - P(Z < a)\).
Memory Aid: Think of the table like a "Left-Hander's Tool." It only tells you the area to the left. If you want the area to the right, you have to subtract from 1!

6. Working Backwards: Finding \(\mu\) and \(\sigma\)

Sometimes, the exam will give you the probability and ask you to find the mean or standard deviation. This is like being given the "answer" and asked for the "question."

Step-by-Step for "Backward" Problems:

  1. Identify the probability: Look at the percentage or area given.
  2. Find the \(Z\)-value: Use the percentage points table (the small table) or look in the middle of the main table to find the \(Z\)-score that matches that area.
  3. Set up the equation: Use \(Z = \frac{x - \mu}{\sigma}\) and plug in your \(Z\), \(x\), and whatever else you know.
  4. Solve: Rearrange to find the missing letter.
Simultaneous Equations

If both \(\mu\) and \(\sigma\) are unknown, the question will give you two different pieces of information. You will create two equations using the \(Z\)-formula and solve them simultaneously.
Pro Tip: Usually, the easiest way to solve these is to subtract one equation from the other to cancel out \(\mu\).


7. Summary Checklist

Before you head into your practice questions, make sure you can:

  • Identify the mean (\(\mu\)) and variance (\(\sigma^2\)) from the notation.
  • Standardize a value using the \(Z = \frac{X - \mu}{\sigma}\) formula.
  • Use the tables correctly to find "less than" and "greater than" probabilities.
  • Use symmetry to deal with negative \(Z\)-values (Remember: \(P(Z < -1)\) is the same as \(P(Z > 1)\)).
  • Solve for \(\mu\) and \(\sigma\) when you are given a probability.

Final Encouragement: The Normal Distribution is the "bread and butter" of S1 Statistics. Once you get comfortable with the \(Z\)-formula and reading the tables, you'll find that most questions follow the exact same pattern. Keep practicing, and it will become second nature!