M1 Statistics: The Normal Distribution (Part 1)

Basic Definitions and Properties

Hello everyone! Welcome to one of the most important topics in statistics: the Normal Distribution. Don't let the name intimidate you. It's called "normal" because it describes so many things we see in the real world, from the heights of students in your school to the scores on a test.

In this chapter, we're going to learn what the normal distribution is, understand its key features, and see why it's such a superstar in the world of statistics. Getting these basics right will make the next chapters much easier. Let's get started!


From Counting to Measuring: Continuous Random Variables

First, a quick recap...

Remember discrete random variables? These are variables that you can count.
For example: The number of heads when you flip a coin 5 times (you can get 0, 1, 2, 3, 4, or 5 heads, but you can't get 2.5 heads). Or the number of typos on a page.

Now, let's meet Continuous Random Variables!

A continuous random variable is one that can take on any value within a given range. Think of things you measure, not count.

  • Example 1: The height of a student. It could be 165 cm, 165.1 cm, 165.11 cm, or any value in between.
  • Example 2: The time it takes to run 100 metres. It could be 12.5 seconds, 12.51 seconds, etc.
  • Example 3: The weight of an apple.
Key Idea: Probability is about Area

For a continuous variable, the probability of it being exactly one specific value is practically zero! (What are the chances someone's height is exactly 170.00000... cm?).

Instead, we talk about the probability of the variable falling within a range. For example, "What is the probability that a student's height is between 165 cm and 170 cm?"

We represent these probabilities as the area under a curve. This special curve is called a Probability Density Function (PDF). The most famous PDF of all is the bell-shaped curve of the normal distribution!

Key Takeaway: Discrete vs. Continuous

Discrete: Countable values (e.g., number of students). We use probability mass functions (like for Binomial or Poisson).
Continuous: Measurable values in a range (e.g., height of students). We use probability density functions (like for Normal).


The Star of the Show: The Normal Distribution

The normal distribution is a continuous probability distribution that is symmetrical and has a characteristic bell shape. It's a mathematical model that fits countless real-world situations.

The Notation: Speaking the Language

When a continuous random variable X follows a normal distribution, we write it like this:

$$X \sim N(\mu, \sigma^2)$$

Let's break that down. Don't worry, it's simpler than it looks!

  • X: This is our continuous random variable (e.g., IQ scores).
  • ~: This little squiggle means "is distributed as" or "follows the distribution of".
  • N: This stands for Normal. Easy!
  • ($$\mu$$, $$\sigma^2$$): These are the two all-important parameters that define the specific shape and position of our bell curve.
    • $$\mu$$ (mu) is the mean of the distribution. It tells us the center of the graph.
    • $$\sigma^2$$ (sigma-squared) is the variance of the distribution. It tells us how spread out the data is.

Remember, the standard deviation, $$\sigma$$, is simply the square root of the variance ($$\sigma = \sqrt{\sigma^2}$$). The standard deviation also measures the spread.

Common Mistake Alert!

Always pay close attention to the second number in the bracket! The notation is $$N(\mu, \sigma^2)$$, which uses the variance.
If you are given that heights of students follow $$N(168, 25)$$:
- The mean $$\mu$$ is 168.
- The variance $$\sigma^2$$ is 25.
- The standard deviation $$\sigma$$ is $$\sqrt{25} = 5$$, NOT 25! This is a very common trap in exams.


The Personality Traits of a Normal Curve

All normal distribution curves share four main properties. Understanding these will give you a real feel for how they work.

1. Bell-Shaped and Symmetrical

The graph of a normal distribution is famously known as the "bell curve".
It is perfectly symmetrical about its center, the mean ($$\mu$$). This means the left half of the curve is a mirror image of the right half.

Analogy: Imagine folding the graph along the vertical line at the mean. The two sides would match up perfectly! This symmetry means that the probability of being a certain amount below the average is exactly the same as being that same amount above the average.

2. The "3-in-1" Center: Mean = Median = Mode

Because the curve is perfectly symmetrical and peaks at the center:

  • The Mean (the average value) is at the center.
  • The Median (the middle value that splits the data 50/50) is also at the center.
  • The Mode (the most frequently occurring value) is at the highest point of the curve, which is... you guessed it, the center!

So, for any normal distribution: Mean = Median = Mode = $$\mu$$.

3. The Spread is Determined by Standard Deviation ($$\sigma$$)

The mean ($$\mu$$) tells us where the center of the curve is, but the standard deviation ($$\sigma$$) tells us how "spread out" or "squashed" it is.

  • A small standard deviation ($$\sigma$$) means the data is tightly packed around the mean. This results in a tall and narrow bell curve.
  • A large standard deviation ($$\sigma$$) means the data is more spread out. This results in a short and wide bell curve.

Analogy: Think of two different classes taking the same test. If Class A has a small $$\sigma$$, most students scored very close to the average. If Class B has a large $$\sigma$$, the scores were all over the place - some very high, some very low.

4. The Total Area Under the Curve is 1

This is a fundamental rule for all probability distributions. Since the curve represents all possible outcomes, the total probability must be 100%, or 1.

Therefore, the total area under the entire normal distribution curve is always equal to 1.

This also means the area of each symmetrical half is 0.5. So, there's a 50% chance of an outcome being above the mean, and a 50% chance of it being below the mean.

Did you know?

The normal distribution is also called the Gaussian distribution, named after the brilliant German mathematician Carl Friedrich Gauss, who did extensive work on it in the early 1800s.


Let's Recap!

Quick Review Box
  • Continuous Random Variables can take any value in a range (e.g., height, weight, time).
  • Probability for continuous variables is the area under a curve.
  • The notation $$X \sim N(\mu, \sigma^2)$$ means the variable X follows a normal distribution with mean $$\mu$$ and variance $$\sigma^2$$.
  • Key Properties:
    1. It's bell-shaped and symmetrical about the mean.
    2. The Mean = Median = Mode.
    3. The standard deviation ($$\sigma$$) controls the flatness/spread of the curve.
    4. The total area under the curve is always 1.
  • Watch out! Remember to take the square root of the second parameter to find the standard deviation $$\sigma$$.

Great job! You've now mastered the fundamental concepts of the normal distribution. These ideas are the foundation for everything that comes next. Keep these properties in mind as we move on to learn how to calculate probabilities using this powerful tool.