H2 Maths 6.3 - Normal Distribution: Introduction and Properties

Step 1 - Notation and the bell curve

The normal distribution is a continuous probability distribution shaped like a symmetric bell curve.

Step 1 - Notation and the bell curve

We write X tilde N of mu, sigma squared. Mu is the mean and sigma squared is the variance.

Step 1 - Notation and the bell curve

The parameter sigma is the standard deviation - it controls how wide or narrow the bell is.

Step 2 - Shape: symmetry and location

The bell curve is perfectly symmetric about the mean.

Step 2 - Shape: symmetry and location

This means the mean, median, and mode all coincide at mu.

Step 2 - Shape: symmetry and location

Half the total area lies to the left of mu and half to the right.

Step 3 - Probability is area; P(X = a) = 0

Because the normal distribution is continuous, the probability that X equals any single exact value is zero.

Step 3 - Probability is area; P(X = a) = 0

Probability corresponds to area under the curve between two limits.

Step 3 - Probability is area; P(X = a) = 0

The total area under the entire curve equals one.

Step 4 - The standard normal and standardisation

Any normal variable can be converted to the standard normal Z, which has mean zero and variance one.

Step 4 - The standard normal and standardisation

The standardising formula is Z = (X − μ) / σ.

Step 4 - The standard normal and standardisation

This single conversion lets you read off all normal probabilities from a Z-table or your graphing calculator.