A variable \(X\) is normally distributed if:

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

• \(\mu\) = mean
• \(\sigma\) = standard deviation
• \(\sigma^2\) = variance

The curve representing a normal distribution is called the normal curve.

The normal distribution has the following properties:

• Bell-shaped curve
• Symmetrical about the mean
• Mean = median = mode
• Total area under the curve = 1

The graph is centered at \( \mu \) and spread depends on \( \sigma \).

To calculate probabilities we convert values to the standard normal variable \(Z\).

• \(X\) = value of the variable
• \(\mu\) = mean
• \(\sigma\) = standard deviation

After finding \(Z\), probabilities are obtained from normal distribution tables.

For a normal distribution:

This helps estimate probabilities quickly.

The heights of students in a school are normally distributed:

\(X \sim N(170, 6^2)\)

Find the probability that a student is taller than 178 cm.

Step 1: Standardise

Step 2: Use tables

Therefore:

• Always write the distribution first: \(X \sim N(\mu,\sigma^2)\).
• Draw a quick normal curve sketch.
• Standardise using the \(Z\) formula.
• Use normal tables carefully.
• Remember: the area under the curve represents probability.