The normal distribution – The normal distribution

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 \( \sigma \).

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

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

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

For a normal distribution:

Interval	Approximate Percentage
\(\mu \pm 1\sigma\)	68%
\(\mu \pm 2\sigma\)	95%
\(\mu \pm 3\sigma\)	99.7%

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

\[ Z = \frac{178-170}{6} \] \[ Z = 1.33 \]

Step 2: Use tables

\[ P(Z < 1.33) = 0.9082 \]

Therefore:

\[ P(X > 178) = 1 - 0.9082 \]

• 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.