The normal distribution – Modelling with the normal distribution

If a variable \(X\) follows a normal distribution:

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

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

The distribution is:

• Symmetrical about the mean
• Bell-shaped
• Total probability = 1

To find probabilities we convert values into a standard normal variable:

::contentReference[oaicite:0]{index=0}

Where:

• \(X\) = observed value
• \(\mu\) = mean
• \(\sigma\) = standard deviation

After calculating \(Z\), we use normal distribution tables.

The weights of apples in a farm are normally distributed with mean \( \mu = 150 \) g and standard deviation \( \sigma = 20 \) g.

Find \(P(X > 180)\).

---

Step 1: Standardise

\[ Z = \frac{180 - 150}{20} \] \[ Z = 1.5 \]

Step 2: Use tables

\[ P(Z < 1.5) = 0.9332 \]

Therefore:

\[ P(Z > 1.5) = 1 - 0.9332 \]

Exam scores follow \(N(70,10^2)\).

Find the mark exceeded by the top 10% of students.

---

From tables:

\[ P(Z > z) = 0.10 \] \[ z = 1.28 \]

Now convert back to \(X\):

\[ X = \mu + z\sigma \] \[ X = 70 + (1.28)(10) \]

The top 10% scored about 83 marks or more.

• Always write the distribution first: \(X \sim N(\mu,\sigma^2)\).
• Draw a quick sketch of the normal curve.
• Standardise using the Z formula.
• Use tables carefully (left-tail probabilities).
• For upper probabilities use \(1 - P(Z