The normal distribution: properties, applications, approximations

What is the Normal Distribution?

The normal distribution is a smooth, bell‑shaped curve that shows how values of a variable are spread out. Think of it as a perfectly balanced seesaw: most values cluster around the middle (the mean), and the further you go from the centre, the less likely you are to see them.

In LaTeX: $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\;e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Key Properties

• Mean (μ) – the centre of the curve.
• Standard Deviation (σ) – how spread out the data are.
• Symmetry – left side mirrors right side.
• Empirical Rule (68‑95‑99.7%) –
  • 68% of data lie within ±1σ of μ.
  • 95% within ±2σ.
  • 99.7% within ±3σ.
• Area under the curve = 1 – total probability is 100%.

📚 Tip: Remember the “68‑95‑99.7” rule by picturing a “normal” classroom: most students (68%) are average, a few (27% total) are above or below, and almost none (0.3%) are extreme.

Real‑World Applications

1. Test Scores – Most exam marks follow a normal curve.
2. Human Heights – Average height with a spread around it.
3. Measurement Errors – Small random errors tend to be normal.
4. Stock Returns – Daily returns often approximated by a normal distribution.

Example: If the mean height of boys in a school is 170 cm with σ = 5 cm, then about 68% of boys are between 165 cm and 175 cm.

Approximations to the Normal Distribution

Some discrete distributions can be approximated by a normal curve when the sample size is large enough. This is handy for quick calculations.

1. Binomial → Normal

For a binomial distribution with parameters $n$ and $p$, if $np \ge 5$ and $n(1-p) \ge 5$, we can use: $$\mu = np,\quad \sigma = \sqrt{np(1-p)}$$ Then $P(X=k) \approx P\!\left(\frac{k-0.5-\mu}{\sigma} < Z < \frac{k+0.5-\mu}{\sigma}\right)$

2. Poisson → Normal

For a Poisson distribution with mean λ, if λ ≥ 10: $$\mu = \lambda,\quad \sigma = \sqrt{\lambda}$$ Use the same continuity correction as above.

📚 Exam tip: Always check the conditions ($np$ and $n(1-p)$ or λ) before using the normal approximation.

Exam Tips & Quick Checks

• 🔍 Identify μ and σ from the problem statement.
• 📐 Use the Empirical Rule for quick probability estimates.
• 🧮 Standardise using Z‑scores: $$Z = \frac{x-\mu}{\sigma}$$
• 📊 Look for continuity correction when approximating binomial or Poisson.
• 🗂️ Check the conditions for approximations: $np \ge 5$, $n(1-p) \ge 5$, λ ≥ 10.
• 📝 Show all steps – examiners love clear reasoning.

Quick Reference Table