Importance of normal distribution in engineering

The normal distribution is the most widely used distribution in engineering because many measured quantities have a symmetric bell-shaped variation around an average value.

Importance

1. Natural variation: dimensions, weights, resistance, voltage, strength of materials and measurement errors generally follow the normal curve.
2. Quality control: control charts and tolerance limits (μ ± 3σ) are based on it. It tells what percentage of items will be defective.
3. Central limit theorem: averages of large samples are approximately normal whatever the population, so tests and confidence intervals use it.
4. Approximation: the Binomial and Poisson distributions are approximated by it for large n or λ.
5. Signals and noise: random noise in communication channels is modelled as Gaussian noise.
6. Reliability and design: to decide specifications, safety margins, and the number of spares.

Example

The diameter of shafts produced by a machine is normal with mean 50 mm and standard deviation 0.2 mm. Shafts between 49.6 mm and 50.4 mm are accepted.

z = (49.6 − 50)/0.2 = −2 and z = (50.4 − 50)/0.2 = +2

P(accepted) = P(−2 < Z < 2) = 2 × 0.4772 = 0.9545

So about 95.45% of shafts are accepted and about 4.55% are rejected, which helps the engineer judge whether the machine setting is good enough.