AdvancedTopic 82 of 104Engineering Mathematics
Normal distribution
The bell curve that arises from many small independent effects; central to quality control.
In depth
The normal distribution N(μ,σ²) has density (1/σ√2π)e^{-(x-μ)²/2σ²}. About 68%, 95% and 99.7% lie within 1, 2 and 3 σ. The z-score z=(x-μ)/σ standardises. Measurement noise and manufacturing variation are often approximately normal.
Key points
- Symmetric and bell-shaped
- Sum of normals is normal
- Six Sigma targets ±6σ
ExampleBolt lengths with mean 50 mm, σ=0.1 mm: 95% fall in 49.8–50.2 mm.
What is Normal distribution?
The bell curve that arises from many small independent effects; central to quality control.
Why is it important to understand normal distribution?
The normal distribution describes the sum of many small random effects, so it is the default model for measurement scatter and manufacturing variation.
Why is the normal distribution so common?
The central limit theorem makes sums of many small independent effects approximately normal, so it appears widely in measured data.
When should I not assume normality?
When the data are skewed, bounded or have heavy tails, such as failure times or extreme loads. Check with a histogram or a probability plot first.
How is Normal distribution used in practice?
Typical applications include setting control limits for a machined dimension using three sigma, modelling the scatter in a material strength test and estimating the probability a bolt length falls outside its tolerance.
What should I learn before Normal distribution?
Study "Binomial and Poisson distributions" first (topic 81 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Normal distribution?
Symmetric and bell-shaped. Sum of normals is normal. Six Sigma targets ±6σ.
Can you give an example of Normal distribution?
Bolt lengths with mean 50 mm, σ=0.1 mm: 95% fall in 49.8–50.2 mm.
What comes after Normal distribution?
The next topic is "Central limit theorem", which builds directly on this one.
How difficult is Normal distribution?
It is classed as Advanced in Engineering Mathematics (topic 82 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.