Normal distribution and z-table
The normal distribution is the symmetric, bell-shaped continuous probability distribution defined by its mean μ and standard deviation σ. Many natural and manufacturing measurements follow it, and it underpins nearly every hypothesis test and capability index on the CQE exam.
The standard normal distribution is the special case with μ = 0 and σ = 1. Any normal random variable X can be converted to the standard normal with the z-transformation:
Once you have z, you look up the cumulative probability P(Z ≤ z) in the z-table (area to the left of z under the standard normal curve).
Worked example 1 — one-sided probability
Fill weight: μ = 500 g, σ = 20 g. Find P(X < 530).
Worked example 2 — two-sided (between two values)
Same process. Find P(480 < X < 520).
This confirms the empirical rule: about 68.3% of values lie within ±1σ of the mean.
Quick-reference: empirical rule
| Range | % of data |
|---|---|
| μ ± 1σ | 68.27 % |
| μ ± 2σ | 95.45 % |
| μ ± 3σ | 99.73 % |
| μ ± 1.96σ | 95.00 % (two-sided 95% CI) |
| μ ± 1.645σ | 90.00 % (two-sided 90% CI) |
Practice problem
A part dimension: μ = 25.0 mm, σ = 0.5 mm, USL = 26.0 mm. What fraction exceeds USL?
Show answer
z = (26.0 − 25.0) / 0.5 = 2.00. P(Z > 2.00) = 1 − 0.9772 = 0.0228 = 2.28%.
Common exam traps
- The z-table gives area to the LEFT. P(Z > z) = 1 − P(Z < z). Do not forget to subtract from 1 for upper-tail problems.
- Negative z-values. The standard normal is symmetric: P(Z < −1.5) = 1 − P(Z < 1.5) = 1 − 0.9332 = 0.0668.
- Rounding before looking up z. Round z to two decimal places for most tables. A z of 1.505 rounds to 1.51, not 1.5.
Enter X, μ, σ to get z and P(X ≤ value); or enter z directly. Leave X blank to compute from z.
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Try 5 free CQE practice questions on this exact topic. Every wrong answer comes with an explanation. No account needed.
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