Educational explanation, not professional design advice. Last reviewed 11 November 2026. Disclaimer
In plain English
No process makes every part exactly the same. If you measure 30 shafts from a lathe, the diameters will vary a little around an average. Two numbers describe this very well: the mean (the average — where the process is centred) and the standard deviation (how much the parts typically differ from that average — the spread).
Many measured quantities follow a bell-shaped curve, the normal distribution, with most values close to the mean and fewer far away. If you know the mean and standard deviation, you can estimate how many parts will fall outside a tolerance — before the customer finds them.
Technical definition
The mean x̄ is the sum of the values divided by their number n. The sample standard deviation s is the square root of the sum of squared deviations from the mean divided by n − 1; dividing by n − 1 (not n) corrects for estimating the spread from a sample. The normal distribution with mean μ and standard deviation σ is symmetric; a value’s z-score, z = (x − μ) ÷ σ, says how many standard deviations it lies from the mean, and the standard normal cumulative function Φ(z) gives the share of values below it.
How it works and where it is used
For a normal distribution, the share of values within ±1, ±2 and ±3 standard deviations of the mean is fixed:
| Range | Share inside | Share outside |
|---|---|---|
| μ ± 1σ | 68.27% | 31.73% |
| μ ± 2σ | 95.45% | 4.55% |
| μ ± 3σ | 99.73% | 0.27% (2.7 per 1000) |
Quality engineers compare the spread with the tolerance using capability indices. Cp compares the tolerance width with six standard deviations; Cpk also takes into account how well the process is centred, using the nearer tolerance limit. A Cpk of 1.33 or more is a common target; below 1.0, a noticeable share of parts will be out of tolerance. Control charts use the same idea, with limits at ±3σ around the mean, to spot when a process changes.
Formula
x̄ = Σx ÷ n s = √[Σ(x − x̄)² ÷ (n − 1)] z = (x − μ) ÷ σ
Cp = (USL − LSL) ÷ 6s Cpk = min[(USL − x̄) ÷ 3s, (x̄ − LSL) ÷ 3s]
USL, LSL = upper and lower specification (tolerance) limits. The share of values above a limit is 1 − Φ(z) for the z of that limit.
Worked example
Ten shafts with a tolerance of 25.00 ± 0.05 mm (LSL 24.95, USL 25.05) measure: 25.02, 24.98, 25.01, 25.03, 24.97, 25.00, 25.02, 24.99, 25.01, 25.00 mm.
- Mean x̄ = 25.003 mm; sample standard deviation s = 0.0189 mm
- Upper limit: z = (25.05 − 25.003) ÷ 0.0189 = 2.49 → share above = 0.64%
- Lower limit: z = (25.003 − 24.95) ÷ 0.0189 = 2.81 → share below = 0.25%
- Expected out of tolerance ≈ 0.89%, about 9 shafts in 1000
- Cp = 0.10 ÷ (6 × 0.0189) = 0.88; Cpk = 0.047 ÷ (3 × 0.0189) = 0.83
The process is slightly off-centre and its spread is too wide for this tolerance. Reducing the spread (better tooling, fixturing or temperature control) would help more than re-centring alone. Ten parts give only a rough estimate; a capability study normally uses at least 30–50 parts.
In the workshop and in practice
These calculations are used in incoming inspection, process capability studies, statistical process control (SPC), measurement system checks and when deciding whether a machine can hold a tolerance before a job is quoted. They also apply to cycle times, weights, fill volumes, test results and any other repeated measurement.
Common mistakes and misconceptions
- Using the population formula for a sample. Dividing by n instead of n − 1 underestimates the spread of the process.
- Assuming every process is normal. Skewed data, tool wear trends or mixed batches give more parts outside tolerance than the normal curve predicts. Plot the data first.
- Judging capability from a handful of parts. The standard deviation from 5–10 values is very uncertain.
- Confusing the standard deviation with the standard error. The standard error (s ÷ √n) describes how precisely the mean is known, not the spread of the parts.
Related calculators
- Mean, Median & Mode Calculator — Mean (average), median, mode, range, minimum, maximum, sum and count of a list of numbers.
- Standard Deviation & Variance Calculator — Sample and population standard deviation, variance, coefficient of variation and standard error of a data list.
- Z-Score Calculator — z = (x − μ) ÷ σ and back, with percentile, upper-tail and two-tailed probabilities.
- Normal Distribution Calculator — P(X < x), P(X > x), P(a < X < b), outside a range, or the value for a probability, for any mean and σ.
Related topics
- Percentages and Ratios in Engineering Work — The everyday arithmetic of shares, changes, efficiencies, mixes and scales — and the traps to avoid.
References
- NIST/SEMATECH e-Handbook of Statistical Methods
- ISO 22514-2: Statistical methods in process management — Capability and performance
- Montgomery, Introduction to Statistical Quality Control