Normal distribution and standard deviation: what they are
The normal distribution is the bell-shaped curve that describes how measurements from a stable process spread around the mean. The standard deviation (σ) quantifies that spread: the larger it is, the wider the curve and the more parts sit far from the target.
On the shop floor, these two numbers answer a basic quality question: given what the process does today, what share of parts will fall outside tolerance? They underpin Cp and Cpk, SPC control charts and the sigma level.
Visual map of the normal distribution
The map shows the curve with 1, 2 and 3 sigma bands, the standard deviation formula and the 25 mm shaft example.
Mean, standard deviation and the 68-95-99.7 rule
The mean (x̄) is the center of the curve. The sample standard deviation (s) is computed from the collected measurements:
x̄ = Σx ÷ n s = √[ Σ(x − x̄)² ÷ (n − 1) ]
In a normal distribution, the share of parts within each band around the mean is fixed. It does not depend on the unit measured, be it millimeters, grams or seconds:
| Band | Range in the example (mean 25.00; σ 0.02) | Inside the band | Outside the band |
|---|---|---|---|
| 1σ | 24.98 to 25.02 | 68.27% | 317 per 1,000 |
| 2σ | 24.96 to 25.04 | 95.45% | 46 per 1,000 |
| 3σ | 24.94 to 25.06 | 99.73% | 2.7 per 1,000 |
This is the empirical rule: about 68% within ±1σ, 95% within ±2σ and 99.73% within ±3σ. That is why the natural spread of a process is said to span 6σ.
Example: a 25 mm shaft diameter
Illustrative example. A shaft has specification 25.00 ± 0.05 mm (limits 24.95 and 25.05). Ten consecutive parts were measured: 25.02; 24.98; 25.01; 25.03; 24.99; 25.00; 24.97; 25.02; 25.01; 24.99.
- Mean: sum = 250.02, so x̄ = 250.02 ÷ 10 = 25.002 mm.
- Sum of squared deviations: Σ(x − x̄)² = 0.00336.
- Sample variance: 0.00336 ÷ (10 − 1) = 0.000373.
- Standard deviation: s = √0.000373 = 0.0193 mm.
To keep round numbers, assume a process with mean 25.00 and σ = 0.02 mm. The upper limit is (25.05 − 25.00) ÷ 0.02 = 2.5 standard deviations from the mean, and so is the lower limit.
z = (limit − mean) ÷ σ = 0.05 ÷ 0.02 = 2.5
In the standard normal table, the area beyond z = 2.5 on each side is 0.62%. Adding both sides, 1.24% of parts (about 12,400 per million) fall outside tolerance. At 1,000 shafts per day, that is roughly 12 scrapped or reworked parts.
The same example gives Cp: tolerance ÷ 6σ = 0.10 ÷ 0.12 = 0.83. Below 1, the process does not fit the tolerance; the Cp and Cpk calculation shows how to center and tighten it.
How to use it on the shop floor
- Collect 30 to 50 measurements under stable conditions, using a check sheet.
- Build a histogram and check that it looks like a bell.
- Compute x̄ and s (spreadsheet or statistical calculator).
- Convert each specification limit to z and look up the area in the normal table.
- If the out-of-spec share is high, fix centering (mean) first, then spread (σ).
When the normal distribution does not apply
Not every characteristic is symmetric. Data with a natural limit at zero (flatness, runout, surface roughness, repair time) are skewed. Defect counts follow other models, such as the Poisson. In those cases, the 68-95-99.7 rule over- or understates risk. For attribute data, the p chart is the right path.
| Situation | Typical behavior | What to do |
|---|---|---|
| Diameter, length, weight | Close to normal | Use z and the normal table |
| Flatness, runout | Skewed, limit at zero | Transform the data or use another distribution |
| Defective parts per lot | Binomial | p or np chart |
| Two shifts mixed | Bimodal | Stratify and analyze separately |
Common mistakes
- Confusing σ with s. σ is the true process value; s is the estimate from the sample and gets more reliable with more data.
- Dividing by n instead of n − 1 in s for a small sample, which understates the spread.
- Using the normal curve without checking the histogram or with an unstable process. Without stability, σ does not describe the future.
- Ignoring gauge resolution. If the measurement system adds a lot of variation, the computed σ includes the instrument error. See MSA and Gauge R&R.
- Mixing lots, machines or shifts in one sample and assuming the resulting σ belongs to the process.
- Looking only at the mean. Two lines with the same mean can have very different σ, and the worse one makes scrap.
Frequently asked questions
What is a normal distribution?
It is the symmetric, bell-shaped curve around the mean that describes the natural variation of many process characteristics, such as dimensions and weights.
What is standard deviation?
It measures how far values typically lie from the center. It has the same unit as the measurement (mm, g, s).
What is the difference between sample and population standard deviation?
The sample version (s) divides by n − 1 and estimates spread from a sample. The population version (σ) divides by n and applies when you have all the data.
What does the 68-95-99.7 rule mean?
In a normal distribution, about 68.27% of values fall within ±1σ, 95.45% within ±2σ and 99.73% within ±3σ of the mean.
How do I calculate the share of parts out of tolerance?
Convert each limit to z = (limit − mean) ÷ σ, look up the area beyond z in the standard normal table and add both sides. In the example, z = 2.5 gives 1.24%.
How many measurements do I need to estimate the standard deviation?
As a practical reference, 30 to 50 under stable conditions. To set the size with a margin of error, see the article on sample size.
Sources
- MONTGOMERY, D. C. Introduction to Statistical Quality Control. Wiley.
- WERKEMA, M. C. C. Ferramentas Estatísticas Básicas para o Gerenciamento de Processos. Belo Horizonte: Fundação Christiano Ottoni.
- WALPOLE, R. E. et al. Probability and Statistics for Engineers and Scientists. Pearson.
- ISO 3534-1. Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in probability.
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About the author
Lean Manufacturing & Behavioral Management Specialist
Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.
