How Many Resistors Meet Spec?

A factory's 1 kΩ resistors scatter around their target in a bell curve. The fraction within ±5% is the area under the curve between the limits, found by converting each limit to a z-score: how many standard deviations it sits from the mean.

How it works

The normal distribution and z-scores

In one sentence

A normal random variable $X \sim N(\mu, \sigma^2)$ has the bell-shaped density $\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}$ . Standardizing with $z = (x-\mu)/\sigma$ gives the standard normal distribution, so $P(L \le X \le U) = \Phi\!\left(\frac{U-\mu}{\sigma}\right) - \Phi\!\left(\frac{L-\mu}{\sigma}\right)$ .

General formula

{\htmlData{sym=0}{z}} = \frac{{\htmlData{sym=1}{x}} - {\htmlData{sym=2}{\mu}}}{\htmlData{sym=3}{{\sigma}}}, \qquad P({\htmlData{sym=4}{L}} \le X \le {\htmlData{sym=5}{U}}) = {\htmlData{sym=6}{\Phi}}\!\left(\frac{{\htmlData{sym=5}{U}} - {\htmlData{sym=2}{\mu}}}{\htmlData{sym=3}{{\sigma}}}\right) - {\htmlData{sym=6}{\Phi}}\!\left(\frac{{\htmlData{sym=4}{L}} - {\htmlData{sym=2}{\mu}}}{\htmlData{sym=3}{{\sigma}}}\right)

Every normal curve is the same shape, stretched and shifted. A z-score measures distance in standard deviations, so one table of areas works for all of them.

What the symbols mean

$z$
z-score
How many standard deviations a value sits from the mean.
$x$
A resistance
One resistor's actual value, in ohms.
$\mu$
Process mean
The average resistance the machine produces.
$\sigma$
Process spread
The standard deviation of the resistances.
$L$
Lower spec limit
The smallest acceptable value: 1000 Ω minus the tolerance.
$U$
Upper spec limit
The largest acceptable value: 1000 Ω plus the tolerance.
$\Phi$
Standard normal CDF
The area under the standard bell curve to the left of a z-score.

Resistor values scatter in a bell-shaped normal distribution with mean $\mu$ and standard deviation $\sigma$ .
Convert each spec limit to a z-score, $z = (x - \mu)/\sigma$ : how many standard deviations it lies from the mean.
The standard normal CDF $\Phi(z)$ gives the area to the left of $z$ , so the share in spec is $\Phi(z_U) - \Phi(z_L)$ .
Limits at $\pm 3\sigma$ keep 99.73% in spec (2,700 defects per million); "six sigma" quality puts them $6\sigma$ away.

Learn the math

The Normal DistributionStats · Lesson 21, Continuous Distributions and the Normal Curve Using the Normal DistributionStats · Lesson 22, Continuous Distributions and the Normal Curve Measurement Error: Accuracy and PrecisionEng Stats · Lesson 1, Engineering Data and Variation Sums of Random Variables and Tolerance Stack-UpEng Stats · Lesson 7, Combining Variation Monte Carlo Tolerance AnalysisEng Stats · Lesson 9, Combining Variation Tolerance IntervalsEng Stats · Lesson 15, Intervals for Engineering Decisions Testing Variability: χ² and F TestsEng Stats · Lesson 19, Testing Engineering Claims Process Capability: Cp and CpkEng Stats · Lesson 29, Statistical Process Control

Learn the math

The Normal DistributionStats · Lesson 21, Continuous Distributions and the Normal Curve

Using the Normal DistributionStats · Lesson 22, Continuous Distributions and the Normal Curve

Measurement Error: Accuracy and PrecisionEng Stats · Lesson 1, Engineering Data and Variation

Sums of Random Variables and Tolerance Stack-UpEng Stats · Lesson 7, Combining Variation

Monte Carlo Tolerance AnalysisEng Stats · Lesson 9, Combining Variation

Tolerance IntervalsEng Stats · Lesson 15, Intervals for Engineering Decisions

Testing Variability: χ² and F TestsEng Stats · Lesson 19, Testing Engineering Claims

Process Capability: Cp and CpkEng Stats · Lesson 29, Statistical Process Control