§1Populations, samples and why statistics
The population is everything the process will ever make; the sample is the dozen parts on the bench. Statistics is the discipline of saying something honest about the first from the second.
Two habits make the rest of the page work. First, distinguish variables data (a measured value — a diameter, a hardness) from attributes data (pass/fail counts); variables data carries far more information per part. Second, sample randomly: measuring five consecutive parts tells you about five minutes of the process, not about the shift. All formulas below assume the sample fairly represents the population — no arithmetic can repair a biased sample.
Contents§2Describing a sample
Two numbers summarise a sample: where it sits (the mean) and how much it spreads (the standard deviation). The running dataset below is used for the rest of the page.
| # | mm | # | mm | # | mm | # | mm |
|---|---|---|---|---|---|---|---|
| 1 | 25.003 | 4 | 25.001 | 7 | 25.004 | 10 | 25.002 |
| 2 | 24.998 | 5 | 24.995 | 8 | 24.999 | 11 | 24.996 |
| 3 | 25.007 | 6 | 25.010 | 9 | 25.006 | 12 | 25.005 |
n = 12: x̄ = 25.0022 mm, s = 0.0046 mm, range 0.015 mm. The divisor n − 1 (not n) compensates for estimating the mean from the same data — with it, s estimates the population spread without systematic optimism.
The median (middle value) resists outliers better than the mean; when the two disagree noticeably, the data is skewed or contaminated and deserves a look before any formula is applied.
Contents§3Distribution curves
Stack a histogram of enough readings and a shape emerges. The shape is a fingerprint of the process.
A stable process fed by many small independent influences — tool deflection, temperature drift, material variation, none dominant — piles its output into the symmetric bell of the normal distribution; the central limit theorem all but guarantees it. Departures diagnose causes: a skewed tail suggests a one-sided constraint (a stop, a dwell); two humps suggest two machines or two setups mixed in one batch; a truncated edge suggests 100 % inspection has already removed one tail. Plot the histogram before trusting any statistic — the formulas of §4 onward assume the bell.
Contents§4The normal curve and z-values
One curve, fully described by its mean μ and standard deviation σ. Convert any reading to “how many σ from the mean” and a single table answers every probability question.
| z | Φ(z) | z | Φ(z) |
|---|---|---|---|
| 0.0 | 0.5000 | 1.6 | 0.9452 |
| 0.1 | 0.5398 | 1.7 | 0.9554 |
| 0.2 | 0.5793 | 1.8 | 0.9641 |
| 0.3 | 0.6179 | 1.9 | 0.9713 |
| 0.4 | 0.6554 | 2.0 | 0.9772 |
| 0.5 | 0.6915 | 2.1 | 0.9821 |
| 0.6 | 0.7257 | 2.2 | 0.9861 |
| 0.7 | 0.7580 | 2.3 | 0.9893 |
| 0.8 | 0.7881 | 2.4 | 0.9918 |
| 0.9 | 0.8159 | 2.5 | 0.9938 |
| 1.0 | 0.8413 | 2.6 | 0.9953 |
| 1.1 | 0.8643 | 2.7 | 0.9965 |
| 1.2 | 0.8849 | 2.8 | 0.9974 |
| 1.3 | 0.9032 | 2.9 | 0.9981 |
| 1.4 | 0.9192 | 3.0 | 0.9987 |
| 1.5 | 0.9332 | ||
| Φ(−z) = 1 − Φ(z). Fraction outside ±z is 2[1 − Φ(z)]. | |||
§5Fraction outside limits
Given the sample’s mean and spread, the normal table predicts scrap before the batch is run.
Limits 24.990–25.010, x̄ = 25.0022, s = 0.0046. z_low = (24.990 − 25.0022)/0.0046 = −2.66; z_high = (25.010 − 25.0022)/0.0046 = +1.71.
Below: Φ(−2.66) = 0.4 %. Above: 1 − Φ(1.71) = 4.3 %. Total ≈ 4.7 % outside limits — nearly all oversize.
The forecast is not just a number — it says the mean sits 2.2 µm high. Re-centring the process to 25.000 would drop both z-magnitudes to ±2.17 and the predicted scrap to about 3 % total; reducing s is the harder, slower project. Centre first.
§6How many tests are enough
Every extra test costs time; too few tests cost the truth. The sample-size formula makes the trade explicit before the testing starts.
With s = 0.0046 from the pilot sample: n = (1.96 × 0.0046 / 0.002)² = 20.1 → 21 measurements. Halving E would quadruple n — precision of the mean is bought at a quadratic price.
For small samples the normal z understates uncertainty because s itself is only an estimate; Student’s t replaces z, fattening the margin. The correction matters below n ≈ 30 and vanishes above:
| ν = n − 1 | t | ν = n − 1 | t |
|---|---|---|---|
| 2 | 4.3027 | 12 | 2.1788 |
| 3 | 3.1824 | 15 | 2.1314 |
| 4 | 2.7764 | 20 | 2.0860 |
| 5 | 2.5706 | 25 | 2.0595 |
| 6 | 2.4469 | 30 | 2.0423 |
| 8 | 2.3060 | ∞ | 1.9600 |
| 10 | 2.2281 | ||
| Use t in place of 1.96 when s comes from the same small sample; at ν = ∞ the t collapses to the normal z. | |||
§7Comparing two averages
Machine B’s sample mean will always differ from machine A’s. The test statistic asks the only fair question: is the gap large compared with the scatter?
Lathe A: x̄ = 25.0022, s = 0.0046, n = 12. Lathe B: x̄ = 25.0062, same s and n. z = 0.0040 / √(2 × 0.0046²/12) = 2.14 — beyond 1.96, so the 4 µm offset is real, not sampling luck, and B needs a tool offset. Had z landed at 1.2, the honest verdict would be “no detectable difference at this sample size”.
With big samples, trivial differences become “significant”. Always state the difference in millimetres next to the verdict; the tolerance, not the z-value, decides whether anyone should care.
§8Machinability and hardness
Hardness is the quickest, cheapest test that predicts how a material will cut — imperfectly, but usefully.
Across a family of similar steels, allowable cutting speed falls roughly in inverse proportion to Brinell hardness: a batch arriving at 300 HB instead of the usual 200 HB warrants a first-guess speed reduction of about a third, then adjustment by trial. The relation is a correlation with scatter, not a law — microstructure, inclusions and work-hardening behaviour shift it, and two materials of equal hardness can machine very differently (compare a free-machining leaded steel with an austenitic stainless at the same HB). Treat hardness as the x-axis of a scatter plot built from your own shop’s data; the regression through that cloud, with its residual spread quantified by the tools of §2, is worth more than any handbook constant. The Machining Operations page of this Library carries the theme into speeds, feeds and Taylor tool-life economics.
Contents§9Quick reference
The working core of the page on one card rack.
Sample
x̄ = Σx/n
s = √(Σ(x−x̄)²/(n−1))
Standardise
z = (x − μ)/σ
±1σ 68.3 % · ±2σ 95.4 % · ±3σ 99.7 %
Outside limits
P = Φ(z_low) + 1 − Φ(z_high)
Sample size
n = (zs/E)²
use t below n ≈ 30
Two means
z = (x̄₁−x̄₂)/√(s₁²/n₁+s₂²/n₂)
real if |z| > 1.96
Hardness rule
speed ∝ 1/HB (first guess)
then trust your own scatter plot