How far off is a straight-line Φ(z)?

Six piecewise-linear approximations of the standard normal CDF on 0 ≤ z ≤ 3, checked against a double-precision reference. Drag on any chart to move z.

e(z) = Φ̂(z) − Φ(z)

Cumulative probability

At this z

Model shown:

Signed error e(z), percentage points

Model A sits at or below zero everywhere: the normal CDF is concave for z > 0, so every chord lies under the curve.

Relative error in the tail Q(z) = 1 − Φ(z), %

A 1 pp error in Φ near z = 2.5 doubles the defect rate. Use these models for yield, not for ppm.

Process yield check

Enter specification limits and process parameters. The exact column uses the double-precision reference.

All models on 0 ≤ z ≤ 3

Computed in your browser on 3,001 points. Values in probability units; multiply by 100 for percentage points.

Field rule: 50 · 84 · 98 · 100

  1. Remember Φ at z = 0, 1, 2, 3: 0.50 0.84 0.98 1.00
  2. Add slope × distance into the segment: 0.34 on [0,1], 0.14 on [1,2], 0.02 on [2,3].
  3. Negative z: use 1 − Φ̂(|z|). Two-sided coverage: 2Φ̂ − 1.

Worst case: 2.35 pp in Φ at z ≈ 1.45, so up to 4.7 pp in two-sided coverage.

Spreadsheet formula (Model C)

Cell A2 holds z. Returns #N/A outside [−3, 3].

=IF(ABS(A2)>3,NA(),0.5+SIGN(A2)*IF(ABS(A2)<=1,0.34*ABS(A2),IF(ABS(A2)<=2,0.34+0.14*(ABS(A2)-1),0.48+0.02*(ABS(A2)-2))))