Z Index Infinity Calculator

Formula Used

Z_Lower = (Mean − LSL) / Standard Deviation

Z_Upper = (USL − Mean) / Standard Deviation

Z Index = min(Z_Lower, Z_Upper)

If one spec limit is blank, that side uses Infinity

Lower Defect Rate = Φ((LSL − Mean) / Standard Deviation)

Upper Defect Rate = 1 − Φ((USL − Mean) / Standard Deviation)

Yield = 1 − (Lower Defect Rate + Upper Defect Rate)

Expected Rejects = Lot Size × Total Defect Rate

Shifted Sigma = Z Index + Shift Adjustment

This method fits manufacturing capability studies where the process follows an approximately normal distribution. When one limit does not apply, the blank side is effectively open-ended, so its Z distance becomes Infinity and its defect probability becomes zero.

How to Use This Calculator

Enter the process mean from your measured production data.
Enter the process standard deviation using the same units.
Provide a lower limit, an upper limit, or both.
Leave one side blank when only one manufacturing limit matters.
Enter lot size to estimate expected rejects for that batch.
Set the shift adjustment if your quality program uses it.
Choose output precision for cleaner reporting.
Click calculate to show the result above the form.
Review Z distances, yield, ppm, Cpk, and the normal curve.
Use the CSV or PDF buttons for reporting and sharing.

Example Data Table

Case	Mean	Std Dev	LSL	USL	Z Lower	Z Upper	Z Index	Interpretation
Two-sided shaft diameter	50.00 mm	2.50 mm	45.00 mm	56.00 mm	2.00	2.40	2.00	Lower side is closer to failure.
Upper-only coating thickness	14.20 µm	0.80 µm	∞	16.00 µm	∞	2.25	2.25	One-sided process check.
Lower-only tensile strength	620 MPa	25 MPa	580 MPa	∞	1.60	∞	1.60	Lower strength threshold controls risk.

FAQs

1. What does Infinity mean in this calculator?

Infinity appears when one specification side is left blank. That side is treated as open-ended, so it does not constrain the process or add defect probability.

2. When should I leave one specification blank?

Leave one side blank for one-sided manufacturing checks, such as minimum strength, maximum thickness, or any process with only one valid acceptance boundary.

3. What is the Z Index showing me?

The Z Index is the smallest sigma distance between your process mean and the active specification limits. It highlights the side most likely to fail first.

4. Why are yield and ppm included?

Yield shows the percentage expected to pass. PPM converts the same risk into defects per million, which is common in manufacturing quality reporting.

5. Does this calculator assume a normal distribution?

Yes. The defect rates and yield are estimated from a normal model. For strongly skewed or mixed processes, use a distribution model that better fits your data.

6. What is the shift adjustment used for?

Some quality programs report a shifted sigma value. The calculator adds your selected shift to the Z Index so you can compare results to that reporting style.

7. How is Cpk handled with one-sided limits?

When one side is blank, that capability side becomes Infinity. The reported Cpk then follows the active side only, which is appropriate for one-sided tolerance studies.

8. Can I use this for batch-level reject estimates?

Yes. Enter the lot size and the calculator estimates expected rejects by multiplying the lot size by the total defect rate.