Ppl and Ppu are one-sided performance indices. They measure the distance from the process mean to each spec limit using the overall standard deviation. Ppl covers the lower limit, Ppu covers the upper. Together they form Ppk: Ppk = min(Ppl, Ppu).
The Formulas
Ppu = (USL - X̄) / 3σo
Ppl = (X̄ - LSL) / 3σo
Where:
- USL is the upper specification limit
- LSL is the lower specification limit
- X̄ is the process mean
- σo is the overall standard deviation (n divisor)
These are identical to Cpu and Cpl, except they use σo (overall, n divisor) instead of σ (sample, n-1 divisor). This makes Ppl and Ppu long-term metrics that include all sources of variation.
Ppl/Ppu vs Cpl/Cpu
| Index Pair | Standard Deviation | What It Captures |
|---|---|---|
| Cpl / Cpu | Sample σ (n-1 divisor) | Short-term margin to each spec limit |
| Ppl / Ppu | Overall σo (n divisor) | Long-term margin to each spec limit |
If Cpu is 1.5 but Ppu is 0.9, the process has enough short-term margin on the upper side but loses it over time. Something between batches is pushing the distribution toward USL. Conversely, if Cpl and Ppl are similar, the lower-side margin is stable.
Directional Drift Detection
The real value of one-sided indices (both Cp and Pp families) is diagnosing which direction your process is drifting.
| Signal | Meaning | Action |
|---|---|---|
| Ppu dropping, Ppl stable | Process mean drifting toward USL over time | Investigate upward drift: calibration, temperature, component lots |
| Ppl dropping, Ppu stable | Process mean drifting toward LSL over time | Investigate downward drift |
| Both dropping | Overall variation increasing | Reduce σ, don't adjust mean |
| Ppu < Cpu | Long-term upper margin worse than short-term | Between-batch variation biased upward |
Reading Ppl and Ppu in TofuPilot
Open the Process Control page, select a numeric measurement, and switch to the Capability tab. The second (teal) KPI row shows Ppk, Pp, Ppl, and Ppu.
Compare the teal row to the purple row above it. If Ppu is significantly lower than Cpu, or Ppl is significantly lower than Cpl, the gap tells you where long-term variation is hurting you most.
Click any index to toggle it on the daily trend chart. Plotting Ppl and Ppu together over weeks reveals whether the process is drifting toward one spec limit over time.
When One-Sided Specs Apply
For measurements with only an upper limit (leakage, noise, response time), TofuPilot shows Ppu and "--" for Ppl. For measurements with only a lower limit (signal strength, gain), TofuPilot shows Ppl and "--" for Ppu. This is the correct behavior. There's no opposing limit to measure against.