OpenSpecEngineering

Finds how well a process fits its tolerance: the capability indices Cp and Cpk from the short-term spread, the performance indices Pp and Ppk from the long-term spread, and the expected parts per million (PPM) outside each specification limit. Quality and manufacturing engineers use it for production part approval, supplier qualification and capability studies [2]. The live panel below the Solution draws the process curve against the limits as you type and shades the out-of-spec tails.

Equations

Cp = (USL − LSL) / 6σ_w

Cpk = min(USL − x̄, x̄ − LSL) / 3σ_w

Pp = (USL − LSL) / 6σ_o

Ppk = min(USL − x̄, x̄ − LSL) / 3σ_o

PPM above = 10⁶·Φ((x̄ − USL)/σ_o)

PPM below = 10⁶·Φ((LSL − x̄)/σ_o)

USL and LSL are the upper and lower specification limits and x̄ the process mean, all in the same unit. σ_w is the within-subgroup standard deviation, usually R̄/d₂ or s̄/c₄ from a control chart. σ_o is the overall standard deviation of all the readings; enter 0 to use σ_w. Φ is the standard normal cumulative distribution [1]. Cp is what the process could do if it were centered; Cpk adds the penalty for being off center. The PPM uses the long-term spread, since that is what the customer receives.

Limits and common mistakes

• The indices mean something only for a stable process. Check the control chart first.
• The PPM assumes a normal distribution. Skewed data, such as flatness or runout, need a fitted distribution or a transformation.
• Don’t put the overall standard deviation in the within field. Cpk then equals Ppk and the short-term figure looks worse than it is.
• Many customers require Cpk ≥ 1.33 for an ongoing process and more for a new or safety-critical one. A small sample gives a wide confidence band on any index.
• The live panel uses σ_w, so its curve and its Cp and Cpk are short-term values.

Worked example

A turned shaft diameter is toleranced 25.00 ± 0.05 mm. A capability study gives a mean of 25.012 mm, a within-subgroup standard deviation of 0.011 mm and an overall one of 0.014 mm. Does it meet Cpk 1.33, and how many parts will be out?

1. Enter Upper Specification Limit (USL) 25.05 and Lower Specification Limit (LSL) 24.95.
2. Enter Process Mean (x̄) 25.012.
3. Enter Within-Subgroup Std Dev 0.011 and Overall Std Dev 0.014.
4. Press Solve. The Solution shows: Cp (potential, within): 1.51515
   Cpk (actual, within): 1.15152
   Pp (overall): 1.19048
   Ppk (overall): 0.904762
   Expected PPM above USL: 3,320.94 ppm
   Expected PPM below LSL: 4.74297 ppm
   Expected total PPM out of spec: 3,325.69 ppm

Check: the mean sits 0.038 mm below the USL, and 0.038 / (3 × 0.011) = 1.15. The spread is good enough (Cp 1.52), but the process runs high. Centering it at 25.000 mm would raise Cpk to Cp, above 1.33.