SME
Manufacturing Processes · Problem type
Process capability: Cₚ and Cₚₖ
Use this for “A shaft is 25.000 ± 0.030 mm. The process runs at a mean of 25.006 mm with σ = 0.008 mm. Find Cₚ, Cₚₖ and the share outside.”
Chart
X is normal with μ = 25.006 mm and σ = 0.008 mm.
P(X ≤ 24.97 or X ≥ 25.03) = 0.0014.
This is an example. Type your own values to start a new problem.
Cₚ
=
USL (mm)
−
LSL (mm)
6
×
σ (mm)
μ
Process mean
example · mm
Cₚₖ
Capability with the offset
calculated
pₒᵤₜ
Share outside the near limit
calculated · %
Formulas
Cₚ
=
USL − LSL
6 ×
σ
1.25
=
25.03 − 24.97
6 × 0.008
Cₚₖ
=
min(USL −
μ
,
μ
− LSL)
3 ×
σ
1 = min(25.03 − 25.006, 25.006 − 24.97) ÷ (3 × 0.008)
pₒᵤₜ = 100 × Q(3 × Cₚₖ)
0.135 = 100 × Q(3 × 1)
Assumptions
•
The size is normally distributed with a steady mean and standard deviation.
•
Cₚ compares the tolerance with the 6σ spread; Cₚₖ also counts how far the mean is off centre.
•
Cₚₖ ≥ 1.33 is a common target; below 1 some parts fall outside the limits.
Step-by-step
WE KNOW
USL = 25.03 mm, LSL = 24.97 mm, μ = 25.006 mm, σ = 0.008 mm
FIND
potential capability (Cₚ), capability with the offset (Cₚₖ), share outside the near limit (pₒᵤₜ)
Step 1 · Find potential capability (Cₚ)
Use Cₚ = (USL − LSL) ÷ (6 × σ)
Divide
the
tolerance
width
by
the
process
spread
of
6
σ
.
Cₚ
=
25.03 − 24.97
6 × 0.008
Cₚ
=
0.06
0.048
Cₚ = 1.25
Step 2 · Find capability with the offset (Cₚₖ)
Use Cₚₖ = min(USL − μ, μ − LSL) ÷ (3 × σ)
Take
the
mean’s
distance
to
the
nearer
limit
and
divide
by
half
the
spread,
3
σ
.
Cₚₖ = min(25.03 − 25.006, 25.006 − 24.97) ÷ (3 × 0.008)
Cₚₖ
=
min(0.024, 0.036)
0.024
Cₚₖ
=
0.024
0.024
Cₚₖ = 1
Step 3 · Find share outside the near limit (pₒᵤₜ)
Use pₒᵤₜ = 100 × Q(3 × Cₚₖ)
The near limit is 3Cₚₖ standard deviations from the mean; Q is the normal curve’s area beyond it, as a percent.
pₒᵤₜ = 100 × Q(3 × 1)
pₒᵤₜ = 100 × 0.00135
pₒᵤₜ = 0.135%
Check
Put the numbers back in. Both sides should match.
1.25
=
25.03 − 24.97
6 × 0.008
✓
1 = min(25.03 − 25.006, 25.006 − 24.97) ÷ (3 × 0.008) ✓
0.135 = 100 × Q(3 × 1) ✓
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Tolerance stack-up: worst case and RSS
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Clearance, transition or interference?
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