Cpk & Defect Rate (PPM) Calculator
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Cpk & Defect Rate Calculator
Calculate Process Capability Index (Cpk) and estimate the Parts Per Million (PPM) defect rate based on your process specifications.
Please check your inputs: LSL must be less than USL, and Standard Deviation must be greater than 0.
Cpk (Capability Index)
–
Cp (Potential Capability)
–
Expected Yield
–
Defects Per Million (PPM)
–
Cpu (Upper Capability)
–
Cpl (Lower Capability)
–
Sigma Level (Approx)
–
What is Cpk and Why Does it Matter?
The Process Capability Index (Cpk) is a vital statistical tool used in Six Sigma and Quality Control manufacturing. It measures how well a process is performing relative to its specification limits. Unlike a simple average, Cpk accounts for both the centering of the process and the variation (spread) within it.
A higher Cpk value indicates a more capable process with fewer defects. The standard benchmark in many industries (particularly automotive) is a Cpk of 1.33, while Six Sigma standards aim for a Cpk of 2.0.
Key Metrics Explained
- Cp: The potential capability of the process if it were perfectly centered. It only looks at the spread of the data relative to the tolerance width.
- Cpk: The actual capability. It considers how close the process mean is to the nearest specification limit.
- PPM (Parts Per Million): The estimated number of defective parts for every one million parts produced based on the normal distribution curve.
Formulas Used in This Calculator
This tool utilizes standard statistical formulas to derive the capability indices and defect rates assuming a normal distribution.
Cp = (USL – LSL) / (6 * σ)
Cpu = (USL – Mean) / (3 * σ)
Cpl = (Mean – LSL) / (3 * σ)
Cpk = min(Cpu, Cpl)
Calculating Defect Rates (PPM)
To calculate the PPM, we determine the area under the normal distribution curve that falls outside the specification limits (tails).
The Z-score for the Upper Limit is: Z_upper = (USL – Mean) / σ
The Z-score for the Lower Limit is: Z_lower = (Mean – LSL) / σ
Using the Cumulative Distribution Function (CDF), we sum the probabilities of failure on both sides and multiply by 1,000,000.
Interpreting Your Results
Understanding the Cpk value is crucial for process improvement:
- Cpk < 1.0: The process is not capable. A significant number of defects are being produced. Immediate action is required to reduce variation or center the process.
- 1.0 ≤ Cpk < 1.33: The process is marginally capable. While relatively stable, there is little margin for error.
- Cpk ≥ 1.33: The process is capable and meets standard industry requirements.
- Cpk ≥ 2.0: Six Sigma quality level. Defect rates are virtually non-existent (approx 3.4 PPM).
Example Calculation
Imagine a manufacturing process for a metal rod with a target diameter of 10mm.
- USL: 10.5 mm
- LSL: 9.5 mm
- Mean: 10.1 mm (The process is slightly off-center)
- Standard Deviation: 0.1 mm
Using the calculator above, you would find:
- Cp: 1.67 (The process spread is tight enough)
- Cpk: 1.33 (Limited by the shift towards the upper limit)
- PPM: ~32 defects per million parts.
// Standard Normal Cumulative Distribution Function
// Approximation using Abramowitz and Stegun 26.2.17
function stdNormalCDF(x) {
var t = 1 / (1 + 0.2316419 * Math.abs(x));
var d = 0.3989423 * Math.exp(-x * x / 2);
var prob = d * t * (0.3193815 + t * (-0.3565638 + t * (1.781478 + t * (-1.821256 + t * 1.330274))));
if (x > 0) prob = 1 – prob;
return prob;
}
function calculateCpk() {
// Get inputs
var usl = parseFloat(document.getElementById('usl').value);
var lsl = parseFloat(document.getElementById('lsl').value);
var mean = parseFloat(document.getElementById('mean').value);
var sigma = parseFloat(document.getElementById('sigma').value);
var errorMsg = document.getElementById('errorMessage');
// Basic Validation
if (isNaN(usl) || isNaN(lsl) || isNaN(mean) || isNaN(sigma)) {
errorMsg.style.display = 'block';
errorMsg.innerText = "Please enter valid numbers for all fields.";
return;
}
if (lsl >= usl) {
errorMsg.style.display = 'block';
errorMsg.innerText = "Error: Lower Specification Limit (LSL) must be less than Upper Specification Limit (USL).";
return;
}
if (sigma USL
var probHigh = 1.0 – stdNormalCDF(zUpper);
// Probability of defect < LSL
var probLow = stdNormalCDF(zLower);
var totalDefectRate = probHigh + probLow;
var ppm = totalDefectRate * 1000000;
var yieldPercentage = (1 – totalDefectRate) * 100;
// Calculate Approximate Sigma Level (Long term usually adds 1.5 shift, but standard calc is Z_min + 1.5 or just 3*Cpk)
// Here we display the "Process Sigma" which is 3 * Cpk
var sigmaLevel = 3 * cpk;
// Update UI
document.getElementById('resultCpk').innerText = cpk.toFixed(3);
document.getElementById('resultCp').innerText = cp.toFixed(3);
document.getElementById('resultCpu').innerText = cpu.toFixed(3);
document.getElementById('resultCpl').innerText = cpl.toFixed(3);
// Format PPM with commas
document.getElementById('resultPpm').innerText = Math.round(ppm).toLocaleString();
document.getElementById('resultYield').innerText = yieldPercentage.toFixed(4) + "%";
document.getElementById('resultSigmaLevel').innerText = sigmaLevel.toFixed(2);
// Visual feedback for Cpk
var cpkElement = document.getElementById('resultCpk');
if (cpk < 1.0) {
cpkElement.style.color = '#c0392b'; // Red
} else if (cpk < 1.33) {
cpkElement.style.color = '#f39c12'; // Orange
} else {
cpkElement.style.color = '#27ae60'; // Green
}
}