Capability Analysis

This section contains details about the computation of the statistics in the Capability Analysis report.

Variation Statistics

All capability analyses use the same formulas. Options differ in how sigma (σ) is computed:

Long-term

Uses the overall sigma. This option is used for

statistics, and computes sigma as follows:

Note: By default, the capability indices in the Long Term Sigma report use the Cp labeling that is used in the other sigma reports. To use Ppk labeling in the Long Term Sigma report, select the File > Preferences > Platforms > Distribution > PpK Capability Labeling preference.

Specified Sigma

Enables you to enter a specific, known sigma for computing capability indices. Sigma is user-specified, and is therefore not computed.

Moving Range

Enables you to enter a moving range span to compute sigma as follows:

where

is the average of the moving ranges

d2(n) is the expected value of the range of n independent normally distributed variables with unit standard deviation.

Short Term Sigma, Group by Fixed Subgroup Size

If r is the number of subgroups of size nj and each ith subgroup is defined by the order of the data, sigma is computed as follows:

where

This formula is commonly referred to as the Root Mean Square Error, or RMSE.

Quantile Sigma

(Appears only for capability analyses based on fitted distributions.) Uses the square root of the variance of the fitted distribution. For more information about the relationship between the parameters in the Parameter Estimates report and the expected value and variance of the fitted distributions, see Continuous Fit Distributions and Discrete Fit Distributions.

Capability Indices for Normal Distributions

This section provides details about the calculation of capability indices for normal data.

For a process characteristic with mean μ and standard deviation σ, the population-based capability indices are defined as follows.

LSL is the lower specification limit

USL is the upper specification limit

T is the target value

For sample-based capability indices, the parameters are replaced by their estimates. The estimate for σ uses the method that you specified in the Capability Analysis window. See Variation Statistics.

If either of the specification limits is missing, the capability indices containing the missing specification limit are reported as missing.

Tip: A capability index of 1.33 is often considered to be the minimum value that is acceptable. For a normal distribution, a capability index of 1.33 corresponds to an expected number of nonconforming units of about 6 per 100,000.

Confidence Intervals for Capability Indices

Note: Confidence intervals for capability indices appear only in the Long Term Sigma report.

The 100(1 - α)% confidence interval for Cp is calculated as follows:

where

is the estimated value for Cp

is the (α/2)th quantile of a chi-square distribution with n - 1 degrees of freedom

n is the number of observations

The 100(1 - α)% confidence interval for Cpk is calculated as follows:

where

is the estimated value for Cpk

is the (1 - α/2)th quantile of a standard normal distribution

n is the number of observations

The 100(1 - α)% confidence interval for CPM is calculated as follows:

where

is the estimated value for CPM

is the (α/2)th quantile of a chi-square distribution with γ degrees of freedom

n is the number of observations

is the mean of the observations

T is the target value

s is the long-term sigma estimate

Note: The confidence interval for CPM is computed only when the target value is centered between the lower and upper specification limits.

Lower and upper confidence limits for CPL and CPU are computed using the method of Chou et al. (1990).

The 100(1 - α)% confidence limits for CPL (denoted by CPLL and CPLU) satisfy the following equations:

where

tn-1(δ) has a non-central t-distribution with n - 1 degrees of freedom and noncentrality parameter δ

is the estimated value for Cpl

The 100(1 - α)% confidence limits for CPU (denoted by CPUL and CPUU) satisfy the following equations:

where

tn-1(δ) has a non-central t-distribution with n - 1 degrees of freedom and noncentrality parameter δ

is the estimated value for Cpu

Capability Indices for Nonnormal Distributions

This section describes how capability indices are calculated for nonnormal distributions. These generalized capability indices are defined as follows:

LSL is the lower specification limit

USL is the upper specification limit

T is the target value

Pα is the α*100th percentile of the fitted distribution

For the calculation of Cpm, μ and σ are estimated using the expected value and square root of the variance of the fitted distribution. For more information about the relationship between the parameters in the Parameter Estimates report and the expected value and variance of the fitted distributions, see Continuous Fit Distributions and Discrete Fit Distributions.

Sigma Quality Statistics

The Sigma Quality statistics for each Portion (Below LSL, Above USL, and Total Outside) are calculated as follows: