## Calculating Capability Indexes

Datapage+ can calculate four different capability indices based on each estimate of the process standard deviation.

The capability indices that are based on the control chart estimates are Cp, Cpk, Cr and PC%.

The capability indices that are based on the sample standard deviation are Pp, Ppk, Pr and PP%.

Each calculation depends on an estimate of the standard deviation and on the lower and upper specification limits (LSL and USL).

The Calculation for each follows.

Index | Calculation | Applies to | Comments
Pp | `((USL-LSL))/(6*S_(sampl\e))` | All Cases | Notes 1 and 2
Ppk | `(M\i\n((USL - mu),(mu - LSL)))/(3*S_(sampl\e)` | All Cases | Notes 1 and 3
Pr | `(6*S_(sampl\e))/((USL-LSL))` | All Cases | Notes 1 and 2
PP% | `(100*6*S_(sampl\e))/((USL-LSL))` | All Cases | Notes 1 and 2
Cp | `((USL - LSL)) / (6*S_(XMR2))` `((USL - LSL)) / (6*S_(XMR3))` `((USL - LSL)) / (6*S_(RX))` `((USL - LSL)) / (6*S_(XBR))` `((USL - LSL)) / (6*S_(XTR))` | Sub Size 1, XMR(2) Chart Sub Size 1, XMR(3) Chart Sub Size 1, RX Chart Sub Size > 1, XBar&R Chart Sub Size > 1, XBar&S Chart

Sub Size > 1, Xtilde&R Chart

Notes 2 and 4

Cpk | `(M\i\n((USL - mu),(mu - LSL)))/(3*S_(XMR2)` `(M\i\n((USL - mu),(mu - LSL)))/(3*S_(XMR3)` `(M\i\n((USL - mu),(mu - LSL)))/(3*S_(RX)` `(M\i\n((USL - mu),(mu - LSL)))/(3*S_(XBR)` `(M\i\n((USL - mu),(mu - LSL)))/(3*S_(XBS)`

`(M\i\n((USL - mu),(mu - LSL)))/(3*S_(XTR)`

Sub Size 1, XMR(2) Chart

Sub Size 1, XMR(3) Chart

Sub Size 1, RX Chart

Sub Size > 1, XBar&R Chart

Sub Size > 1, XBar&S Chart

Sub Size > 1, Xtilde&R Chart

Notes 3 and 4

Cr | `(6*S_(XMR2)) / ((USL - LSL))` `(6*S_(XMR3)) / ((USL - LSL))` `(6*S_(RX)) / ((USL - LSL))` `(6*S_(XBR)) / ((USL - LSL))` `(6*S_(XBS)) / ((USL - LSL))`

`(6*S_(XTR)) / ((USL - LSL))`

Sub Size 1, XMR(2) Chart

Sub Size 1, XMR(3) Chart

Sub Size 1, RX Chart

Sub Size > 1, XBar&R Chart

Sub Size > 1, XBar&S Chart

Sub Size > 1, Xtilde&R Chart

Notes 2 and 4

PC% | `(100*6*S_(XMR2)) / ((USL - LSL))` `(100*6*S_(RX)) / ((USL - LSL))` `(100*6*S_(XBR)) / ((USL - LSL))` `(100*6*S_(XBS)) / ((USL - LSL))` `(100*6*S_(XTR)) / ((USL - LSL))` | Sub Size 1, XMR(2) Chart Sub Size 1, XMR(3) Chart Sub Size 1, RX Chart Sub Size > 1, XBar&R Chart Sub Size > 1, XBar&S Chart

Sub Size > 1, Xtilde&R Chart

Notes 2 and 4

Notes:

1. The capability indices that are calculated using the sample standard deviation, `S_(Sample)`, are Pp, Ppk, Pr, and PP%. Different calculations are performed when a non-normal distribution is used to model the data. These calculations are located here: (Link it)

2. The indices Pp, Pr, PP%, Cp, Cr, and PC% are functions of the total tolerance zone `((USL - LSL))`. The value of these indices is questionable when applied to a unilaterally toleranced process such as true position, flatness, etc. In these cases there is not really a tolerance zone because 0.0 is a mathematical limit and not a specification limit. Use of non-normal distributions may be useful in unilaterally toleranced situations.

3. The indices Ppk and Cpk are calculated using the specification limit that is closest to the mean. For unilaterally toleranced processes, the calculations are applied only to the actual tolerance. For example, in the case of true position or flatness Cpk and Ppk are calculated as `((USL - mu)) / (3*S_(???))`. If this alteration is not made, computations of Cpk and Ppk will indicate the process is less capable when in fact it is getting closer to the target of zero. This often causes the apparently impossible condition of Cpk (or Ppk) being larger than Cp (or Pp). The problem here is not in the calculation of Cpk and Ppk, but the non-validity of Cp and Pp.

4. The capability indices that are calculated using the control chart estimates; `S_(XMR2)`, `S_(XMR3)`, `S_(RX)`, `S_(XBR)`, `S_(XBS)`, and `S_(XTR)`, are Cp, Cpk, Cr, and PC%.

5. A common point of confusion is using the sample standard deviation in the capability calculations and to check the calculations of Cp, Cpk, Crk, and PC%; when these values are actually used to calculate Pp, Ppk, Pr, and PP%.