We need to calculate a few numbers so that we can

assess the process capability to understand

more how or if the process can meet expected specifications.

Cpk is a process performance capability index that is used to

determine if the natural process limits lie inside of the specification limits.

It is calculated using the minimum of the Z upper or the Z lower limits.

The Z values will be calculated in a following example.

The next calculation is the CP or the capability index that shows how well

Cpk could be if the process is centered in the specification.

The calculation is given by using the range of the tolerance,

the max minus the mean divided by

six sigma hat or six times the estimated standard deviation.

Let's do an example.

Suppose we have a product being packaged with weight specifications between 1.0 or

the lower specification level and 1.10 or the upper specification level in pounds.

You can consider this as the tolerance range.

Packages shipped within this range is the desired target.

Data was collected over a period of 30 days,

and assuming the distribution is normally distributed or producing a bell-shaped curves,

the control chart designed used

a subgroup sample size of four and took readings every two hours.

The data shows that X double bar equal 1.065

pounds and then R bar equals 0.05.

Now let's calculate the estimated standard deviation from the X bar.

The formula is sigma hat equals R bar over D2,

which is the statistical constant for a subgroup of four.

So we see sigma hat equals 0.024 pounds.

Here, we must calculate the Z values for the upper and lower ranges.

The distance from the upper specification limit to the process average is

1.10 minus 1.065 equals 0.035.

Dividing this by the standard deviation estimate gives us 1.46.

This can be thought of as the number of standard deviations between

the process average and the upper specification limit.

This is normally represented by Z sub U.

We can also calculate the lower Z value using a similar formula to be around 2.70.

Now, we can use the Z values and our Cpk formula.

We choose the minimum of the two numbers,

1.46 and 2.70, which is 1.46 and dividing it by three, giving us 0.49.

The CP calculation uses the tolerance range of 1.10 minus

1.00 or .10 divided by six sigma hat or .024,

that we calculated earlier,

and it yields us 0.69.

Now that we have CP and Cpk identified,

we can apply some rules of thumb.

If CP equals 2 and Cpk equals 1.5,

then these are the values we want when a process is achieved six sigma quality.

If either CP or Cpk is greater than or equal to 1.33,

this indicates the process is capable.

If either CP or Cpk values of around 1.0 means that

the process barely made specification and will produce 0.27% defects statistically.

If either CP or Cpk is less than 1,

that indicates the process is producing units

outside of engineering specifications or the tolerance.

And an abnormally high CP or Cpk, such as greater than 3,

indicates either the specification is too loose or there

exists an opportunity to move to a less expensive process.

Some other characteristics.

Basically you want CP and Cpk to be as high as possible, but not over 3.

The CP value does not change as the process

is being centered unless something in the process changes.

The CP will equal Cpk if the process is perfectly centered.

Cpk is always equal to or smaller than CP and if Cpk becomes negative,

the process average is outside one of the upper or lower control limits.

For our example, we just calculated CP equals 0.69 and Cpk equals 0.49.

They are not equal and they were both under 1.0.

This indicates that the process is producing units outside of engineering specifications,

or in other words, we are producing nonconforming parts right now.

Let's see what we can do to improve this.