# Process Capability Studies Using One-Sided Specifications

Process capability studies calculate the process capability indices Cp and Cpk.  These statistics tell us how well the process is meeting specifications or requirements.  Capability studies are designed for two-sided specifications that have an upper and a lower limit.  Adjustments must be made when we have a one-sided specification and/or a boundary limit in place of a specification limit.  In this article we will look in detail at two real-world examples of capability analysis using one-sided requirements and boundary limits.

Capability studies are designed for continuous data that is normally distributed and comes from a stable process. We must confirm normality and stability before we conduct these studies.  Capability indices may also be calculated for non-normal data.  We would choose this type of analysis if the data are continuous and stable but distributed in some fashion other than normal, say in a Weibull distribution.

Cpk is more commonly used than Cp because it considers process centering.  We calculate two values for Cpk, one using the upper specification limit and the other using the lower specification limit.  We then choose the smallest of the two as the Cpk that is most representative of the capability of the process to stay within specification.  The smallest value results when the center of the distribution is closer to one of the specification limits.

It is possible to run capability studies using only one specification limit.  Some experts calculate only one Cpk value and use that as representative of the process.  This may make sense in some but not all cases.  Other experts advise against conducting process capability studies when there is a one-sided specification.

Instead of an upper or a lower specification limit, a process may have a boundary limit. You should only define a boundary limit if it is not possible for data to exceed that limit. Here is a real-world example.

Conversion rate for customer orders is calculated by dividing the number of orders confirmed by the number of customer contacts.  The requirement/specification for conversion rate is 40%.  If we consider 40% to be a minimum requirement, we can use 40% as the lower specification limit.  Because conversion rate cannot exceed 100%, we can define the upper limit as a boundary of 100%.

The underlying data is discrete – the number of confirmed orders and the number of customer contacts are both discrete data elements.  The resulting conversion rate is a percentage (or proportion).  Percentage data may take on any value and is commonly treated in practice as continuous data, especially in cases like this when the underlying data (number of orders and number of contacts) is quantitative.  Again, be aware that not all experts agree with this approach.

Data on the total number of customer contacts and the number of confirmed orders was collected for 50 consecutive days of operation and the conversion rate was calculated for each day.  The resulting mean conversion rate was .2059, or 20.59%.  The distribution of the 50 data points was tested for normality using a probability plot with the following results (NOTE:  This plot and all subsequent analysis was done using Minitab):

The null hypothesis is no difference between the data and a normal distribution.  The alternative hypothesis is that there is a difference between the data and a normal distribution.  The calculated p value of .252 is greater than the alpha decision value of .05 for 95% confidence, so we fail to reject the null hypothesis and conclude that the data are normally distributed.

I next created I-MR Control Charts for Conversion Rate.  The I-MR charts were used because the 50 data points are individual values and conversion rate is being treated as continuous data.  The resulting charts show that the mean is stable.  The moving range chart shows just one out of control point – point 39 fell more than three standard deviations from the center line.  I am comfortable relying on the results of capability analysis based on these control charts.

I next ran a normal capability analysis with a lower spec limit of .40 and an upper boundary of 1.00.  Here is the resulting analysis:

The resulting Cpk value is -1.42.  A negative value implies that the output of the process falls outside of the specification.  Graphically we can see that all output falls below the target level of .4 or 40% conversion rate.  It is expected that 1M out of every 1M values, or 100% of the process output, will fall below the lower specification limit of .40 which defines the minimum acceptable performance.

Another possibility is to use binomial capability analysis for conversion rate.  The binomial distribution is useful when the result in question is binary – i.e. where there are only two possible results.  In this case the result of a customer contact is binary – either a confirmed order or no order.

Binomial process capability is designed to evaluate the capability of a process to stay below a target level of defectives.  A desired conversion rate of 40% or more confirmed orders is the same as a desired defective rate of 60% or fewer failures to confirm an order.

I ran a Binomial Capability analysis using the number of calls with no order as the defective and total number of calls to the customer as the non-constant sample size.  The result tells the same story as the previous capability analysis.  The mean percentage of defectives is 79% and the process is not at all capable of 60% or fewer defectives.

The resulting P control chart shows that the process is not at all stable.  It is interesting to note that the I-MR chart for conversion rate has much wider control limits and shows the process to be in stable, i.e. in control.  The P chart for defectives has much narrower control limits and shows the process to be unstable, i.e. out of control.

The Rate of Defectives plot is generated by Minitab when the sample size is not equal, which is the case here as the number of calls per day varies widely.  It is a scatter plot of the % defectives vs. sample size.  If the data fall randomly around the center line, we can conclude that the data follow a binomial distribution. This is clearly the case for sample sizes above 2000 calls per day, but not so for the days where the number of customer contacts was under 2000.   The number of calls per day ranged from 902 to 3535.

The calculated Upper and Lower Confidence intervals for the rate of defectives are 79.25% and 78.80% respectively.  We can be highly confident that the percentage of defectives will not fall below 60%.

The cumulative % defective chart should stabilize over time, which is the case here.  This indicates that we have enough samples for a stable estimate of the % defective.

Process Z is the point on a standard normal distribution such that the area to the right of that point is equal to the average proportion of defective units.  The higher the process Z the better the process performance, and a process Z of 2 or more is ideal.  In this case the process Z is -.8073.  The negative process Z indicates that the process output falls outside the target.

From a practical standpoint, both the normal capability analysis and the binomial capability analysis tell us the same story – the process is completely incapable of meeting the minimum requirement of 40% conversion rate which is equivalent to the requirement of a maximum of 60% failure to convert.

Another real-world example of one-sided limits involves confirmed orders per agent per hour.  The target here is 4 orders per agent per hour.  Data was collected for 50 days and the resulting mean orders per agent per hour was 2.292.

The underlying data is discrete and quantitative.  The resulting statistic of confirmed orders per agent per hour can take on any value and a fractional result makes sense, so confirmed orders per hour per agent were treated as continuous data.

The data for 50 days was tested for normality with the following results:

The null hypothesis is no difference between the data and a normal distribution.  The alternative hypothesis is that there is a difference between the data and a normal distribution.  The calculated p value of .596 is greater than the alpha decision value of .05 for 95% confidence, so we fail to reject the null hypothesis and conclude that the data are normally distributed.

I also created I-MR control charts to check the stability of this process.  The resulting control charts show that the process mean has two points out of control.  Point 13 indicates 9 points in a row on the same side of the centerline.  Point 31 indicates 4 out of 5 points more than one standard deviation from the centerline.  The moving range chart is stable.

The question now is what to use for specification limits or boundaries if we wish to conduct a normal capability analysis.  If we use four as an upper spec limit, and zero as a lower boundary, we will determine the capability of the process to stay below four confirmed orders per agent per hour. This is the opposite of what is expected of the process.

If we enter four as a lower specification limit, we are faced with the problem of what to use as an upper specification limit.  Minitab requires an upper specification or boundary and a lower specification or boundary before the analysis can be completed.  For the purposes of forcing Minitab to perform the analysis, I entered a lower spec limit of four and an upper spec limit of ten.  Here is the resulting output:

The results are the same as the previous analysis for conversion rate.  We have a negative Cpk of -1.25, and visually we can see that the entire output of the process falls below the requirement of four orders per agent per hour.  The minor lack of stability indicated on the individual value control chart is of lesser concern because the process is clearly not capable of meeting the requirement.