Welcome. In this tutorial we will be constructing control
charts for proportions, otherwise known as the p-chart.
The p-chart is used to monitor attributes and applies to categorical or qualitative
variables. They are generally used to analyze the proportions
of non-conforming or defective items in a process.
The centreline for the p-chart is p-bar which is found by taking the total of defective
or non-conforming items and dividing it by the total number of items sampled.
The upper control limit, UCL, is p-bar + 3 times square root of p-bar times q-bar divided
by n-bar. The lower control limit, LCL, is similar with
a minus sign. q-bar is 1 minus p-bar.
That is, proportion of non-defective items. And n-bar is the average of the sample sizes.
For our first example, we consider these process data consisting of the number of defective
items in a sample of 200 items collected every day for 15 days.
We want to calculate the control limits and determine if the process is in statistical
control. The total number of defective items over the
15 days is 156, while the total number of items sampled is 200 times 15 which gives
3,000. We calculate p-bar by dividing the total number
of defective items by the total sample size, and that gives .052.
Consequently, q-bar is 1 minus .052 which gives 0.948.
The average of the sample sizes, n-bar, is 200, since every sample size is 200.
Now that we have the 3 required values, we can now calculate UCL and LCL.
So, plugging the values into the UCL formula we have .099.
And for LCL, we have .005. To ensure better accuracy, you want to take
the square root immediately after evaluating these values.
And also note that this 3 multiplies the result after taking the square root.
Don’t take it as a cube root. Now that we have the control limits, let us
now calculate the proportions and construct the p-chart.
The proportions are found by dividing the number of defectives by sample size.
So for Day 1, the proportion is 11/200 which gives .055.
We do the same for the other days so we have all information required for the chart.
And here is the p-chart. Everything looks okay. No point beyond the
control limits and there is no unwanted pattern.
We can say confidently that the process is in statistical control.
Let’s look at a second example where the sample sizes are not all the same.
The sum of the nonconformers or defectives is 119
While the total sample size is 2249. That gives us an overall proportion or p-bar
of .053. And that makes q-bar .947. The average sample size is total sample size
divided by number of samples which gives 224.9.
We now have the values required to calculate the limits.
The upper control limit is .098 And the lower control limit is .008.
Now that we have the control limits, we can go ahead and calculate the proportions.
The proportion of non-conformers for day 1 is 10/224 which gives .045.
We do the same for days 2 to 10. Now let’s look at the p-chart.
We can see here that the proportion of defectives for day 9 is higher than the upper limit.
Therefore the process is out of control. We will recommend checking for the cause of
this unusually high value and addressing it. And that’s p-chart.
Thank you!