P chart

P chart
See also
Np chart Control chart CUSUM chart Quality loss function Overall equipment effectiveness Attribute control chart Sequential sampling Pareto chart Six sigma

P chart, also known as a fraction nonconforming control chart, is a graphical tool which was developed in industry to help with interpreting and reducing sources of variability in manufacturing processes. The definition of fraction nonconforming (defective) (p) is usually exemplified by the ratio of the number of nonconforming items in the population to the total number of items in that population. Investigated item may have one or more quality characteristics that are examined simultaneously. If at least one of the attributes does not conform to standard, the item is classified as nonconforming .

Genesis

In the 1920s, W.A. Shewhart developed the concept of statistical control chart for the purpose of improving the reliability of telephone transmission systems. The conception arose from observating the operators. Shewhart noticed that they were overreacting and making inappropriate changes in settings as a response to indicator variations that were simply random. Those decisions were wasteful and inducted more variation in the process, making the system less stable. Shewhart's theory of variation provides that quality is inversely proportional to variability. It also states that understanding the variability of some indicators may show the operator when and how reduce it .

At that time, the concept of using the two-point moving range for measuring the dispersion of a set of individual values didn't occurred yet. Shewhart faced the problem how to create a process behavior chart for individual values based on counts. Then he decided to use theoretical limits based on a probability model. Therefore, he could estimate both the central line and the three-sigma distance with only one location statistic.

Design of the P chart

There are three parameters of fraction nonconforming control chart that must be specified: the sample size, the frequency of sampling, and the width of the control limits. Preferably, we should have some general guidance for selecting those parameters.

Formulas for the points on the Chart

Let's suppose we have k samples, each of n_i size. Let D_i stand for the number of nonconforming units in the i^th sample. The i^th proportion p_i is calculated as the following ratio:

$p_{i}={\frac {D_{i}}{n_{i}}}$.

P Chart Center Line

In the P Charts procedure, the center line proportion may be implement directly. It can also be estimated from a series of samples. If it is estimated from the samples we use the formula for the centerline proportion:

${\bar {p}}={\frac {\sum _{i=1}^{k}D_{i}}{\sum _{i=1}^{k}n_{i}}}$.

You can reduce this formula if all the samples are the same size.:

${\bar {p}}={\frac {\sum _{i=1}^{k}D_{i}}{kn}}={\frac {\sum _{i=1}^{k}p_{i}}{k}}$

P Chart limits

To calculate the lower and upper control limits for the P chart use these formulas:

$LCL={\bar {p}}-m{\sqrt {\frac {{\bar {p}}(1-{\bar {p}})}{n_{i}}}}$: $UCL={\bar {p}}+m{\sqrt {\frac {{\bar {p}}(1-{\bar {p}})}{n_{i}}}}$

where m stands for multiplier (usually set to 3) chosen to control the probability of false alarms (out-of-control signals when the process is under control).

There are two steps that one should take into concideration to avoid potential pitfalls:

• Ensure that there are enough observations taken for each sample,
• Be wary of differences in the number of observations from each sample.

Examples of P chart

• A P chart is often used in manufacturing to track the number of defects in a sample over time. For example, a manufacturing company may use a P chart to track the number of defects in a batch of parts over a given period. The company can then use the chart to identify any trends in the level of defects and take action to improve quality.
• Another example of a P chart is its use in healthcare to track the number of patients who do not receive the recommended care based on clinical guidelines. The chart can be used to track the number of patients who do not receive the recommended care over a given period of time, allowing healthcare providers to identify any trends and take action to improve care.
• Finally, a P chart can also be used in the service industry to track customer satisfaction. The chart can be used to track the number of customers who are not satisfied with the service they received over a given period of time, allowing the service provider to identify any trends and take action to improve customer satisfaction.

Advantages of P chart

The P chart has several advantages for use in industry:

• It is a simple control chart to interpret and provides an easy way to track the fraction of nonconforming items in a population.
• It can be used to compare different populations of items with one another and to determine whether an improvement has been made.
• It provides an objective measure of how well a process is performing and helps identify areas that need to be improved.
• It can be used to evaluate the effectiveness of corrective action taken to reduce the fraction of nonconforming items.
• It allows for the use of statistical process control to monitor quality performance over time.

Limitations of P chart

P charts are an effective tool for measuring and monitoring product quality, however, there are certain limitations to using them. These include:

• P charts can only be used to monitor a process when the items in the population are independent of each other. This means that if the items are related to each other, the P chart may not be able to accurately measure the quality of the process.
• P charts can only be used to measure the fraction of nonconforming items in a population. It cannot be used to measure the degree of nonconformance, as this requires a different type of chart.
• P charts may not be able to accurately measure the quality of a process if the population size fluctuates. This is because the chart is based on a fixed population size, so any changes in the population size will affect the accuracy of the chart.
• P charts are most effective when the population size is large. For smaller population sizes, the chart is less accurate, as there is less data available to make accurate predictions.
• Finally, P charts can only be used to measure the quality of a process over a certain period of time. If the process is changing over time, the P chart may not be able to accurately measure the quality of the process.

Other approaches related to P chart

P chart is one of the most commonly used graphical tools for monitoring process variation, and there are several other approaches related to it:

• The C chart is a type of control chart used to monitor the number of nonconformities (or defects) per unit in a given set of data. It is used when the number of nonconformities is being counted for a given process.
• The U chart is a control chart used to monitor the number of nonconformities per unit of area or length. It is used when the nonconformities are being counted over a given area or length of a product or process.
• The NP chart is a type of control chart used to monitor the proportion of nonconforming items in a population. It is used when the proportion or fraction of nonconforming items in a population is being examined.
• The X chart is a control chart used to monitor the average value of a given process. It is used when the average value of a process is being examined.

In summary, the P chart is one of the most commonly used graphical tools for monitoring process variation, and there are several other approaches related to it such as the C chart, U chart, NP chart, and X chart. These charts are used to monitor different aspects of a process such as the number of nonconformities, the proportion of nonconforming items, and the average value of a process.

References

• Duclos A., Voirin N. (2010), The p-control chart: a tool for care improvement, “International Journal for Quality in Health Care”, Vol. 22, Nr 5
• Hou C. D., Shao Y. E., Haung S. (2013), A Combined MLE and Generalized P Chart Approach to Estimate the Change Point of a Multinomial Process, "Applied Mathematics & Information Sciences", Vol. 7, Nr 4
• Montgomery D.C. (2012), Introduction To Statistical Quality Control, 7th Edition, Wiley, Arizona State University
• Nist/Sematech e-Handbook of Statistical Methods (2012), Proportions Control Charts “Engineering Statistics handbook”
• Pandurangan A. (2011), Fuzzy Multinomial Control Chart With Variable Sample Size, "International Journal of Engineering Science and Technology", Vol. 3, Nr 9
• Ryan T.P. (2011), Statistical Methods for Quality Improvement, Wiley, Smyrna
• Shah S., Shridhar P., Gohil D. (2010), Control chart: A statistical process control tool in pharmacy, "Asian Journal of Pharmaceutics", Vol. 4, Nr 3
• Wheeler D. J., (2011), What About p-Charts?, Quality Digest

 

Author: Anna Kasprzyk