## a) Proportion Defective (p charts) and the Number of Defects (c charts)

*Proportion Defective (p-chart)*

P-Chart is the most flexible control chart and it is mostly used to monitor the proportion or the percentage of the goods or items defective in a provided sample (Finch, 2011). In order to use p-chart, sample items are divided into two different groups namely good and bad while also referring as defective or non-defective, and conforming or non-conforming to the specifications (Gygi et al., 2005). Many students get stuck with such dissertation writing task and are compelled to buy dissertation. In p-chart, the computation of centerline is carried out as the average proportion defective in the population, which is acquired by taking a number of samples of observations at random as well as calculating the average value of amongst all the samples (Stout, 2001). In p-chart, for constructing the upper and lower control limits, the following formula is used:

In the above formula,

= the sample proportion defective

z= standard normal variable

= the standard deviation of the average proportion defective (Kazmier& Kazmier, 2009)

This method can be applied when a production manager at the manufacturing plant of tires found numbers of defective tires. If students are asked to write thesis on such topics, they can also buy coursework online. In order to monitor the defective prodcution of tires, p-chart can be calculated using random samples of 20 observations. If the observation sample is as mentioned above, then the calucaltion will be carried out as follows:

*Number of Defects (c-chart)*

The C-chart is also a control chart for the defects, as it is used to track the number of defects per unit sampled. The use of C-chart becomes important when product is with complex properties or has many defects (Sharma, 2011). In this view, it is probable to create control charts either for total number of defects or for the defect density (May& Spanos, 2006). Thus, when using c-chart, it is assumed that the existing of defects in samples of constant size is showed properly through the Poisson distribution as shown below:

In the above, x= total number of defects

C= centerline

This method can be applied when production manager of automobile found 516 defects in 26 tires during the evaluation of the quality, but not all defects can be classified as unacceptable. Therefore, other attributes would be considered in the unit and–chart would be used in the situation as shown below:

This determines the centerline for the c chart therefore; the upper and lower limit can be calculated as follows:

- For calculating the upper and lower control limits and R, the following steps are carried out:

Since the independent sample results are already recorded for each subgroup, therefore subgroup average will be calculated for each subgroup using the following formula:

Upon calculating the average for n= 3

Upon calculating the average for n= 4

Upon calculating the average for n= 5

Now, R will be calculated as follows:

Now, in order to calculate the upper and lower control limits, average range or RBar will be calculated, which is as follows:

Now, in order to calculate Xdbar, the following will be calculated:

Now, calculating upper and lower control limits for n= 3

Similarly, calculating upper and lower control limit for n= 4

Now, calculating upper and lower control limit for n= 5

Now calculating the upper and lower control limit for n= 3

Calculating the upper and lower control limit for n= 4

Calculating the upper and lower control limit for n= 5

- Using the information from Tables 2 and 3 and the chart limits previously calculated, Control charts for this case have been created namely R charts and X bar charts.

*Interpreting R Charts*

The above R chart shows four distinct values plotted while one value appears more than 25 percent of the time. The too often repeated value shows that there is inadequate resolution of the measurement and it has adverse effects on the control limits. In the beginning, the process is not in the control because two consecutive control measurements exceed the +2sd control limit. However, at the end of the process, R chart shows that control measurement is on the line of mean.

*Interpreting X Bar Charts*

In the above X bar chart, it can be seen that single control limit exceeds the +3sd control limit depicting violation of Westgard rule 1_{3s}. However, X bar does not reject the process at all, as the upper control limit of the process is shown on the control line.