A p-chart is used to monitor the fraction of defectives in the output of a process.

Range chart
Mean chart
P-chart
C-Chart

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Answer (Detailed Solution Below)
Proportions Control Charts
What is used to monitor the proportion defective in the output of a process?
For which of the following would a P chart be used?
Which of the following chart is monitoring how the fraction defective fluctuate during the production process?
What chart control the number of defects per unit of output?

Answer (Detailed Solution Below)

Option 3 : P-chart

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10 Questions 30 Marks 10 Mins

Explanation:

The P-chart is also known as the “fraction defectives” or “fraction non-conforming chart,” because it is used to monitor and control the fraction produced in a process that is defective or non-conforming.

Control charts are statistical tool, showing whether a process is in control or not.

Two types of process data:

Variable is a continuous data, things we can measure; Example includes length, weight, time, temperature, diameter, etc.
Attribute is a discrete data, things we count; Examples include number or percent defective items in a lot, number of defects per item etc.

Types of Control Charts:

Variable charts are meant for variable type of data. X bar and R Chart, X bar and sigma chart, chart for the individual units

Attribute charts are meant for attribute type of data. p chart, np chart, c chart, u chart, U chart

$A p-chart is used to monitor the fraction of defectives in the output of a process.$

Control charts for the variable type of data (X bar and R charts)

In the x bar chart, the sample means are plotted in order to control the mean value of a variable
In R chart, the sample ranges are plotted in order to control the variability of a variable

Control charts for Attribute type data (p, c, u charts)

p-charts calculates the percent defective in sample; p-charts are used when observations can be placed in two categories such as yes or no, good or bad, pass or fail etc.
c-charts counts the number of defects in an item; c-charts are used only when the number of occurrences per unit of measure can be counted such as number of scratches, cracks etc.
u-chart counts the number of defects per sample; The u chart is used when it is not possible to have a sample size of a fixed size

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6. Process or Product Monitoring and Control
6.3. Univariate and Multivariate Control Charts
6.3.3. What are Attributes Control Charts?

Proportions Control Charts

$p$ is the fraction defective in a lot or population The proportion or fraction nonconforming (defective) in a population is defined as the ratio of the number of nonconforming items in the population to the total number of items in that population. The item under consideration may have one or more quality characteristics that are inspected simultaneously. If at least one of the characteristics does not conform to standard, the item is classified as nonconforming.

The fraction or proportion can be expressed as a decimal, or, when multiplied by 100, as a percent. The underlying statistical principles for a control chart for proportion nonconforming are based on the binomial distribution.

Let us suppose that the production process operates in a stable manner, such that the probability that a given unit will not conform to specifications is $p$. Furthermore, we assume that successive units produced are independent. Under these conditions, each unit that is produced is a realization of a Bernoulli random variable with parameter $p$. If a random sample of $n$ units of product is selected and if $D$ is the number of units that are nonconforming, then $D$ follows a binomial distribution with parameters $n$ and $p$ so that

The binomial distribution model for number of defectives in a sample $$ P(D=x) = \left( \begin{array}{c} n \\ x \end{array} \right) p^x (1-p)^{n-x} \, , \,\,\,\,\,\, x = 0, \, 1, \, \ldots, \, n \, , $$ where $$ \left( \begin{array}{c} n \\ x \end{array} \right) $$ denotes a "combination", referring to $n$ things taken $x$ at a time. The mean of $D$ is $np$ and the variance is $np(1-p)$. The sample proportion nonconforming is the ratio of the number of nonconforming units in the sample, $D$, to the sample size $n$, $$ \hat{p} = \frac{D}{n} \, .$$

The mean and variance of this estimator are $$ \mu = p $$ and $$ \sigma_{\hat{p}}^2 = \frac{p(1-p)}{n} \, . $$ This background is sufficient to develop the control chart for proportion or fraction nonconforming. The chart is called the $p$-chart. $p$ control charts for lot proportion defective If the true fraction conforming $p$ is known (or a standard value is given), then the center line and control limits of the fraction nonconforming control chart is $$ \begin{eqnarray} UCL & = & p + 3\sqrt{\frac{p(1-p)}{n}} \\ \mbox{Center Line} & = & p \\ LCL & = & p - 3\sqrt{\frac{p(1-p)}{n}} \, . \end{eqnarray} $$ When the process fraction (proportion) $p$ is not known, it must be estimated from the available data. This is accomplished by selecting $m$ preliminary samples, each of size $n$. If there are $D_i$ defectives in sample $i$, the fraction nonconforming in sample $i$ is $$ \hat{p}_i = \frac{D_i}{n} \, , \,\,\,\,\, i = 1, \, 2, \, \ldots, \, m \, , $$ and the average of these individuals sample fractions is $$ \bar{p} = \frac{\sum_{i=1}^m D_i}{mn} = \frac{\sum_{i=1}^m \hat{p}_i}{m} \, . $$ The $\bar{p}$ is used instead of $p$ in the control chart setup. Example of a $p$-chart A numerical example will now be given to illustrate the above mentioned principles. The location of chips on a wafer is measured on 30 wafers.

On each wafer 50 chips are measured and a defective is defined whenever a misregistration, in terms of horizontal and/or vertical distances from the center, is recorded. The results are

Sample	Fraction	Sample	Fraction	Sample	Fraction
Number	Defectives	Number	Defectives	Number	Defectives

1	0.24	11	0.10	21	0.40
2	0.30	12	0.12	22	0.36
3	0.16	13	0.34	23	0.48
4	0.20	14	0.24	24	0.30
5	0.08	15	0.44	25	0.18
6	0.14	16	0.16	26	0.24
7	0.32	17	0.20	27	0.14
8	0.18	18	0.10	28	0.26
9	0.28	19	0.26	29	0.18
10	0.20	20	0.22	30	0.12

The reader can download the data as a text file.

Sample proportions control chart The corresponding control chart is given below:

$A p-chart is used to monitor the fraction of defectives in the output of a process.$

What is used to monitor the proportion defective in the output of a process?

A c-chart reflects the number of defects per item. A p-chart is used to monitor the proportion defective in the output of a process.

For which of the following would a P chart be used?

A p-chart is used to record the proportion of defective units in a sample. A c-chart is used to record the number of defects in a sample.

Which of the following chart is monitoring how the fraction defective fluctuate during the production process?

p control charts are used to monitor the variation in the fraction of defective items in a group of items.

What chart control the number of defects per unit of output?

The u-chart is a quality control chart used to monitor the total count of defects per unit in different samples of size n; it assumes that units can have more than a single defect. The y-axis shows the number of defects per single unit while the x-axis shows the sample group.