# Np chart

Np chart, also known as a number nonconforming control chart, is a tool used in statistical quality control used to monitor the amount of of nonconforming units in a sample[1].

## Usage of Np chart

Control charts are powerful tools for constructing and sustaining the statistical control of a producing process. Classic control charts are audited by taking random, n-sized samples from the running procedure for a h times. After inspecting the items in the sample and recording measurements of interest, computed data (regarding the appropriate time order) are presented on a graphical visualization. To compare the results with predetermined values of control limits two limit controls are necessary: an upper control limit (UCL) and a lower control limit (LCL). If the plotted calculated statistic marks between them it is considered that the process is in the in-control state. In another case that may be a signal that some process shift has appeared and a rectifying actions should be taken immediately to eliminate the cause of the discrepancy and bring this process back into control[2].

The fraction nonconforming is usually understood as the ratio of the number of nonconforming items in a population to the total number of items in that population. The items usually have a number of quality characteristics that are examined at the same time by the inspector. If the item does not accommodate to standard on one or more of these attributes, it is classified as nonconforming. An np control chart make it possible to look at variation in yes/no type attributes data. There are only two possible results: either the item is defective or it is not defective[3].

The np Control Chart is an adaptation of the p chart but it bases on the number nonconforming rather than the fraction nonconforming. It is helpful in situations when it is easier for someone to interpret process performance in terms of concrete numbers of units in place of the somehow more abstract proportion[4].

The np-chart is used with data compiled in subgroups that are the same size. Np charts present how the process which is measured by the number of nonconforming items, fluctuates over time. Usage of np-charts is convenient for determining if the process is stable and foreseeable, likewise to observe the effects of process improvement approaches[5].

## Designing Np chart

Designing a control chart means making the choices of n, h, UCL and LCL to be used during the process monitoring[6].

### Gathering the data

Gather the data:

1. Select the subgroup size (n) – this size must be constant.
2. Choose the frequency with which you will collect the data.
3. Select the number of subgroups (k).
4. Check out every item in the subgroup and mark it as defective or non-defective.
5. Establish np (number of defective items) for each subgroup.

### Mapping out the data

Map out the data:

1. Choose the scales for the chart.
2. Mark the values of the nps for every subgroup on the chart.
3. Draw straight lines between consecutive points.

### Computing the CL, UCL and LCL

Compute the center line, upper control line and lower control line:

1. Calculate the center line (CL). $$n\bar{p}=\frac{\sum np}{k}$$
2. Draw the CL on the control chart as a solid line.
3. Compute the control limits for the np chart. Upper control limit: $$UCL=n\bar{p}+3\sqrt{n\bar{p}(1-(n\bar{p}/n))}$$ Lower control limit: $$LCL=n\bar{p}-3\sqrt{n\bar{p}(1-(n\bar{p}/n))}$$.
4. Mark the control limits on the chart as dashed lines.

### Interpreting the results

Use those tests for statistical control:

• Points beyond the control limits,
• Length of runs test,
• Number of runs test.

## Footnotes

1. Hashemian S.M., Noorossana R., Keyvandarian A., Shekary A.M. 2016, pp. 769
2. Kooli I., Limamb M. 2015, pp. 483
3. Montgomery D.C. 2012, pp. 316-330
4. Montgomery D.C. 2012, pp. 316-330
5. Montgomery D.C. 2012, pp. 316-330
6. McNeese B. 2009

## References

• Bashiri, M., Amiri, A., Asgari, A., Doroudyan, M. H. (2013), Multi-objective efficient design of np control chart using data envelopment analysis, "International Journal of Engineering", Vol. 26, Nr 6
• Chong, Z. L., Khoo, M. B., Castagliola, P. (2014), Synthetic double sampling np control chart for attributes, "Computers & Industrial Engineering", Vol. 75
• Faraz, A., Heuchenne, C., Saniga, E. (2017), The np Chart with Guaranteed In‐control Average Run Lengths, "Quality and Reliability Engineering International', Vol. 33 Issue: 5, pp. 1057– 1066
• Hashemian S.M., Noorossana R., Keyvandarian A., Shekary A.M. (2016), Performance of adaptive np-chart with estimated parameter, "International Journal of Quality & Reliability Management", Vol. 33 Issue: 6, pp. 769
• Ho, L. L., Costa, A. F. B. (2011), Monitoring a wandering mean with an np chart, "Production", Vol. 21, Nr 2
• Kooli I., Limamb M. (2015), Economic design of attribute np control charts using a variable sampling policy "Applied Stochastic Models in Business and Industry", Vol. 31, pp. 483
• McNeese B. (2009), np Control Charts, "SPC for Excel"
• Montgomery D.C. (2012), Introduction To Statistical Quality Control, 7th Edition, Wiley, Arizona State University, pp. 316 – 330
• Morais M.C. (2016), An ARL-Unbiased np-Chart, "Economic Quality Control", Vol. 31 Issue: 1, pp. 11–21
• Ye ZS, Xie M. (2015), Stochastic modelling and analysis of degradation for highly reliable products, "Applied Stochastic Models in Business and Industry", Vol. 31 Issue: 1, pp. 16-36

Author: Anna Kasprzyk