**The Lot Plot is the graphical tool developed by D. Shainin allowing to accept or not a batch of production by sampling.**

## Introduction

Lot Plot is a statistical method of controlling ´ acceptance by sampling. Developed by Dorian Shainin in the years 1940, she uses graphical analysis of sampled variable data to determine whether a batch of potentially defective parts must be accepted or discarded for control 100 %.

The development by Walter A. Shewhart of the control chart method demonstrated the application of statistical techniques to the manufacturing industry and illustrated the effectiveness of the presentation and graphical analysis. Shainin integrated these concepts into the development of the Lot Plot .

In 1946, Shainin was able to demonstrate to the Bureau of Aeronautics of the U. S. Navy that the Lot Plot method was more efficient than a 100 % control. Following this demonstration, the Navy agreed to standardize the Lot Plot method. The latter was soon erected as a standard in many industries.

On the advice of his friend and mentor Joseph M. Juran, Shainin turned to the world of the Council. In 1952, Shainin joined Rath & Strong, Inc., a management consulting firm based in Lexington, Massachusetts, as executive vice president. Position

This technique applies only if the variable is measurable quantitatively.

## 1 – Carry out the levy

Sampling is always the same : We take a sample of 50 parts that we divide into 10 groups of 5.

## 2-Calculate the average

The average and the extent of each of the 10 subgroups are calculated.

## 3 – Calculate the characteristics of the histogram chart

In the first step, the average and the extended are calculated.

For class width, Shainin suggests taking in theory a quarter of the extent of the first subgroup. However, in practice it is advisable to have between 7 and 16 for a refined representation.

## 4 – Build the histogram

For each new measure, a cross is put in the corresponding box. This allows the histogram to be built automatically.

If the histogram appears as non-normal, you must stop the measurements and try to understand why (change of batch, tooling…).

## 5 – Calculating acceptance limits

Thanks to the elements calculated previously, the limits are calculated at 3 σ. The formulas are as follows :

**Upper Lot Limit :**X_{cross}+ 3r_{cross}/d_{2}**Lower Lot Limit :**X_{cross}-3r_{cross}/d_{2}

D_{2} is the coefficient of Duncan. This one is always equal to 2.326, since we still have 5 pieces per subgroup for 50 parts in total.

## 6-Acceptance criteria

The criteria and the reaction to have are based on the type of distribution that we get. Shainin has identified 11 types of criteria :

Distribution type | Description | Reaction | Illustration |
---|---|---|---|

Normal | All data is in the specifications. | Accept | |

The calculated limits are in the specifications. | Accept | ||

ULL or LLL is out of specification. Calculer le nombre de valeurs en dehors des spécifications. | Determine if rejection or not | ||

Not normal | The limits are in the specifications. Les résultats sont faussés. | Investigate and / or redo | |

One or both limits are out of specification. | Batch set aside and sorted | ||

Binomial | Values are "misplaced". We can investigate to see if it is an erroneous measure. Otherwise, it will be necessary to redo a control and evaluate the quality of the batch. | Investigate and / or redo |

## Example

We want to analyze a batch of plastic bottle. We have measured the length of these, and we have the following data in hundredth. The specifications in the plan are from 10 to 15.

- Calculating the global average = 12.93
- Calculating the global range = 2.9
- ULL = 16.67
- LLL = 9.19

**Conclusion :**

First of all, we have an aberrant measure at 20 hundredths. After investigations it is simply a measurement error. We decide to redo the measure, and we find 14 hundredths. We remake the new calculations and find :

- Calculating the global average = 13.06
- Calculating the range = 3.5
- ULL = 13.06 + 3.5/2,326 = 17.58
- LLL = 13.06-3.5/2,326 = 8.55

Nevertheless, a distribution is identified **Binomial** And the ULL and LLL values are outside of the specifications.

## Source

E. G. Schilling, D. V. Neubauer (2009) -Acceptance sampling in quality control

W. G. Ireson (1954) – Some practical aspects of the batch pbatch Sampling Acceptance plan