## What is Operating Characteristic (OC) Curve?

The Operating Characteristic (OC) curve is a graph used in acceptance sampling. It describes how well an acceptance plan distinguishes between good and bad lots. A curve is related to a specific plan, which is a combination of sample size and acceptance level.

Table of Contents

Here, the x-axis shows the percentage of defectives and the y-axis shows the probability of acceptance of the entire lot.

## Parameters of OC curve

**Producer’s Risk (a)****Consumer’s Risk (b)****Acceptable Quality Level (AQL)****Lot Tolerance Percentage Defective (LTPD)**

### Producer’s Risk (a)

Refers to the probability that a sampling plan would reject a good lot (non-defective) of products, which forces the producer to replace the rejected lot. It is also called a Type I error.

### Consumer’s Risk (b)

Refers to the probability that the consumer would accept a bad lo, which contains a large number of defects. Once accepted, the consumer would find it hard to get the bad lot replaced by the producer. It is also called a Type II error.

### Acceptable Quality Level (AQL)

Refers to the maximum percentage of defects that consumers are willing to accept. AQL is considered the satisfactory quality level and is also called Acceptance Quality Limit.

### Lot Tolerance Percentage Defective (LTPD)

Refers to the percentage of defects that is higher than AQL and is therefore not accepted by the consumers. LTPD has a low probability of acceptance. This is also known as the Limiting Quality Level (LQL). The LQL is expressed as a proportion of defectives, while the LTPD is expressed as a percentage.

Figure shows an OC curve for a sample of 50 items (n) taken from a batch of 2000 (N) and using a critical acceptance number (c) of 2. The critical acceptance number (c) of 2 means that the batch will be accepted if there are two or fewer defectives in the sample. From the OC curve, you can see that there is about a 23% probability of accepting a batch that contains 8% of defective items.

## Objectives of OC Curve

The OC Curve refers to a graph of attributes of a sampling plan considered during project management. It displays the percentage of lots or batches that are expected to be acceptable under the specified sampling plan and for the specified process quality.

- The main objectives of the OC curve are:
- To help in the selection of sampling plans.
- To help in the selection of plans, which are effective in reducing risks.
- To help in lowering the cost of inspection

## Types of OC Curve

**There are three types of OC curves:**

### Type A Curve

Gives the probability of acceptance of an individual lot coming from finite production. In other words, this curve gives the probability of acceptance of a defective fraction as a function of a finite lot. For example, in a lot of 200 items, 0.5 or 1% of items are defective.

### Type B Curve

Gives the probability of acceptance of lots coming from a continuous process. In other words, it gives the probability of acceptance of a lot as a function of the continuous product quality. These curves are based on the assumption that the size of the lot is infinite.

### Type C Curve

Gives the long-run percentage of a product accepted during the sampling phase.

## Plotting the Oc Curve

**There are two ways to construct an OC curve:**

- Binomial equation
- Poisson distribution
- The binomial equation is:

n = Number of items sampled (called trials)

p = Probability that an x (defect) will occur at any one trial

Pa= Probability of exactly x results in n trials

When the sample size (n) is large and the percent defective (p) is small, the Poisson distribution can be used as an approximation of the binomial formula. This is useful because binomial equations can become quite complex, and because cumulative Poisson tables are readily available.

Now, let’s consider an example where you will calculate the probability of acceptance in a single sampling plan and then construct an OC curve using this probability.

An organization has submitted a lot of goods having a size of N for quality inspection. The inspector has taken a sample of 90 items with acceptance number 2 and the lot fraction defective as 0.01. Determine the probability of acceptance of the lot by the inspector.

Here, n = 90, c = 2, and p = 0.01

The formula to calculate the probability of acceptance of the lot is as follows:

If you input the different values of p in equation (1), then you will get different values of probability of acceptance Pa, which are shown in Table:

Fraction Defective, p | Probability of Acceptance, Pa |
---|---|

0.005 | 0.989371 |

0.01 | 0.938056 |

0.02 | 0.731175 |

0.03 | 0.491012 |

0.04 | 0.296970 |

0.05 | 0.166431 |

0.06 | 0.087984 |

0.07 | 0.044387 |

0.08 | 0.021537 |

0.09 | 0.010106 |

Therefore, the OC curve distinguishes between the accepted level and rejected level of fraction defective in a lot. For example, if 100 lots from a process that manufactures 1% defective products are submitted to the above sampling plan, then 94 lots will be accepted while 6 lots will be rejected.