# How can Acceptance Number and Reject Number be larger than Sample Size on Z1.4 Table?

#### Katana_Clarity

##### Starting to get Involved
I am pretty new to Sampling and I am trying to figure things out. Please oblige if my question is dumb but I am assuming Sample Size means the items from the lot that will be inspected. Acceptance Number means the least number of items acceptable within the Sample to make the lot acceptable and reject number means the most number of items that can be non conforming to reject the lot as a whole.

Well, if acceptance number and rejectance number are both derived from the sample, how can sample size be smaller than acceptance and rejection number as seen in the image for Sample Size Code D, AQL of 65. Sample Size comes to 8 and acceptance number is 10, how is that possible?

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
First, the acceptance number is the maximum number of defects in the sample for the lot to be acceptable.

Second - welcome to negotiated statistics. The tables are not strictly based on stats. If they were the lot size would be irrelevant. (Except when the lot size is extremely small). In the example you show, the sample size is impossible so you would have to increase the sample size to a possible one. But in all honesty, an. AQL of 65 is simply absurd in this day and age.

#### Steve Prevette

##### Deming Disciple
Staff member
Super Moderator
My interpretation of this situation is based upon Acheson Duncan's Quality Control and Industrial Statistics (also https://www.amazon.com/Quality-Control-Industrial-Statistics-Fifth/dp/0256035350 ) which reprints this table as part of the chapter on MIL-STD-105D on page 229. Basically the numbers are all similar down the diagonals. Note 10 and 11 run down and to the left.

Along that diagonal you would need 11 defects to reject. On that row with a sample size of 3 (and more importantly a Code Letter of B) you would never end up rejecting such a sample. Which means you really need to increase the Lot or Bunch size by accumulating sequential items into a bigger "lot". Note that the code letter comes from Table 1 (page 228 in Duncan). By the way, the table accept and reject values are counts of defects, not defects per 100 items. It is the AQL which is based upon "maximum percent defective (or the maximum number of defects per 100 units)" (page 222 of Duncan). So, yes, you can have more defects than units. But again, who really wants there to be 400 defects per 100 units, as pointed out by Bev.

One does need to realize that these plans (MIL STD 105 based) ASSUME that you are making Lot after Lot after Lot, and with time you end up gaining sufficient samples to judge the production by.

My copy of Duncan all 1,123 pages, is quite worn to pieces after 33 years of use.

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#### Katana_Clarity

##### Starting to get Involved
First, the acceptance number is the maximum number of defects in the sample for the lot to be acceptable.

Second - welcome to negotiated statistics. The tables are not strictly based on stats. If they were the lot size would be irrelevant. (Except when the lot size is extremely small). In the example you show, the sample size is impossible so you would have to increase the sample size to a possible one. But in all honesty, an. AQL of 65 is simply absurd in this day and age.
How can I increase the sample size, when my lot size is 115 which puts me in Sample Size Code Letter D?

But in all honesty, an. AQL of 65 is simply absurd in this day and age.
Isn't AQL of 65 here saying that I will tolerate a process average of 65% defective in 100 items?

#### Katana_Clarity

##### Starting to get Involved
My interpretation of this situation is based upon Acheson Duncan's Quality Control and Industrial Statistics (also https://www.amazon.com/Quality-Control-Industrial-Statistics-Fifth/dp/0256035350 ) which reprints this table as part of the chapter on MIL-STD-105D on page 229. Basically the numbers are all similar down the diagonals. Note 10 and 11 run down and to the left.

Along that diagonal you would need 11 defects to reject. On that row with a sample size of 3 (and more importantly a Code Letter of B) you would never end up rejecting such a sample. Which means you really need to increase the Lot or Bunch size by accumulating sequential items into a bigger "lot". Note that the code letter comes from Table 1 (page 228 in Duncan). By the way, the table accept and reject values are counts of defects, not defects per 100 items. It is the AQL which is based upon "maximum percent defective (or the maximum number of defects per 100 units)" (page 222 of Duncan). So, yes, you can have more defects than units. But again, who really wants there to be 400 defects per 100 units, as pointed out by Bev.

One does need to realize that these plans (MIL STD 105 based) ASSUME that you are making Lot after Lot after Lot, and with time you end up gaining sufficient samples to judge the production by.

My copy of Duncan all 1,123 pages, is quite worn to pieces after 33 years of use.
Which means you really need to increase the Lot or Bunch size by accumulating sequential items into a bigger "lot". Note that the code letter comes from Table 1 (page 228 in Duncan).
So in this case, I should change the lot size to find Ac and Re numbers lower than sample number?

By the way, the table accepts and reject values are counts of defects, not defects per 100 items. Can you expand on this?

#### Steve Prevette

##### Deming Disciple
Staff member
Super Moderator
There are a lot of good instructional videos and power points out there on MIL STD 105E. The interesting thing is - the MIL STD was free, but ANSI took it, reprinted it almost word for word, and charges you for the privilege of having an ANSI sampling plan. Oh well.

Here is one thing out there: How_to-read_the_ANSI_tables_for_single_sampling.pptx (qualityinspection.org)

For the question of "the table accepts and reject values are counts of defects, not defects per 100 items " you need to understand what Table 1 does (gives you your sample letter) which is based upon the AQL which is a percent or Rate. But once I am in table 2, those numbers represent how many defects can I detect (the count of defects) until I need to reject the result.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
How can I increase the sample size, when my lot size is 115 which puts me in Sample Size Code Letter D?
The answer is that you cannot increase your sample size and so you must inspect 100%.

Isn't AQL of 65 here saying that I will tolerate a process average of 65% defective in 100 items?
Yes it is. My comment is that who in their right mind would think that a 65% defective process or lot is acceptable? This means that you would actually think it’s OK to ship a lot that is 65% defective. You couldn’t clock the time it would take for me to disqualify you as a supplier. Most companies are looking for defect rates in the <1% or even ppm range (where sample inspection can’t help you).