Selecting Representative Samples

Dr. Wayne A. Taylor

For the OC curve, AQL, and LTPD of a sampling plan to be valid, one must obtain representative samples of the lot.  Failure to select representative samples can seriously compromise the sampling plan’s protection.  Consider the analogy of dealing cards to obtain a fair hand.  If the deck is thoroughly shuffled, the cards can be dealt off the top of the deck.  However, if the deck is not shuffled, care must be taken to select each card at random.  Likewise, for defects that occur randomly throughout the lot, it does not matter how the sample is selected.  But when defects occur in runs or clusters, it is important to select a representative sample.  For purpose of example, suppose we are inspecting a printing operation using the single sampling plan with sample size 20 and accept number zero.

One way to obtain a representative sample is to select a random sample.  This involves randomly selecting the first sample and then randomly selecting each additional sample from the units remaining in the lot.  Randomly selecting a spot in the lot and then grabbing 20 units is not a random sample.  While the first unit is selected randomly, the remaining units are not.  Suppose the printer went out of alignment and printed 100 consecutive off-center labels before correcting itself.  Grabbing a handful of samples results in one chance of finding the run of defects.  Selecting a random sample provides 20 chances of finding the run.  In practice, random samples may be difficult to select. So what are the alternatives?

One alternative is stratified sampling.  Suppose the product is placed in 10 tote pans.  Then we can select two units from each tote pan, one from each end.  This ensures the 20 units are spread out across the lot giving us 20 chances of finding the run of defects.

Another alternative is periodic sampling.  Suppose the lot consists of 2000 units.  Then selecting every hundredth unit spreads the units out across the lot providing a representative sample.  Care must be taken when using periodic sampling on machines with multiple stations.  One should not select every fortieth tablet from a 40-well tableting machine.  All samples will come from the same well.  However if the period of the samples and the number of stations do not have a common divisor,  each station is sampled an equal number of times.

One method of selecting samples that does not ensure a representative sample is carton sampling.  Suppose that the 20 units must be selected after the product has been packed in cartons.  To avoid opening too many cartons, four cartons are selected and five units taken from each.  Again, if the defects occur randomly, any sample will do.  However, if the defects occur in runs, the sampling plan’s protection may be seriously compromised.  Suppose a run of 100 defects occurs and is packaged into a single carton.  Then one has four chances of selecting this carton instead of the 20 chances provided by a random sample.

Many confuse stratified sampling with carton sampling.  Both involve dividing the lot into subgroups.  In stratified sampling, samples are taken from each subgroup.  In carton sampling, no samples are taken from certain subgroups (cartons). Stratified sampling should be encouraged, while carton sampling should be avoided whenever possible by selecting the samples before they are placed in the cartons.

Carton sampling takes many guises as shown by the following examples.  To evaluate how representative your sample is, you should determine the percentage of the total period or lot represented by the sample.

Example 1: A lot is contained in two 24″ tall bins. Half the required units are selected from the top of each bin.  The samples are selected from the top inch of the bins.  Think of each inch as representing a carton.  Then the sample only represents 2 inches (cartons) of the total 48 inches (cartons).  This provides 4% coverage of the lot.

Example 2: Product is assembled, placed inside a pouch and then placed in a tote pan to be sealed.  Generally, within one minute, it is removed from the tote and sealed.  A sample of 32 units is required every hour.  Four times during the hour, eight units are selected from the tote pan awaiting sealing.  Since the tote contains less than one minute’s production, the 32 units all come from four minutes (cartons) of the 60 minute period.  This provides approximately 8% coverage of the period.  An alternative is to have the sealer operator take a periodic sample by setting aside a unit every two minutes.  This provides 100% coverage.

If the defects are likely to occur in runs or clusters, nonrepresentative samples can seriously compromise the protection provided by the sampling plan.  To determine if you are at risk, determine the coverage of the lot provided by your current scheme and when low, ask if defects are likely to occur in runs or clusters.

Appeared in FDC Control, Food Drug & Cosmetic Division ASQC, No. 105, February 1995, p. 4-5