If you bought a mobile phone, you would expect that it had been fully tested before it went on sale. However, if you bought a carton of milk or a bar of chocolate you would not expect that it had been individually
tested, although it might have been inspected in some way. Chemical or biological analysis of items often requires modifying them by, for example, adding some reagent and observing the effect, or measuring
pH. So, instead of putting each item through a range of separate tests, random samples are assessed. If the samples are in line with expectations then all the other items are likely to be so as well.
Testing items that vary discreetly
If 10% of the items produced by a manufacturing process are defective and 90% are without defect, then the chance that an item chosen at random is not defective is 90%. However if two items are chosen then the chance that no item is defective is 0.90 2 = 0.81 (or 81%) and the chance that one or both are defective is therefore 1-0.81 (or 19%). If five items are chosen then the chance that no item is defective is 0.90 5 = 0.59 (or 59%) and the chance that one or more are defective is therefore 1 − 0.59 (or 41%). In order to have a better than 50:50 chance of detecting a defective item from this production line a sample of 7 (or more)
would need to be selected. (1 − 0.907 = 0.52 or 52%).
If, however, 99.99% of the products are perfect then to have a better than 50:50 chance of detecting at least one defective item, almost 7000 samples would need to be checked! (1 − 0.99996932 = 50.004%).
Testing items that vary continuously
In reality many manufacturing processes produce items that vary slightly in some way (dimensions, mass, resistance etc.) but are generally within certain limits. Repeated physical measurements of sample items tend
to follow what is called a normal distribution about the average or mean value. The amount of variation in measurements could be due to
1. variations in the manufacturing process(es)
2. variations in the measurement process.
The spread of values can be represented graphically as a histogram (if the values are given, for example, to the nearest millimetre).
If more and more measurements are taken with finer intervals the shape of the histogram approaches that of the normal distribution curve.