How many beans?
What do you expect the contents of this can of beans to weigh?
Don't be silly - we're still in the EU!
Before we joined the Common Market/EEC/EU every can of beans had to contain at lest the declared weight. Manufactures would aim for 16oz but declare 15.5oz.
Consumers knew that if they tipped the beans out and they weighed less than 15.5oz they had been cheated.
Trading Standards knew that if they tipped the beans out and they weighed less than 15.5oz the consumer had been cheated
Manufacturers knew that if they checked a sample from each production batch they could be sure they were not cheating anyone.
Life was simple wasn't it?
Then we joined the Common Market and the introduction of this symbol.
With this came:
fill temperature allowances
and action limitsThe situation now is:
Consumers have no way of knowing whether they are being cheated or not
Trading Standards have a monumental task if they want to check whether a manufacturer is complying with weights and measures regulations.
Manufacturers have to maintain extensive and complicated records for Trading Standards to audit.
If you are interested in the details then peruse the example given below in the guide to the legislation. (This is for sweets and may be a little more complicated than for canned beans)
An example on setting target quantities and action limits
A packer produces 200 g bags of sweets; the variation due to the packing process is dependent on the size of sweet.
The process variation, stated as a standard deviation, is either 4 g, 5 g or 6 g.
The packing line produces 4,000 bags an hour and the packer monitors the average
weight of the product by taking samples of five bags from the line every 30 minutes.
The average of each sample is calculated and used to determine when action is needed.
What should be the lowest target quantity the packer should aim for, and what should
the minimum action limit (1 in 1,000) for the sample mean be, for each of the three processes?
Lowest Target Quantity
1.1. Process variation
1.1.1. The average range or standard deviation can be used to consider this parameter.
The standard deviation (SD) is more robust, as it is based on all the data available. In either case, the following assumes that the distribution of the contents of the packages is ‘normal’. If this assumption is incorrect the variation from normality also has to be considered.
1.1.2. In the scenario the variability of the filling process is given as a standard deviation. In order to ensure all three packers’ rules are met, the highest of the following has to be used to address this variability:
Qn T1+2s T2+3.72s
Qn is the nominal (labelled) quantity
s is the standard deviation of the filling process,
T1 is Qn – TNE
T2 is Qn –2 TNE
TNE is the tolerable negative error for the nominal quantity
WELMEC document 6.4 at paragraph E2.5 deals with this matter as above. It is treated differently in the Packers Code at paragraph C15, but the result is the same.
For the various process variations the results are:-
variation Qn T1+2s T2+3.72s Largest
4 200 199 196.88 200
5 200 201 200.6 201
6 200 203 204.32 204.32
As can be seen from this table that which of the packers’ rules is critical depends on the variability of the filling process. In this exercise the critical rule is:-
For s=4 g the first rule, ensuring the average is correct,
For s=5 g the second rule, ensuring that the number of packages having a greater negative error than the tolerable negative error isacceptable, and
For s=6 g the third rule, ensuring that there are no packages with a negative error greater than twice the tolerable negative error produced.
1.1.3. The above ‘largest’ quantities would be acceptable as a target quantity if there were no other issues that needed addressing. The scenario gave information about the sampling plan the packer is using and so the adequacy of this needs to be considered.
1.2. Sampling Allowance
1.2.1. Generally the packer’s sampling plan is considered to be equivalent to the
reference test if at least 50 items are sample d during the time that 10,000 packs
are filled (with a minimum time of 1 hour and a maximum time of 1 day or shift).
See the Packers Code, paragraph C22, or WELMEC 6.4, paragraph E.3
1.2.2. The scenario is that the packing line produces 4,000 bags an hour and
the packer monitors the average weight of the product by taking samples of five bags
from the line every 30 minutes. The average of each sample is calculated and
used to determine when action is needed.
From this information, the time taken to fill 10,000 packs, referred to as the
Production Period, is equal to 10,000/4,000 hr = 2.5 hr.
The number of samples (of size 5, i.e. n = 5) taken during this period, sampling
every half hour, is k = 2.5/0.5 = 5.
Therefore the number of items sampled during the
production period (time taken to produce 10,000 packs) is kn = 5x5 = 25. As this is less than 50 a sampling
allowance is needed to ensure that the packers system is as efficient as the
reference test in detecting non-compliance.
1.2.3. The appropriate allowance, which is used to enhance the target quantities
established in 1.1.2 above, is obtained by looking up the tables in the Packers
Code, Table C1, or WELMEC 6.4, Table E.3.
The allowances are based on 3 control system:-
using Shewhart Control with Action
Limit only (1 in 1,000),
-using Shewhart Control with Warning Limit (1 in 40) & Action Limit (1 in 1,000), and
-using Cusum control with h=5, f=0.5 (as per BS 5703)
The exercise indicates that only an Action Limit is used which is referred to as
Procedure A in the tables.
1.2.4. Looking at n=5 and k=5 for procedure A gives a sampling allowance
factor, z = 0.20. This factor is multiplied with the standard deviation of the
packing process to produce an allowance, which is added to the targets
determined in 1.1.2 above.
1.2.5. The minimum target quantity taking into account process variability and
SD Qn T1+2s T2+3.72s Largest zs=0.20s Min Qt
4 200 199 196.88 200 0.8 200.8
5 200 201 200.6 201 1 202
6 200 203 204.32 204.32 1.2 205.52
1.3. Other Allowances
1.3.1. The scenario does not give any indications that other allowances are
necessary but other issues that need to be considered include:-
-a ‘wandering average’,
-storage allowance, particularly for desiccating products,
-tare variability, where the quantity determination assumes a constant tare
-temperature, if the product is filled hot or cold and the volume changes when
determined at 200C
Theses are considered in the Packers Code in paragraphs C21 to C25 and
WELMEC 6.4 in paragraphs E.5.1 to E.5.4.
2. Action Limits
1.4. The scenario asks for the ‘minimum action limit (1 in 1,000) for the
sample mean’. The distribution of the sample mean (of samples of size n) is
related to the distribution of the individual items (the process variability).
1.5. If the process variability (standard deviation) is s
and the number of items in a sample is n
then the standard deviation of the distribution of the means, sometimes referred to as the standard error of the means, is s/n
1.6. The action limit, with a chance of 1 in 1,000 of exceeding, as with the
normal distribution comes at three times the standard error away from the target
quantity. Only the lower action limit is needed for legal metrology, although
upper limits may be set, for example for safety reasons (aerosols), or economic
reasons (duty on alcohol) –but the corresponding limit must be no nearer the
target quantity than the lower one.
So for legal metrology the Action Limit should be no lower than Qt -3 s/n
(If there was a warning limit of 1 in 40 being used this would be at 2 times the
An example is given in the Packers’ Code at paragraph C34, it is also covered in
WELMEC 6.4 at paragraph E.7.2.
Using the data from the exercise this gives:
Process Variation,s Minimum Target Quantity (g) Minimum Action Limit forthe Mean (g)
4 g 200.8 195.4
5 g 202.0 195.3
6 g 205.5 197.5
So that's all clear now!