[G_R__E_G]

I'm hoping some of you math geniuses can help me with a project I'm working on. We now have a program here that importers/distributors of retail products help to pay for the eventual recycling of the packaging materials they put into the marketplace. Various types of materials have different prices per ton. They leave it up to individual companies to determine their own methodology for determinging these numbers as long as we can defend our methodology. If we can't defend it they force us to pay based upon retail sales which tends to not work out in our favour. What we did was weigh all of the packaging materials from a sample set of products in each of our product categories. The 2000 or so products we report on are divided into approximately 30 categories so we are not just looking at a random group - the products are similar. When doing the calculations we took a sample group of a least 5% of the items from each category as it was felt that this would give a reasonable statistical representation of each category. The government agency we report to is "okay" with what we're doing but they are asking how we determined that a 5% sample size is big enough. Is there a mathematical calculation that I can do to give the accuracy of my sample? Or is there a backwards calculation I can do to determine how big of sample I need to hit a particular accuracy?

I'll give an example category as this might help:

Portable audio has 57 skus. We weighed 3 of them and found the average weight in grams for 5 materials in these 3 items. Overall for the year we sold a total of approx. 40,000 of these 57 skus and therefore multiplied the averages by 40K to come up with overall weight.

Any help is much appreciated (and helps justify my being on the forums at work )

[helene_t]

The standard error (SE) of the mean is SD/sqrt(n) where SD is the sample standard deviation and n is the sample size. Now you sample from a finite population, i.e. the standard error applies only to the 54 non-sampled skus. You can adjust for that by dividing the square of the SE by the fraction of non-sampled skus:

SE = sqrt((N-n)*SD^2/(n*N))

where N is the total number of skus.

Actually if sales figues per sku are available the adjustment factor should be the sales mass of the non-sampled sku divided by the sales mass of the sampled skus.

Also if you have gross items weights available (those must be much cheaper to assess than the material contents) you can apply the formula to material content per kilogram rather than material content per item. If there is correlation between gross weight and material content this reduces the sample standard deviation.

Conventionally, the uncertainty margins are set to plus/minus 2*SE. Now if the government agency is satisfied with an accurate estimate of your total liabilities (the liabilities per material may be of minor importance), you first computer SEs per material in monetary (liability) terms and then compute the SE of the gross liability by

sqrt(SE1^2+SE2^2+SE3^2......)
where SE1, SE2 etc are the standard error of your liability from material 1, material 2 etc.

So basically the required sample size depends on the standard deviations and on each product category's contribution to your total liabilities. The total number of skus is of minor importance - other things being equal, a product category divided into 1000 skus require appr. the same sample size as one divied into 100 skus. Thus your 5% rule is not rational. A rule like "5 skus per category" is better, but you may take larger sample size when
- the SD is large. The theory of "optimal stratification" says that the sample size per category must be proportional to the SD
- the contribution from the particular category to your gross liabilites is large.

[pdmunro]

A couple of examples.

http://www.measuring..._continuous.htm
http://www.isixsigma...nt/c000709a.asp