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Sample Size Considerations

When improving a product or process, the question arises "How many prototypes should we build to prove product/process viability?" Being able to quantitatively answer this question can yield a significant improvement to both the eligibility and the value of potential SR&ED claims.

Some industries have their own standards and procedures to dictate the number of test parts that need to be manufactured before a design change can be accepted as reliable and within tolerances. When available, this is an excellent metric for SR&ED.

When an industry has no available standards, then the answer can be found in basic statistics. The equations required depend on which of 4 typical industrial situations are relevant:

  1. an average value of a product property of interest must be determined with confidence
  2. after a product / process change, one wishes to prove that a property of interest has actually improved from a previous value
  3. one wishes to prove that a given percentage of their parts will be within specified tolerances
  4. one wishes to prove that the variance on a property has reduced

A thorough discussion of these situations, including examples, can be found in our document Sample Size Calculations for SR&ED, which will soon be available for download from this site. However for convenience, a useful Rule of Thumb equation for determining if specs are within tolerances (case 3) is shown below:

N = 1  ·  (1 + r)  ·  χ2α,4   +   1
4 (1 − r) 2

where:

N is the number of samples / prototypes needed for the test,

χ2α,4 is the upper critical value of the chi-square distribution with probability α and 4 degrees of freedom (available from online tables),

α is the significance level of the statistical test (conversely, (1 − α)% is how confident you are in the test results),

r is the proportion of the population that is (1 − α)% likely to fall between the smallest and the largest results from the N samples.

This equation works well when little is known about the variance of the property of interest. Sample values for N have been calculated in the following chart based on common r and α values:

αr = 0.75r = 0.80r = 0.85r = 0.90r = 0.95r = 0.99
0.101418243776388
0.051722304693473
0.0124304164130661

Implementation Example:

Company B has just made changes to their process for making silicon wafers. They want to be 95% sure that 90% of the wafers they produce will fall within specified thickness limits.

They let α = 0.05, and r=0.90. From the table above they see that they need 46 samples.

They produce 46 sample wafers from their new process, and measure the thickness of each sample as they proceed. They make note of the smallest and largest values of thickness, and compare these to their lower and upper tolerance limits.

If their measured values fall within the tolerance limits, they are 95% certain that at least 90% of their product will meet specification. If some of their measured values fall outside their tolerance limits, they are 95% certain that less than 90% of their product will meet specifications, and perhaps more modifications are required to their process.