At the printing company I work for, we have different materials that we print on; part numbers for those materials are assigned based on the unique supplier, material type, and material widths (so the same supplier could send us two different widths of the same material type, and they'd get different part numbers). With each part number, our supplier sends us a statistical summary of their quality control data - how many samples were taken (n), the average thickness of the samples, and the standard deviation. For the most common parts, we might get a few hundred different averages, Ns, and standard deviations. I'm trying to group those summaries together. Since the sample size is so high, I'm fine averaging the average thicknesses for each part number; the numbers are closely grouped enough that it shouldn't be a significant difference. My question is, how do I summarize the standard deviation? The quality tests are standardized and the instruments used to measure the thickness are constantly calibrated, so I have no reason to think that the measurements would be inconsistent from one shipment to the next.

Question

At the printing company I work for, we have different materials that we print on; part numbers for those materials are assigned based on the unique supplier, material type, and material widths (so the same supplier could send us two different widths of the same material type, and they&#039;d get different part numbers). With each part number, our supplier sends us a statistical summary of their quality control data - how many samples were taken (n), the average thickness of the samples, and the standard deviation. For the most common parts, we might get a few hundred different averages, Ns, and standard deviations. I&#039;m trying to group those summaries together. Since the sample size is so high, I&#039;m fine averaging the average thicknesses for each part number; the numbers are closely grouped enough that it shouldn&#039;t be a significant difference. My question is, how do I summarize the standard deviation? The quality tests are standardized and the instruments used to measure the thickness are constantly calibrated, so I have no reason to think that the measurements would be inconsistent from one shipment to the next.

delalbaef · Accepted Answer

The standard unbiased estimator for the underlying variance would be the pooled variance,       s    p    2   defined by the formula      s    p    2    =                    ∑                  i          =          1                k            (              n        i            −      1      )              s        i        2                            ∑                  i          =          1                k            (              n        i            −      1      )        .The pooled standard deviation would then be       s    p   (take the square root of the pooled variance). Here       n    i   is the sample size of the   ith sample, and       s    i   is the   ith sample standard deviation.Edit: I should also mention that the usual unbiased way to average the means (the method you described, simply averaging them, is unbiased, but is not the minimum variance unbiased estimator) is  μ  =                    ∑                  i          =          1                k                    n        i                    μ        i                            ∑                  i          =          1                k                    n        i            where       μ    i   is the   ith sample mean.

At the printing company I work for, we have different materials that we print on; part numbers for t

Answered question

Answer & Explanation

New Questions in College Statistics