Two plating lines and the shape of defect counts
A metal-finishing plant monitors microscopic pits per 10 cm2 on plated panels from two lines. In the same week, Line A produced n = 120 sampled panels with mean 7.4 pits, median 6.9, SD 2.1, skewness +1.35, and excess kurtosis +3.8. Line B produced n = 120 panels with mean 7.1, median 7.0, SD 2.0, skewness +0.12, and excess kurtosis -0.15. Two Line A panels measured 19 and 21 pits; a quality engineer wants to delete them as outliers before analysis. Another engineer proposes reporting mean +/- 1 SD and using a two-sample t-test on means to decide which line is better. The supplier contract pays a bonus when a "typical panel" has fewer than 7 pits, and the plant must decide which summary and test to present at a supplier review next week. No measurement protocol changed between lines.
1. Interpret the skewness and excess kurtosis for each line, and explain what they imply about the mean, median, and spread. Should the two extreme panels be deleted? Recommend the summary statistics for the supplier review.
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Skewness is the standardized third moment; +1.35 on Line A indicates a long right tail, so the mean (7.4) is pulled above the median (6.9). Excess kurtosis +3.8 means heavier tails and a sharper peak than a normal distribution, consistent with the 19- and 21-pit panels. Line B is nearly symmetric (skewness +0.12) with near-normal tail weight (excess kurtosis -0.15), so its mean and median nearly coincide.
Deleting the two panels is not justified merely because they are large. First check measurement error, plating-bath contamination, or data-entry problems. If no assignable cause is found, retain them and report a sensitivity analysis. For the review, report median and IQR (or median and 5th-95th percentiles) for both lines; also report mean and SD for comparability, but label Line A's mean as tail-sensitive. Do not rely on mean +/- 1 SD as a "typical range" for a skewed, heavy-tailed line.
2. Is the proposed two-sample t-test on means a sound basis for deciding which line is better? Explain the role of sample size and distribution shape, and propose a more defensible analysis.
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A two-sample t-test compares means, not "typical panels." With n = 120 per line, the central limit theorem makes the sampling distribution of each mean approximately normal even with skewness, so the test is not automatically invalid. However, Line A's heavy tail inflates its variance and can make its mean unstable; the usual test also assumes independent samples and, in its pooled form, similar variances.
A more defensible primary analysis is a bootstrap or permutation test for the difference in medians, or a Mann-Whitney test if the question is stochastic ordering. Report the effect size with a confidence interval, not just a p-value. If the business decision depends on total pits, the mean is relevant; if it depends on a typical panel, the median is relevant. Prespecify the primary endpoint and do not choose the test after seeing which one gives significance. Also check whether the two extreme panels are isolated special causes before pooling.
3. The contract phrase "typical panel has fewer than 7 pits" must be converted into a measurable criterion. Which statistic and decision rule would you propose, and how does the skewness of Line A affect the answer?
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Convert "typical" into a prespecified statistic. I would propose median pits per panel < 7, with a 95% confidence interval for the median, and a secondary criterion of the proportion of panels with fewer than 7 pits, also with a confidence interval. For Line A, the median is 6.9, so it meets the median rule narrowly; its mean of 7.4 does not, which shows why the mean is a poor contract statistic under right skew. For Line B, median 7.0 fails a strict "<7" rule.
The decision rule should include sampling uncertainty: require the upper confidence bound for the median to be below 7, not just the point estimate. Because Line A has heavy tails, also cap the proportion above a high threshold, for example 15 pits, so a supplier cannot meet the median while shipping occasional severe defects. This makes the criterion resistant to the few extreme panels without ignoring them.