How can I see Kurtosis in a box plot? How do I know whether a distribution is leptokurtic or platykurtic by only having the box plot?

I find boxplots highly misleading for assessing tails, so I would not do this. In particular, the “obvious” way to assess kurtosis is to consider how many “outlier” points there are, but it means nothing to have, say, $200$ outliers on the plot. If there are $200$ outliers in a sample of $500$, maybe it’s fair to consider the tails heavy. If there are $200$ outliers in a sample of $5000$, perhaps the tails are not so heavy. However, the boxplot gives no sense of what proportion of points are extreme, just the count.
