The sketch part was fine, mean to the right of median, long right tail, 95th percentile way out there.
First, sketch a right-skewed distribution and correctly place the mean, median, and 95th percentile. Then explain that the sampling distribution of the sample mean will be approximately normal (by the Central Limit Theorem) with the same mean but smaller variance, and discuss how the median and 95th percentile behave differently.
Pro tip: Emphasize that the mean of the sampling distribution equals the population mean, but the median and 95th percentile of the sampling distribution are not equal to their population counterparts; they converge to the population mean as sample size increases. This shows deep understanding of the CLT and its implications.
Draw a right-skewed distribution of daily comment counts. Mark the mean to the right of the median, and the 95th percentile far to the right, indicating a long tail.
Describe that when repeatedly sampling groups of users and computing the average, the distribution of these averages will be approximately normal (by the CLT) with the same mean as the population but reduced variance (standard error = σ/√n).
State that the mean of the sampling distribution equals the population mean. The median and 95th percentile of the sampling distribution will be closer to the mean than in the population, and as sample size increases, they converge to the population mean.
Relate this to A/B testing: when comparing average comment counts between groups, the sampling distribution of the mean is normal, enabling standard hypothesis tests. However, metrics like the 95th percentile require different methods (e.g., bootstrapping) due to non-normality.
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