Start by clarifying that comments per daily active user is a discrete count variable, then describe the expected distribution as right-skewed with a long tail, likely overdispersed. Explain why Poisson is a poor fit due to its equidispersion assumption and inability to capture excess zeros and heavy tails, and suggest alternatives like negative binomial or zero-inflated models.
Pro tip: Mention that in practice, you'd validate the distribution by plotting the empirical histogram and computing the variance-to-mean ratio; if it's substantially greater than 1, Poisson is inappropriate. Also, note that user heterogeneity and behavioral clustering often lead to overdispersion.
State that comments per daily active user is a count variable, hence discrete, not continuous. It can only take non-negative integer values.
Explain that the distribution is typically right-skewed: most users comment zero or few times, while a small fraction of highly active users comment many times, creating a long right tail.
Poisson assumes mean equals variance (equidispersion) and independence of events. In reality, comments per user often show overdispersion (variance > mean) due to user heterogeneity and clustering, making Poisson a poor fit.
Suggest negative binomial (accounts for overdispersion), zero-inflated Poisson or negative binomial (if excess zeros), or a mixture model to capture user-level heterogeneity.
Emphasize checking the variance-to-mean ratio and plotting the empirical distribution. The choice of distribution affects modeling, inference, and product decisions (e.g., identifying power users).
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I knew the central limit theorem angle but got a bit tangled explaining when the skew persists.
Explain that the bootstrap distribution of the sample mean is the sampling distribution of the mean under resampling with replacement, and its shape is governed by the Central Limit Theorem. Clarify that with 10,000 users, the distribution will be approximately normal regardless of the population distribution, but the approximation improves with larger sample size and is exact if the population is normal.
Pro tip: Emphasize that the bootstrap distribution estimates the sampling distribution of the mean, and its normality depends on the original sample size (10,000), not the number of bootstrap resamples (100,000). The latter only reduces Monte Carlo error.
Describe that each bootstrap sample is drawn with replacement from the original 10,000 users, and the mean is computed. The distribution of these 100,000 means is the bootstrap distribution.
State that the bootstrap distribution is centered at the original sample mean and its standard error is approximately the sample standard deviation divided by the square root of 10,000.
Explain that by the Central Limit Theorem, the distribution of the sample mean is approximately normal for large sample sizes. With n=10,000, it will be very close to normal even if the population is skewed.
Highlight that normality depends on the sample size (10,000) and the population distribution. Larger sample sizes and less skewed populations yield better normal approximations.
Note that 100,000 resamples only reduce Monte Carlo error, making the bootstrap distribution smoother, but do not affect its underlying shape or normality.
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Start by clarifying that the bootstrap distribution of the mean approximates the sampling distribution, so its width reflects the standard error. Then explain that doubling the sample size reduces the standard error by a factor of √2, so the width of the bootstrap distribution decreases by about 29%. Emphasize that this is a direct consequence of the Central Limit Theorem and the √n relationship.
Pro tip: Mention that the bootstrap distribution's spread estimates the sampling variability, not the population variability, so it shrinks with larger n. Also note that while the width decreases, the shape becomes more normal, which is important for inference.
Explain that the bootstrap distribution of the mean is generated by resampling with replacement from the original sample, and its width (e.g., standard deviation or interquartile range) estimates the standard error of the mean.
State that the standard error of the mean is σ/√n, where σ is the population standard deviation. Thus, the width of the bootstrap distribution is proportional to 1/√n.
When n doubles from 10,000 to 20,000, the standard error changes by a factor of 1/√2 ≈ 0.707. So the width decreases by about 29.3% (or is multiplied by 0.707).
Discuss that larger samples provide more precise estimates, reducing variability. Mention that the bootstrap distribution becomes narrower and more normal, which improves the accuracy of confidence intervals and hypothesis tests.
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The mode being zero is obvious once you think about it but I fumbled the p95 discrete-tie question.
Start by defining each statistic and its properties, then compare their stability and decision-making relevance in the context of a product metric. Finally, address the specific challenge of computing p95 for a discrete count variable, discussing methods like interpolation or using the nearest-rank method.
Pro tip: Emphasize that the choice of statistic depends on the metric's distribution and the business question—mean for totals, median for typical user experience, p95 for tail risks—and that for discrete counts, p95 should be interpreted as the value below which 95% of observations fall, often using the nearest-rank method to avoid fractional counts.
Briefly define mean, median, mode, and p95, and explain how each summarizes the data differently. Highlight that mean is sensitive to outliers, median is robust, mode is for categorical or discrete peaks, and p95 captures the tail.
Discuss which statistic is most stable across samples. Typically, the median is more stable than the mean for skewed data, while p95 can be volatile due to tail sensitivity. Mode can be unstable if the distribution is flat or multimodal.
Explain which statistic matters most for decisions. For product metrics, the mean might drive revenue, but the median reflects typical user experience, and p95 highlights worst-case scenarios for performance or satisfaction. The choice depends on the business goal.
Address that p95 for discrete counts can be non-integer. Use methods like nearest-rank (the smallest value with cumulative frequency ≥95%) or interpolation, but be cautious as interpolation may produce impossible values. Discuss the trade-offs and recommend nearest-rank for interpretability.
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Truncation versus rounding bias I covered fine.
Start by distinguishing between integer counts and floating-point means, then discuss the numerical issues that arise during storage and aggregation at scale. Structure your answer around precision loss, overflow, and distributed aggregation challenges, and propose solutions like using higher-precision types, compensated summation, and careful data modeling.
Pro tip: Mention that while integer counts are exact, converting to floating-point for means can introduce rounding errors; using Kahan summation or pairwise summation in distributed settings shows depth. Also, highlight that storing raw counts and computing means on the fly avoids precision loss from pre-aggregated floats.
Explain that per-user counts are integers (exact) but means are real numbers (approximate), leading to potential precision loss when converting or aggregating.
Discuss how storing means as floats can accumulate rounding errors, especially with many users or high counts, and how integer overflow can occur if counts are stored in fixed-width types.
Describe issues in distributed aggregation: summing floats across nodes can lead to non-associativity and precision loss, and merging pre-computed means requires weighted averages to avoid bias.
Propose solutions: use 64-bit integers for counts, compute means from raw counts, employ compensated summation (Kahan) or pairwise summation, and consider using higher-precision floats (e.g., double) or fixed-point arithmetic.
Discuss trade-offs between precision and performance, and recommend storing raw counts and computing means on demand, or using streaming algorithms for approximate means if exactness is not critical.
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The SE formula is straightforward but the discussion around robust estimators versus variance reduction is where it got interesting.
Start by defining the standard error of the sample mean under overdispersion, clarifying that it is larger than the naive i.i.d. formula due to variance inflation. Then, for each robust method (trimmed means, Winsorization, covariate-based variance reduction), explain when it is appropriate, focusing on trade-offs between bias, efficiency, and interpretability in the context of A/B testing at Meta.
Pro tip: Emphasize that in large-scale A/B tests, the choice of method should be driven by the metric's distribution and business impact; for example, trimmed means are often preferred for heavy-tailed revenue metrics to improve power without introducing excessive bias, but always validate with simulation.
Write the formula for the standard error of the sample mean when per-user variance exceeds the mean, such as SE = sqrt( (σ^2 + φμ) / n ) or using the delta method for ratio metrics. Explain that overdispersion inflates the variance and thus the standard error.
Describe trimmed means as removing a fixed percentage of extreme values from both tails. Use when the metric has heavy tails and you want a robust measure of central tendency that reduces the influence of outliers, but be aware of potential bias if the distribution is asymmetric.
Describe Winsorization as replacing extreme values with a specified percentile. Use when you want to retain all data points but limit the impact of outliers, often leading to a more efficient estimator than trimming when the tails are not too extreme.
Describe using pre-experiment covariates (e.g., CUPED) to reduce variance by adjusting for baseline differences. Use when you have relevant pre-period data and want to increase power without altering the metric definition, especially for metrics with high variance.
Discuss trade-offs: trimmed means and Winsorization change the estimand and may introduce bias, while covariate adjustment preserves the estimand but requires valid covariates. Choose based on metric distribution, business goals, and whether the treatment effect is homogeneous.
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