## Sketching distributions and comparing mean, median, and mode This is a conceptual statistics question with two independent scenarios. For each, you sketch the **shape** of a distribution (no exact parameters needed) and reason about its summary statistics. Assume you have a large random sample in each case. ### Constraints & Assumptions - You are sketching **shape only** — relative peaks, skew, and tails matter; precise means and standard deviations do not. - Treat each scenario as drawn from a large random sample, so sampling noise is negligible and the sketch reflects the underlying population shape. - Standard real-world data issues (outliers, heavy tails, mixed subpopulations, censoring) may be present and are fair game to call out. ### Clarifying Questions to Ask - Are we sketching the **probability density / histogram shape**, or the empirical CDF? (Assume density/histogram unless told otherwise.) - Should the two subpopulations in Part A be weighted equally, or by their true population proportions? - For Part B, is "number of connections" capped or censored by the platform (e.g., a displayed "500+" ceiling)? - Should bot, spam, or inactive accounts be included in the Part B population, or filtered out first? ### Part A — Human heights 1. Sketch the distribution of **adult male heights in the US**. 2. Sketch the distribution of **adult female heights in the US**. 3. If you combine men and women into a **single** population, what does the height distribution look like? Explain why. ```hint Per-group shape Height is driven by many small additive genetic and environmental factors. The Central Limit Theorem intuition suggests what canonical single-peaked shape each single-sex group should look like. ``` ```hint Combining two groups Combining two single-sex distributions is not "adding" two normals into one normal — think of it as a **mixture**. Whether you see one peak or two depends on how far apart the two means are *relative to* the spread (separation in units of standard deviation). ``` #### What This Part Should Cover - Names the canonical shape for each single-sex group and the mechanism (many small additive factors → approximately bell-shaped), with appropriate hedging that "real data may have mild skew/tails." - Identifies the combined distribution as a **mixture** of two distributions with different means, not a single normal. - Reasons about *when* the mixture appears bimodal vs. a wide unimodal blob — i.e., dependence on mean separation relative to spread and on the mixing proportions. ### Part B — Number of connections on a professional social network Consider the distribution of the **number of connections (friends) per user** on a professional social network such as LinkedIn. 1. Sketch the distribution of the number of connections. 2. Will the **mean** be close to a typical user's value? Explain what the mean does and does not represent here. 3. Compare **mean vs. median vs. mode** for this distribution and justify the ordering. ```hint Shape Most users have relatively few connections while a small set (recruiters, influencers, power networkers) have very many. What does that asymmetry imply about skew and the right tail? ``` ```hint Ordering the three statistics For a right-skewed distribution, recall the standard inequality among mode, median, and mean — and tie the ordering to *which* statistic the long right tail pulls hardest. ``` #### What This Part Should Cover - Identifies the distribution as **right-skewed / heavy-tailed** with a long right tail, and explains the generative reason (preferential attachment / a few high-degree hubs). - Explains that the mean is inflated by tail users and is therefore **not** representative of a typical user; names the median (or percentiles) as the robust alternative. - States and *justifies* the ordering $\text{mode} < \text{median} < \text{mean}$ by linking each statistic to how the right tail affects it, rather than just asserting the rule. ### What a Strong Answer Covers These dimensions span both parts: - Connects **distribution shape to the right choice of summary statistic** — symmetric vs. skewed, single population vs. mixture — instead of treating mean/median/mode as interchangeable. - Reasons from a **generative mechanism** (additive small effects → bell shape; hub formation → heavy tail; two sexes → mixture) rather than memorized labels. - Surfaces real-world caveats where relevant (skew/heavy tails in heights, censoring/bots in connections) and notes how they would change the sketch or the chosen statistic. ### Follow-up Questions - In Part A, what condition on the two means and standard deviations would make the combined distribution clearly bimodal? Give the rough threshold in units of standard deviation. - In Part B, how would you communicate "typical" connection count to a non-technical stakeholder, and which statistic(s) would you report? - The platform displays connection count as "500+" above a ceiling. How does this censoring change your sketch, and how does it bias the mean vs. the median? - For the right-skewed connections data, what transformation would make the distribution easier to visualize and model, and why?