Statistical Fundamentals and Exploratory Analysis Questions

Core descriptive and exploratory statistical techniques used to summarize data, detect patterns, and generate testable hypotheses. Covers measures of central tendency and dispersion such as mean median and standard deviation, distributional assumptions, frequency and cross tabulation, visualization for exploration, cohorting and segmentation, identifying biases and data quality issues, and designing exploratory analyses to suggest causal hypotheses. Understand when to apply EDA to prepare data for formal tests and how to translate exploratory findings into confirmatory analyses. Candidates should demonstrate ability to summarize quantitative data, detect anomalies, and propose appropriate follow up hypothesis tests or experiments.

HardTechnical

109 practiced

Explain what a p-value represents and list five common misinterpretations. Discuss reproducibility concerns related to p-values and describe better practices for moving from exploratory findings to confirmatory claims (pre-registration, effect sizes, confidence intervals, replication).

Sample Answer

A p-value is the probability, under a specified null hypothesis and model assumptions, of observing data as extreme or more extreme than what was actually observed. It quantifies compatibility between the observed data and the null—smaller p-values mean less compatibility, not proof of a specific alternative.

Five common misinterpretations:1. “p < 0.05 means the result is practically important.” (Statistical significance ≠ effect size.)2. “p < 0.05 means the null is false with 95% probability.” (p-value is not P(H0|data).)3. “A large p-value proves the null.” (Failure to reject ≠ evidence for H0; may be low power.)4. “P-values are stable across replications.” (They vary widely with sampling variability.)5. “Multiple tests don’t affect p-values.” (Multiple comparisons inflate false positives unless adjusted.)

Reproducibility concerns:- p-hacking, selective reporting, and multiple testing produce many false positives.- Publication bias (only significant findings reported) and underpowered studies lead to inflated effect estimates and poor replication rates.- Reliance on a binary threshold (0.05) encourages chasing significance rather than robustness.

Better practices to move from exploratory to confirmatory claims:- Pre-registration: specify hypotheses, analysis plan, primary outcome, and sample size/power before seeing confirmatory data to prevent data-driven choices.- Emphasize effect sizes and uncertainty: report point estimates and their practical meaning, not just p-values.- Report confidence intervals: show range of plausible values and precision; CIs communicate both magnitude and uncertainty.- Replication: attempt independent replication with adequately powered samples; meta-analysis of replications yields more reliable estimates.- Adjust for multiplicity when applicable, use false discovery rate control for many tests, and complement frequentist results with Bayesian posterior probabilities when appropriate.

For business analyses, present effect sizes and CIs alongside p-values, transparently document analytic choices, and treat exploratory findings as hypotheses to be validated before making high-stakes decisions.

MediumTechnical

73 practiced

Describe a robust strategy to detect anomalies in an e-commerce daily revenue time series that exhibits weekly and monthly seasonality. Compare STL decomposition + sigma thresholds, z-scores on detrended residuals, rolling median absolute deviation, and unsupervised ML methods like isolation forest. Which would you choose for production monitoring and why?

Sample Answer

Clarify requirements and constraints:- Detect point and sustained anomalies in daily revenue with weekly and monthly seasonality.- Low false positives for business alerts, explainability, operational simplicity, and ability to run in SQL/BI pipelines.

High-level approach (robust production strategy):1. Preprocess: impute missing days, holiday flags, log-transform revenue to stabilize variance.2. Decompose seasonality: use STL (seasonal-trend-loess) with weekly and monthly periods (e.g., period=7 and a monthly component via seasonal window or multi-seasonal STL variant). Extract trend + seasonal + residual.3. Residual scoring: compute robust score = residual / rolling MAD (median absolute deviation) using a 30–90 day window to adapt to volatility.4. Thresholding & alerts: flag |score| > 3 (tunable) and require persistence (e.g., >=2 consecutive days or percent deviation > X) before paging. Include business rules for holidays/promotions.5. Monitoring pipeline: schedule daily job, store recent residuals and flags, surface in dashboard with explainability (trend and seasonal plots).

Comparison of methods:- STL + sigma thresholds: good for clear decomposition and explainability. Sigma thresholds on residuals are simple but sensitive to outliers and variance changes.- z-scores on detrended residuals: common, but z uses mean/std which is influenced by anomalies; not robust for heavy tails.- Rolling MAD (recommended for residuals): robust to outliers, adapts to changing scale, simple to compute in SQL/python; combining with persistence reduces noise.- Unsupervised ML (Isolation Forest, One-Class SVM): can capture complex patterns and cross-feature interactions (e.g., traffic, AOV), but less interpretable, requires feature engineering, retraining, and careful calibration to avoid false positives—higher operational cost.

Which to choose for production (Data Analyst context):- Primary: STL decomposition + residuals normalized by rolling MAD with persistence rules. This balances interpretability (can show seasonal/trend components), robustness to outliers, ease of implementation in ETL/BI, and low maintenance.- Secondary: augment with Isolation Forest as a supplement for multivariate anomalies (when adding features like sessions, conversion rate). Use it for investigative alerts, not primary paging.

Implementation notes / tuning:- Use log1p for skew, choose MAD window based on business cycle (e.g., 60 days), tune threshold by running historical backtests (precision/recall) and incorporate whitelist for known campaigns/holidays.- Add alert enrichment: % deviation, expected vs actual plot, and probable cause tags (holiday/promo) to speed triage.

This approach yields explainable, robust daily monitoring that an analyst can implement and maintain using SQL/Python and surface in dashboards.

MediumTechnical

63 practiced

You observe average order value differs between two advertising channels. Formally write the null and alternative hypotheses to test whether the difference is zero. Choose an appropriate statistical test, list its assumptions, describe how you would check them, and propose an alternative approach if assumptions are violated or sample sizes are highly imbalanced.

MediumTechnical

79 practiced

Given a dataset with duplicate customer IDs, inconsistent date formats, negative quantities, and extreme outliers in transaction amounts, describe how you would prioritize cleaning steps for conducting EDA. Provide concrete validation checks and explain how to make the cleaning reproducible and auditable.

Sample Answer

Priority approach (high-level):1. Define schema & business rules with stakeholders (unique key, allowed date formats, quantity >=0, realistic amount ranges).2. Run quick sanity checks to catch catastrophic issues (duplicates, parse errors, negative quantities).3. Normalize/correct formats (dates, types) so downstream checks work.4. Handle duplicates and referential integrity.5. Treat outliers after understanding business context.6. Document and make everything reproducible and auditable.

Concrete validation checks (examples):

- Duplicate customers per transaction (SQL):

sql

-- transactions with duplicate transaction_id
SELECT transaction_id, COUNT(*) FROM transactions GROUP BY transaction_id HAVING COUNT(*)>1;

- Date parsing and invalid dates (Python/pandas):

python

df['ts'] = pd.to_datetime(df['date'], errors='coerce')   # coerces invalid -> NaT
invalid_dates = df[df['ts'].isna()]

- Negative quantities:

sql

SELECT * FROM transactions WHERE quantity < 0;

- Outliers (IQR/Z-score):

python

q1,q3 = df['amount'].quantile([0.25,0.75])
iqr = q3-q1
outliers = df[(df['amount'] < q1 - 1.5*iqr) | (df['amount'] > q3 + 1.5*iqr)]

Reproducible & auditable process:- Encapsulate cleaning in version-controlled scripts/notebooks; parameterize (input path, thresholds).- Use data-testing tools (Great Expectations, dbt tests) to codify rules; run in CI before deployment.- Persist raw immutable snapshot and assign hashes/checksums; write cleaned outputs with metadata (timestamp, commit hash, applied rules).- Log every transformation in a changelog or provenance table (row counts before/after, reasons for row drops).- Produce a data dictionary and summary report (validation results, samples of corrected rows) for stakeholders.

Decision rules & trade-offs:- Prefer non-destructive edits (add cleaned columns) until business signs off.- Flag extreme outliers for manual review if they drive key metrics.- Keep thresholds configurable and record rationale in documentation.

MediumTechnical

71 practiced

Write a PostgreSQL query to compute the 95th percentile (per-user) of session_duration_seconds from sessions(session_id, user_id, duration_seconds int). Include only users with at least 5 sessions. Use SQL constructs available in PostgreSQL and explain how percentile_cont behaves with even/odd counts, ties, and NULL values.

Sample Answer

Approach: compute the per-user 95th percentile using PostgreSQL’s percentile_cont(0.95) WITHIN GROUP aggregate, grouping by user_id and filtering to users with at least 5 sessions via HAVING.

sql

SELECT
  user_id,
  COUNT(*) AS session_count,
  percentile_cont(0.95) WITHIN GROUP (ORDER BY duration_seconds) AS p95_duration_seconds
FROM sessions
WHERE duration_seconds IS NOT NULL
GROUP BY user_id
HAVING COUNT(*) >= 5
ORDER BY user_id;

Key points / reasoning:- percentile_cont(0.95) is a continuous percentile: it interpolates between sorted values when the exact rank is not an integer. For example, with N=4 and p=0.95, it computes a fractional index and returns a weighted average between the neighboring ordered values.- With odd counts where the rank falls exactly on an element, percentile_cont returns that element (no interpolation).- With ties (duplicate values), those duplicates appear in the sorted order; interpolation treats them like any other values, which may produce the same value if neighbors are equal.- NULLs: aggregates ignore NULLs, so we explicitly filter out NULL duration_seconds (or rely on the aggregate’s behavior). COUNT(*) counts rows (including NULLs) — here we used COUNT(*) but filtered NULL durations in WHERE; alternatively use COUNT(duration_seconds) in HAVING to ensure at least 5 non-NULL sessions.- If you prefer to require 5 non-NULL durations:

sql

HAVING COUNT(duration_seconds) >= 5

Alternative: If you need the percentile per user but want the full session rows flagged with whether they exceed the per-user p95, compute the p95 via a windowed subquery and join back.

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