Statistical Analysis

What This Skill Does

• The Statistical Analysis skill provides guidance on descriptive statistics, trend analysis, outlier detection, and hypothesis testing.
• It also offers advice on interpreting statistical claims.
• This skill is useful for data analysts, scientists, and anyone who needs to draw insights from data.

When to Use

• Calculate descriptive statistics for a dataset.
• Analyze trends in data over time.
• Detect outliers in a dataset.
• Perform hypothesis testing.
• Choose the right measure of central tendency.
• Describe the distribution of a dataset.

Key Features

• Provides guidance on choosing the right measure of center.
• Covers measures of spread and variability.
• Recommends reporting key percentiles for business context.
• Explains how to describe data distributions.
• Offers advice on interpreting statistical claims.
• Includes methods for outlier detection.

Statistical Analysis Skill

Descriptive statistics, trend analysis, outlier detection, hypothesis testing, and guidance on when to be cautious about statistical claims.

Descriptive Statistics Methodology

Central Tendency

Choose the right measure of center based on the data:

Always report mean and median together for business metrics. If they diverge significantly, the data is skewed and the mean alone is misleading.

Spread and Variability

Standard deviation: How far values typically fall from the mean. Use with normally distributed data.

• Interquartile range (IQR): Distance from p25 to p75.
• Robust to outliers.
• Use with skewed data. Coefficient of variation (CV): StdDev / Mean. Use to compare variability across metrics with different scales. Range: Max minus min. Sensitive to outliers but gives a quick sense of data extent.

Percentiles for Business Context

Report key percentiles to tell a richer story than mean alone:

p1: Bottom 1% (floor / minimum typical value) p5: Low end of normal range p25: First quartile p50: Median (typical user) p75: Third quartile p90: Top 10% / power users p95: High end of normal range p99: Top 1% / extreme users

Example narrative: "The median session duration is 4.2 minutes, but the top 10% of users spend over 22 minutes per session, pulling the mean up to 7.8 minutes."

Describing Distributions

Characterize every numeric distribution you analyze:

Shape: Normal, right-skewed, left-skewed, bimodal, uniform, heavy-tailed Center: Mean and median (and the gap between them) Spread: Standard deviation or IQR Outliers: How many and how extreme Bounds: Is there a natural floor (zero) or ceiling (100%)?

Trend Analysis and Forecasting

Identifying Trends

Moving averages to smooth noise:

7-day moving average (good for daily data with weekly seasonality)

df['ma_7d'] = df['metric'].rolling(window=7, min_periods=1).mean()

28-day moving average (smooths weekly AND monthly patterns) df['ma_28d'] = df['metric'].rolling(window=28, min_periods=1).mean()

Period-over-period comparison: Week-over-week (WoW): Compare to same day last week Month-over-month (MoM): Compare to same month prior Year-over-year (YoY): Gold standard for seasonal businesses Same-day-last-year: Compare specific calendar day

Growth rates: Simple growth: (current - previous) / previous CAGR: (ending / beginning) ^ (1 / years) - 1 Log growth: ln(current / previous) -- better for volatile series

Seasonality Detection

Check for periodic patterns: Plot the raw time series -- visual inspection first Compute day-of-week averages: is there a clear weekly pattern? Compute month-of-year averages: is there an annual cycle? When comparing periods, always use YoY or same-period comparisons to avoid conflating trend with seasonality

Forecasting (Simple Methods)

For business analysts (not data scientists), use straightforward methods:

Naive forecast: Tomorrow = today. Use as a baseline. Seasonal naive: Tomorrow = same day last week/year. Linear trend: Fit a line to historical data. Only for clearly linear trends. Moving average forecast: Use trailing average as the forecast.

Always communicate uncertainty. Provide a range, not a point estimate: "We expect 10K-12K signups next month based on the 3-month trend" NOT "We will get exactly 11,234 signups next month"

When to escalate to a data scientist: Non-linear trends, multiple seasonalities, external factors (marketing spend, holidays), or when forecast accuracy matters for resource allocation.

Outlier and Anomaly Detection

Statistical Methods

Z-score method (for normally distributed data): z_scores = (df['value'] - df['value'].mean()) / df['value'].std() outliers = df[abs(z_scores) > 3] # More than 3 standard deviations

IQR method (robust to non-normal distributions): Q1 = df['value'].quantile(0.25) Q3 = df['value'].quantile(0.75) IQR = Q3 - Q1 lower_bound = Q1 - 1.5 * IQR upper_bound = Q3 + 1.5 * IQR outliers = df[(df['value'] < lower_bound) | (df['value'] > upper_bound)]

Percentile method (simplest): outliers = df[(df['value'] < df['value'].quantile(0.01)) | (df['value'] > df['value'].quantile(0.99))]

Handling Outliers

Do NOT automatically remove outliers. Instead:

Investigate: Is this a data error, a genuine extreme value, or a different population? Data errors: Fix or remove (e.g., negative ages, timestamps in year 1970) Genuine extremes: Keep them but consider using robust statistics (median instead of mean) Different population: Segment them out for separate analysis (e.g., enterprise vs. SMB customers)

Report what you did: "We excluded 47 records (0.3%) with transaction amounts >$50K, which represent bulk enterprise orders analyzed separately."

Time Series Anomaly Detection

For detecting unusual values in a time series:

Compute expected value (moving average or same-period-last-year) Compute deviation from expected Flag deviations beyond a threshold (typically 2-3 standard deviations of the residuals) Distinguish between point anomalies (single unusual value) and change points (sustained shift)

Hypothesis Testing Basics

When to Use

Use hypothesis testing when you need to determine whether an observed difference is likely real or could be due to random chance. Common scenarios:

A/B test results: Is variant B actually better than A? Before/after comparison: Did the product change actually move the metric? Segment comparison: Do enterprise customers really have higher retention?

The Framework

Null hypothesis (H0): There is no difference (the default assumption) Alternative hypothesis (H1): There is a difference Choose significance level (alpha): Typically 0.05 (5% chance of false positive) Compute test statistic and p-value Interpret: If p < alpha, reject H0 (evidence of a real difference)

Common Tests

Practical Significance vs. Statistical Significance

Statistical significance means the difference is unlikely due to chance.

Practical significance means the difference is large enough to matter for business decisions.

A difference can be statistically significant but practically mean