Statistical Analysis

# 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:

| Situation | Use | Why | |---|---|---| | Symmetric distribution, no outliers | Mean | Most efficient estimator | | Skewed distribution | Median | Robust to outliers | | Categorical or ordinal data | Mode | Only option for non-numeric | | Highly skewed with outliers (e.g., revenue per user) | Median + mean | Report both; the gap shows skew |

**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: ```python # 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: 1. Plot the raw time series -- visual inspection first 2. Compute day-of-week averages: is there a clear weekly pattern? 3. Compute month-of-year averages: is there an annual cycle? 4. 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): ```python 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): ```python 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): ```python outliers = df[(df['value'] < df['value'].quantile(0.01)) | (df['value'] > df['value'].quantile(0.99))] ```

### Handling Outliers

Do NOT automatically remove outliers. Instead:

1. **Investigate**: Is this a data error, a genuine extreme value, or a different population? 2. **Data errors**: Fix or remove (e.g., negative ages, timestamps in year 1970) 3.