/statistical-analysis

name: statistical-analysis
description: Apply statistical methods including descriptive stats, trend analysis, outlier detection, and hypothesis testing. Use when analyzing distributions, testing for significance, detecting anomalies, computing correlations, or interpreting statistical results.
user-invocable: false

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:

# 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):

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:

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. **Genuine extremes**: Keep them but consider using robust statistics (median instead of mean) 4. **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:

1. Compute expected value (moving average or same-period-last-year) 2. Compute deviation from expected 3. Flag deviations beyond a threshold (typically 2-3 standard deviations of the residuals) 4. Distinguish between point anomalies (single