When Normality Fails

Most statistical methods you've learned assume data is approximately normal. But real-world data often isn't:

• Income data is heavily right-skewed (most people earn moderate amounts, a few earn millions)
• Survival times are right-skewed (most patients survive a while, some die quickly)
• Insurance claims have heavy tails (most claims are small, but catastrophic ones are enormous)
• Likert scale data (1-5 ratings) can't be normal — it's discrete and bounded

When normality fails, standard methods can give misleading results: wrong p-values, incorrect confidence intervals, and poor predictions.

Right skew (positive skew) — long tail to the right:

• Mean > Median
• Examples: income, house prices, hospital stays, claim sizes

Left skew (negative skew) — long tail to the left:

• Mean < Median
• Examples: age at death in developed countries, scores on easy exams

Impact on standard methods:

• t-tests can produce incorrect p-values
• Confidence intervals may not have the claimed coverage
• The mean becomes a poor measure of "typical"

Visual methods (most useful):

1. Histogram — Does it look bell-shaped? Skewed? Multimodal?
2. QQ plot (quantile-quantile plot) — Plots your data quantiles against theoretical normal quantiles. Points should fall on a straight line if data is normal. Deviations at the tails indicate non-normality.
3. Box plot — Asymmetric box or many outliers suggest non-normality.

Formal tests:

• Shapiro-Wilk test — Best for small-medium samples
• Kolmogorov-Smirnov test — Compares data to a reference distribution

Warning about formal tests: With large samples, they reject normality for trivially small deviations. With small samples, they lack power to detect meaningful departures. Visual assessment is usually more informative.

Option 1: Transform the data Apply a mathematical function to make data more normal:

• Log transform — Great for right-skewed positive data (income, prices)
• Square root transform — Milder than log, for count data
• Box-Cox transform — Finds the optimal power transformation

Option 2: Use non-parametric methods (next lesson) Methods that don't assume any particular distribution shape.

Option 3: Use robust methods

• Trimmed means (remove top/bottom 5-10% before averaging)
• Median instead of mean
• Robust standard errors

Option 4: Use bootstrapping (lesson after next) Estimate sampling distributions empirically, no normality needed.

Option 5: Rely on CLT With large enough n, the sampling distribution of the mean is approximately normal regardless. But "large enough" can be very large for heavily skewed data.

Outliers are observations that are far from the rest of the data. They can be:

1. Data errors — Typos, measurement malfunctions. Should be corrected or removed.
2. Natural extreme values — Real but rare observations. Often the most interesting data points!
3. From a different population — Mixed data sources (measuring adults and accidentally including children).

Never automatically remove outliers. Always investigate first. The "outlier" might be your most important data point — or it might be a data entry error. Context matters.