Q-Q Plot
Visual method showing deviation patterns by plotting sample vs theoretical quantiles.
Plot Overview & Methodology
What it shows:
A Q-Q (Quantile-Quantile) plot is a graphical tool for comparing two probability distributions by plotting their quantiles against each other. For normality testing, it plots sample quantiles against theoretical normal quantiles. A straight diagonal line indicates that the data follows a normal distribution, while deviations from the line reveal specific patterns of non-normality.
Advantages:
- • Highly visual and intuitive interpretation
- • Shows the nature and location of departures
- • Works effectively with any sample size
- • Identifies specific patterns (skewness, outliers, heavy tails)
- • No assumptions about distribution parameters
Limitations:
- • Subjective interpretation of "straight enough"
- • No objective test statistic or p-value
- • Requires experience to interpret correctly
- • Natural variation can obscure patterns in small samples
Quantile Statistic & Formula Focus
Theoretical Quantile Formula:
Each ordered observation is paired with its expected position under the standard normal distribution.
Theoretical Normal Quantile
Φ-1
= inverse standard normal cumulative distribution function
= rank order of the observation
= sample size
Standardized Sample Quantile:
Standardized Sample Value
= i-th ordered observation
= sample mean
= sample standard deviation
The plot pairs (qi,zi); a straight 45° line indicates normality.