A Q-Q (quantile-quantile) plot compares the quantiles of two distributions.
If the two distributions are identical (or very close), the points on the Q-Q plot will fall approximately along the 45° straight line
A **Q-Q plot** (quantile-quantile plot) is a graphical tool used to compare two probability distributions by plotting their quantiles against each other.
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## **How it works**
- One distribution’s quantiles are on the x-axis, the other’s on the y-axis.
- If the two distributions are similar, the points will fall roughly along the **line \(y = x\)** (the 45° diagonal).
- Deviations from this line indicate how the distributions differ in shape, spread, or tails.
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## **Types of Q-Q plots**
1. **Two-sample Q-Q plot**: Compare two empirical datasets.
2. **Theoretical Q-Q plot**: Compare sample data to a theoretical distribution (e.g., normal Q-Q plot to check normality).
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## **Benefits of Q-Q plots**
1. **Visual check for distribution similarity**
- Quickly see if two datasets come from the same distribution family.
2. **Assess normality**
- Common use: Normal Q-Q plot to check if data is approximately normally distributed.
3. **Identify tails behavior**
- Points deviating upward at the top → right tail of sample is heavier than theoretical.
- Points deviating downward at the top → right tail is lighter.
4. **Detect skewness**
- A curved pattern suggests skew.
5. **Spot outliers**
- Points far off the line may be outliers.
6. **Compare location and scale differences**
- If points lie on a straight line with slope ≠ 1 → scale difference.
- If intercept ≠ 0 → location shift.
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## **Example interpretation**
- **Straight diagonal line**: Distributions are the same.
- **Straight line with slope > 1**: Sample has greater variance.
- **S-shaped curve**: Tails differ (one distribution has heavier or lighter tails).
- **Concave up**: Sample distribution is right-skewed relative to theoretical.
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