Skewed vs. Normal Distribution

Explore how a skewed distribution deviates from a theoretical normal distribution. Adjust the skewness parameter to see the effect on the density curves and the Quantile-Quantile (Q-Q) plot. Toggle between Raw Scores and Standardized Scores (Mean=0, Variance=1) to see how standardization aligns the shapes.

Skewness (α): 4.0

Normal Skewed

X-Axis Unit (Normal): Standardized

Y-Axis Unit (Skewed): Standardized

Boxplots (Normal vs. Skewed)

Compare the medians (center lines), Interquartile Ranges (boxes), and Outliers (circles). Because the skewed distribution is tightly packed in the middle, its box is narrower, which causes its heavy tail to be classified as statistical outliers.

1. Density Curves

Regions in Red show where skewed concentration is higher than normal.

2. Cumulative (CDF)

The horizontal gap between the curves reveals the exact difference in values at any given percentile.

3. Q-Q Plot

Plots the exact horizontal gaps from the CDF. Points in Red indicate Skewed > Normal.

Reading the Curve: Concavity & The CDF Gap

Notice how the Q-Q plot isn't just a straight line, but often forms a distinct curve. This curvature—mathematically called concavity—maps directly to how the horizontal gap between the curves changes on the CDF plot.

Curving Upwards (U-Shape / Positive Skew): When the Q-Q plot accelerates upwards, the Skewed values are growing faster than the Normal values. Look at the top-right of the CDF: the Red curve becomes flatter than the Blue curve and stretches far to the right. Because it climbs slower, the horizontal gap between Blue and Red rapidly widens as you move up. This expanding gap is what plots as a steep upward curve on the Q-Q chart!
Curving Downwards (Arch / Negative Skew): Try setting the Skewness slider to a negative number. The Q-Q plot now arches downwards. On the CDF, look at the bottom-left. The horizontal gap is widest at the lowest percentiles and narrows as you move up. Because the Skewed distribution is catching up to the Normal distribution, the slope on the Q-Q plot flattens out.

The "Density Trap": Why Height Doesn't Equal Concavity

It is tempting to look at the Density plot and assume that when the Red curve dips below the Blue curve, the Q-Q plot should curve downwards. However, lower density just means the data is spread thinly. If the data is spread thin, you have to travel much further along the X-axis to collect the next 5% of data. Therefore, a lower density actually creates a steeper slope on the Q-Q plot, not a dip! Concavity is driven by the widening or narrowing of the horizontal gap on the CDF, rather than which density curve is currently taller.