Q-Q Plot 2D: Testing Normality with a Quantile-Quantile Scatter

Population Shape

Sample

Sample size n 200

Live Readouts

R² = 1.000 Skewness = 0.00 Excess kurtosis = 0.00

How it works

A Q-Q plot checks whether a sample came from a normal population by comparing its sorted values against the quantiles a perfect normal sample would have:

sort sample:      x(1) ≤ x(2) ≤ ... ≤ x(n)
plotting position: p_i = (i - 0.5) / n
theoretical:       q_i = Φ⁻¹(p_i)
standardized:      z_i = (x(i) - mean) / sd
point:             (q_i, z_i)

If the population really is normal, z(i) ≈ q_i for every i, so all points fall on the y = x reference line. Each point's color tracks its residual z_i − q_i, so any curvature away from the line is visible at a glance:

Right-skewed — points sag below the line at low quantiles and arc above it at high quantiles (a "J" curve).
Heavy-tailed — an "S" shape: the lower tail drops steeply and the upper tail shoots up, because extreme values are more common than a normal predicts.
Uniform — an inverse-S: too few extreme values, so both tails flatten toward the center.
Bimodal — a stepped, wavy curve where the two component peaks create a flat plateau in the middle.

R² is the squared Pearson correlation between q_i and z_i — 1.000 only for an exactly straight line. Sample skewness (third standardized moment) flags asymmetry; excess kurtosis (fourth standardized moment, minus 3) flags tail weight relative to the normal's baseline of 0. Drag the plot to pan, scroll or pinch to zoom.

On the line (small residual) Off the line (large residual)

x: theoretical normal quantile
y: standardized sample quantile

Drag to pan · Scroll/pinch to zoom