Quantile-Quantile Plots

Some example QQ plots

Author

Affiliation

Jon Yearsley

School of Biology and Environmental Science, UCD

Introduction

To help you identify different types of distributions from a quantile-quantile plot, we give examples of histograms and quantile-quantile plots for five qualitatively different distributions:

• A normal distribution
• A right-skewed distribution
• A left-skewed distribution
• An under-dispersed distribution
• An over-dispersed distribution

Normally distributed data

Below is an example of data (150 observations) that are drawn from a normal distribution. The normal distribution is symmetric, so it has no skew (the mean is equal to the median).

On a Q-Q plot normally distributed data appears as roughly a straight line (although the ends of the Q-Q plot often start to deviate from the straight line).

Right-skewed data

Below is an example of data (150 observations) that are drawn from a distribution that is right-skewed (in this case it is the exponential distribution). Right-skew is also known as positive skew.

On a Q-Q plot right-skewed data appears as a concave curve.

You can start to understand this pattern by considering the smallest and largest observations (left-hand and right-hand sides of the QQ-plot, respectively):

• the smallest observations are larger than you would expect from a normal distribution (i.e. the points are above the line on the QQ-plot). This means the lower tail of the data’s distribution has been reduced, relative to a normal distribution.
  
• the largest observations are larger than you would expect from a normal distribution (i.e. the points are above the line on the QQ-plot). This means the upper tail of the data’s distribution has been extended, relative to a normal distribution.

Left-skewed data

Below is an example of data (150 observations) that are drawn from a distribution that is left-skewed (in this case it is a negative exponential distribution). Left-skew is also known as negative skew.

On a Q-Q plot left-skewed data appears as a concave curve (the opposite of right-skewed data).

You can start to understand this pattern by considering the smallest and largest observations (left-hand and right-hand sides of the QQ-plot, respectively):

• the smallest observations are smaller than you would expect from a normal distribution (i.e. the points are below the line on the QQ-plot). This means the lower tail of the data’s distribution has been extended, relative to a normal distribution.
  
• the largest observations are smaller than you would expect from a normal distribution (i.e. the points are below the line on the QQ-plot). This means the upper tail of the data’s distribution has been reduced, relative to a normal distribution.

Under-dispersed data

Below is an example of data (150 observations) that are drawn from a distribution that is under-dispersed relative to a normal distribution (in this case it is the uniform distribution). Under-dispersed data has a reduced number of outliers (i.e. the distribution has thinner tails than a normal distribution). Under-dispersed data is also known as having a platykurtic distribution and as having negative excess kurtosis.

On a Q-Q plot under-dispersed data appears S shaped.

You can start to understand this pattern by considering the smallest and largest observations (left-hand and right-hand sides of the QQ-plot, respectively):

• the smallest observations are larger than you would expect from a normal distribution (i.e. the points are above the line on the QQ-plot). This means the lower tail of the data’s distribution has been reduced, relative to a normal distribution.
  
• the largest observations are less than you would expect from a normal distribution (i.e. the points are below the line on the QQ-plot). This means the upper tail of the data’s distribution has been reduced, relative to a normal distribution.

Over-dispersed data

Below is an example of data (150 observations) that are drawn from a distribution that is over-dispersed relative to a normal distribution (in this case it is a Laplace distribution). Over-dispersed data has an increased number of outliers (i.e. the distribution has fatter tails than a normal distribution). Over-dispersed data is also known as having a leptokurtic distribution and as having positive excess kurtosis.

On a Q-Q plot over-dispersed data appears as a flipped S shape (the opposite of under-dispersed data).

You can start to understand this pattern by considering the smallest and largest observations (left-hand and right-hand sides of the QQ-plot, respectively):

• the smallest observations are smaller than you would expect from a normal distribution (i.e. the points are below the line on the QQ-plot). This means the lower tail of the data’s distribution has been extended, relative to a normal distribution.
  
• the largest observations are larger than you would expect from a normal distribution (i.e. the points are above the line on the QQ-plot). This means the upper tail of the data’s distribution has been extended, relative to a normal distribution.