Shape of Distributions (College Board AP® Statistics)

Skewness describes the shape of a distribution

• It describes the symmetry or asymmetry of the distribution

A distribution has positive skew if the shape leans to the left

• Values above the median have a greater spread than values below the median

  • The right tail is longer than the left tail

• The distribution is said to be skewed to the right

  • This is because the data usually cause the mean to be to the right of the median

A distribution has negative skew if the shape leans to the right

• Values below the median have a greater spread than values above the median

  • The left tail is longer than the right tail

• The distribution is said to be skewed to the left

  • This is because the data usually cause the mean to be to the left of the median

A distribution is symmetrical if the left side and right side are reflections of each other about the median

• The mean and median are equal

The skewness is related to the median and the mean of the data set

• In a symmetric distribution

  • the median and the mean are roughly the same

  • median mean

• In a positively skewed distribution 

  • median < mean

• In a negatively skewed distribution 

  • mean < median

A cluster is a region of the distribution where the data is concentrated

• This means there are a lot of data points in a region

A gap is a region of the distribution where there is no data

• Clusters are normally separated by gaps

An outlier is a point that is far away from the majority of the data

• An outlier is very small or very large compared to the rest of the data points

A peak of a distribution occurs at a value or group where the frequency is higher than the nearby values or groups

• A peak occurs at the mode

• Peaks can occur at places other than the mode

A uniform distribution has no peaks

• The frequency is the same for all values or groups

If the frequencies are approximately equal then the distribution is approximately uniform

A unimodal distribution has one main peak

A bimodal distribution has two prominent peaks

• One peak might be higher than the other