Shape & Skewness of Distributions (College Board AP® Statistics): Revision Note

Skewness of distributions

What is skewness?

• 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

How is the skew related to the median and the mean?

• 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

How do I find the skewness from the quartiles?

• If the median is roughly in the middle of the first and third quartiles, then the distribution is approximately symmetric

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• If the median is closer to the first quartile, then the distribution has positive skew

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• If the median is closer to the third quartile then the distribution has negative skew

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Features of a distribution

What are clusters, gaps, outliers and peaks?

• 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

What are uniform, unimodal and bimodal distributions?

• 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