Overview
This lecture explains the concepts of symmetry and skewness in distributions, and how they can be identified and interpreted using histograms, stem plots, and box plots.
Understanding Symmetry and Skewness
- Symmetry refers to a distribution that can be divided into two equal sides of the same shape.
- A distribution is skewed if it is not symmetrical, meaning one tail is longer than the other.
- Skewness refers to the asymmetry in a distribution’s shape.
Types of Skewness
- A distribution is skewed to the left if it has a long tail that trails towards the left.
- A distribution is skewed to the right if it has a long tail that trails towards the right.
Identifying Skewness in Graphical Displays
- Histograms, stem plots, and box plots can all be used to identify symmetry and skewness.
- For stem plots, flipping the plot sideways helps visualize the direction of the tail.
- Skewness in box plots is interpreted using the size of boxes and whiskers:
- If boxes are unequal, the larger side indicates the skew.
- If boxes are equal, the longer whisker shows the direction of skew.
- If boxes and whiskers are equal, the distribution is symmetrical.
Effect of Skewness on Mean and Median
- In a symmetrical distribution, the mean equals the median and both are at the center.
- In a left-skewed distribution, the mean is less than the median; the mean is closer to the left tail.
- In a right-skewed distribution, the mean is greater than the median; the mean is closer to the right tail.
Key Terms & Definitions
- Symmetry — when a distribution’s two sides are mirror images.
- Skewness — the measure of asymmetry in a distribution.
- Left-skewed distribution — has a long tail to the left.
- Right-skewed distribution — has a long tail to the right.
- Histogram — a bar graph that displays frequencies of data intervals.
- Stem plot — a tool for displaying data values in groups.
- Box plot — a graphical display showing data spread and outliers.
- Mean — the arithmetic average of a data set.
- Median — the middle value of a data set.
Action Items / Next Steps
- Practice identifying symmetry and skewness in provided histogram, stem plot, and box plot examples.