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Wize University Statistics Textbook > Exploratory Data Analysis

Skewness and Symmetry

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Skewness and Symmetry

For quantitative data, you can plot them into a histogram and observe how they are distributed. Are the data distributed evenly or are they skewed to one direction?

Consider these three unimodal distributions:

Symmetric Distribution

Mean = Median

Examples of symmetric distributions:
  • Exam grades
  • Time it takes to commute to work

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Left-Skewed Distribution (a.k.a. Negatively Skewed Distribution)


Mean < Median

Examples of left-skewed distributions:
  • Time people spend at a concert (very few people leave early; most people stay to the end)
  • Age of death (very few people die young; many people die old)

Right-Skewed Distribution (a.k.a. Positively Skewed Distribution)


Mean > Median

Examples of right-skewed distributions:
  • Number of bedrooms of homes (most homes have 1 or 2 bedrooms; few have many bedrooms)
  • Salary of employees (few employees earn very high salaries, like top executives)

Practice: Skewness

For each of the following distributions, determine if it is more likely to be symmetric, skewed to the right, or skewed to the left.


(i) Number of pets in a household.
Extra Practice
Skewness and Symmetry
For each variable, determine whether its distribution is more likely to be symmetric, skewed to the right, or skewed to the left.
NHL players' salaries.
Skewness and Symmetry
For each variable, determine whether its distribution is more likely to be symmetric, skewed to the right, or skewed to the left.
Number of pets in a household.
Skewness and Symmetry
Here's a summary of the statistics for the height of students in a kindergarden class:
Shortest student: 83cm
Mean height: 92cm
Skewness and Symmetry
If the graph of a distribution is positively skewed, what must be true about this graph?
Skewness and Symmetry
Twenty-three students wrote a quiz and their marks out of 50 are recorded. Here is the data: 8, 20, 21, 22, 30, 30, 31, 36, 37, 37, 37, 41, 41, 42, 43, 44, 45, 45, 45, 48, 48, 49, 50.
Which of the following is true?
Boxplots: Skewness and Symmetry
The 5 number summary statistics for the number of years a typical family owns their first home before selling it is given by:
0.5  1  3.5  6  300.5\ \ 1\ \ 3.5\ \ 6\ \ 30
A total of 120 families participated in this study.
Skewness and Symmetry
The distribution of a nominal variable can be...
Skewness and Symmetry
True or false? Enter "T" for true and "F" for false.
1. Multimodal distributions could not be skewed.
F
2. If the mean is equal to the median, then the distribution must be symmetric and unimodal.
F
Histograms, Skewness and Symmetry
The histogram refers to the number of emails Melanie responds to during a day at work:
Which of the following is true?
Histograms, Skewness and Symmetry
Melanie was able to respond to 91 emails today. If we add this additional data to the histogram, what will happen?
Skewness and Symmetry
The data "peaks" at a bin near the higher end of a unimodal histogram. Which of the following is true about the distribution?