skewness

Examples of skewness in the following topics:

• Exercises

  • What value of λ in Tukey's ladder decreases skew the most?
  • What value of λ in Tukey's ladder increases skew the most?
  • How does the skew in each of these compare to the skew in the raw data.
  • Which transformation leads to the least skew?

• Shapes of Distributions

  • Figure 1 shows a distribution with a very large positive skew.
  • Distributions with positive skew normally have larger means than medians.
  • The relationship between skew and the relative size of the mean and median lead the statistician Pearson to propose the following simple and convenient numerical index of skew:
  • The following measure of kurtosis is similar to the definition of skew.
  • A distribution with a very large positive skew.

• Skewness and the Mean, Median, and Mode

  • The shape distribution is called skewed to the left because it is pulled out to the left.
  • The mean and the median both reflect the skewing but the mean more so.
  • Again, the mean reflects the skewing the most.
  • If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.
  • Skewness and symmetry become important when we discuss probability distributions in later chapters.

• Comparing Measures of Central Tendency

  • Understand how the difference between the mean and median is affected by skew
  • Differences among the measures occur with skewed distributions.
  • Notice this distribution has a slight positive skew.
  • The large skew results in very different values for these measures.
  • A distribution with a very large positive skew.

• Typical Shapes

  • When a histogram is constructed for skewed data, it is possible to identify skewness by looking at the shape of the distribution.
  • A distribution is said to be positively skewed (or skewed to the right) when the tail on the right side of the histogram is longer than the left side.
  • A distribution is said to be negatively skewed (or skewed to the left) when the tail on the left side of the histogram is longer than the right side.
  • This distribution is said to be negatively skewed (or skewed to the left) because the tail on the left side of the histogram is longer than the right side.
  • This distribution is said to be positively skewed (or skewed to the right) because the tail on the right side of the histogram is longer than the left side.

• The normality condition

  • If the sample size is 10 or more, slight skew is not problematic.
  • Once the sample size hits about 30, then moderate skew is reasonable.
  • Data with strong skew or outliers require a more cautious analysis.

• Histograms and shape

  • If a distribution has a long left tail, it is left skewed.
  • If a distribution has a long right tail, it is right skewed.
  • Can you see the skew in the data?
  • Is it easier to see the skew in this histogram or the dot plots?
  • This distribution is very strongly skewed to the right.

• Examining the Central Limit Theorem

  • The uniform distribution is symmetric, the exponential distribution may be considered as having moderate skew since its right tail is relatively short (few outliers), and the log-normal distribution is strongly skewed and will tend to produce more apparent outliers.
  • We can also relax our condition on skew when the sample size is very large.
  • If we can obtain a much larger sample, perhaps several hundred observations, then the concerns about skew and outliers would no longer apply.
  • Strong skew is often identified by the presence of clear outliers.
  • For example, outliers are often an indicator of very strong skew.

• The Average and the Histogram

  • Two common examples of symmetry and asymmetry are the "normal distribution" and the "skewed distribution. "
  • Skewness is the tendency for the values to be more frequent around the high or low ends of the $x$-axis.
  • When a histogram is constructed for skewed data it is possible to identify skewness by looking at the shape of the distribution.
  • A distribution is said to be negatively skewed when the tail on the left side of the histogram is longer than the right side.
  • There can also be more than one mode in a skewed distribution.

• Transforming data (special topic)

  • When data are very strongly skewed, we sometimes transform them so they are easier to model.
  • The histogram of MLB player salaries is useful in that we can see the data are extremely skewed and centered (as gauged by the median) at about $1 million.
  • Most of the data are collected into one bin in the histogram and the data are so strongly skewed that many details in the data are obscured.
  • Transformed data are sometimes easier to work with when applying statistical models because the transformed data are much less skewed and outliers are usually less extreme.
  • Commmon goals in transforming data are to see the data structure differently, reduce skew, assist in modeling, or straighten a nonlinear relationship in a scatterplot.