skewed
(adjective)
Biased or distorted (pertaining to statistics or information).
Examples of skewed in the following topics:
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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?
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.