This lesson covers Introduction to Skewed Distribution.

Sources:

• https://www.statisticshowto.com/probability-and-statistics/skewed-distribution/
• http://jse.amstat.org/v13n2/vonhippel.html
• https://www.statisticshowto.com/skewness/

What is a Skewed Distribution?

If one tail is longer than another, the distribution is skewed.

• aka Asymmetric or Asymmetrical distributions

Symmetry means:

• one half of the distribution is a mirror image of the other half
• The tails are exactly the same, e.g. normal distribution

Left-Skewed

Long left tail

• aka negatively-skewed distributions
  • long tail in the negative direction on the number line. The mean is also to the left of the peak.
• Mean is to the left of the peak. This is the main definition behind “skewness”, which is technically a measure of the distribution of values around the mean.
• In most cases, the mean is to the left of the median. This isn’t a reliable test for skewness though, as some distributions (i.e. many multimodal distributions) violate this rule. You should think of this as a “general idea” kind of rule, and not a set-in-stone one.

• Box Plot (Left Skewed)

  • Left Whisker is longer than right whisker

  

• Histogram (Left Skewed)

  

Right-Skewed

Long right tail

• aka positive-skew distributions
  • long tail in the positive direction on the number line. The mean is also to the right of the peak.

• Histogram (Right Skewed)

  

• Box Plot (Right Skewed)

  • Right whisker is longer than left whisker.

  

• Example:
  • Numbers: $1, 2, 3$
    • Evenly spaced, with $2$ as the mean
  • Adding a number to the far left: $-10,~ 1,~ 2,~ 3$
    • Left skewed
  • Adding a value to the far right: $1,~ 2,~ 3,~ 10$
    • Right skewed

Exception

• Distribution from a 2002 General Social Survey. Respondents stated how many people older than 18 lived in their household.
• Right-skewed graph, but the mean is clearly to the left of the median.

Compute Skewness

• Measure of lack of symmetry

• A standard normal distribution is perfectly symmetrical and has zero skew.

• Other Zero-skewed distributions:

  • T Distribution
  • Uniform distribution
  • Laplace distribution

• Computation for various distributions (non-zero)

  Distribution	Equation
  Bernoulli distribution.	$\frac{1-2p}{\sqrt{p(1-p)}}$
  Beta distribution.	$\frac{2(b-a)}{2+a+b}\sqrt{\frac{1+a+b}{ab}}$
  Binomial distribution.	$\frac{1-2p}{\sqrt{np(1-p)}}$
  Chi square distribution.	$2\sqrt{\frac{2}{r}}$
  F distribution.	$\frac{2(2n+m-2)}{m-6}\sqrt{\frac{2(m-4)}{n(m+n-2)}}$
  Negative binomial. $\frac{2-p}{\sqrt{r(1-p)}}$	$\frac{2-p}{\sqrt{r(1-p)}}$
  Poisson Distribution.	$\nu^{-1/2}$

Calculation

• Pearson Mode Skewness

  • The mean, mode and median can be used to figure out if you have a positively or negatively skewed distribution.
    • If the mean is greater than the mode, the distribution is positively skewed.
    • If the mean is less than the mode, the distribution is negatively skewed.
    • If the mean is greater than the median, the distribution is positively skewed.
    • If the mean is less than the median, the distribution is negatively skewed.
  • $Skew = \frac{Mean – Mode}{Standard~ Deviation}$
    • Mode Skeweness

• Alternative Pearson Mode Skewness

  • $Skew = 3 * \frac{Mean – Median}{Standard~ Deviation}$
    • Median Skewness

• MS Excel

  • SKEW function is used to calculate the skewness of the sample data
    • Excel uses adjusted Fisher-Pearson standardized coefficient
    • $G = \frac{n}{(n-1)(n-2)} \Sigma(\frac{x_i - \bar{x}}{s})^3$
    • $s$ - STDEV.S in excel for Sample
  • SKEW.P function is used to calculate the skewness of the population data
    • $S_k = \frac{1}{n} \Sigma(\frac{x_i-\mu}{\sigma})^3$

• Data

  • $X = {1,1,~~ 2,2,2,2,2,~~ 3,3,3,3,~~ 5,5,~~ 7, 8}$

    • Plot Histogram
      • Insert -> Histogram
      • Right Click Data Area -> Format Data Series -> Bin Width 0.9
        • Right Skewed
      • $SKEW.P() \implies 1.09899799$

  • $X = 1, 2,~~ 3,3,~~ 5,5,5,~~ 7,7,7,7,~~ 8,8,8,8,8$

    • Plot Histogram

      • Insert -> Histogram
      • Right Click Data Area -> Format Data Series -> Bin Width 0.9
        • Left Skewed
      • $SKEW.P() \implies -0.704386$