Introduction

• a normal distribution to be symmetrical in shape
• real-world data are often have asymmetric distributions
• asymmetry in a distribution is measure by skewness
• kurtosis (peakedness) defines whether a distribution is truly normal
  • or whether it may have so-called fatter or thinner tails

Symmetric Distributions – I

• A distribution is symmetric if the relative frequency or probability of certain values are equal at equal distances from the point of symmetry.
• The point of symmetry for normal distributions is the mean (and at the same time median and mode!)
• The most common symmetric distribution is the normal distribution.
• However, there are a number of other distributions that are symmetric.

A symmetric distribution has the following property:

$$Q_3-Q_2=Q_2-Q_1$$ where $Q_1$, $Q_2$, and $Q_3$ are 1st, 2nd, and 3rd quartiles. Thus the ratio $(Q_3-Q_2)/(Q_2-Q_1)$ can be used as a measure of asymmetry.

Symmetric Distributions – II

Have a look at following histogram:

This distribution meets all of the conditions of being symmetrical.

Skewness

• Skewness is the degree of distortion or deviation from the symmetrical normal distribution.

• Skewness can be seen as a measure to calculate the lack of symmetry in the data distribution.

• Skewness helps you identify extreme values in one of the tails. Symmetrical distributions have a skewness of 0.

Typs of Skewness

Fisher-Pearson coefficient of skewness

For univariate data $x_1,x_2,...,x_n$ the formula for skewness is:

$$g_1=\frac{\frac{1}{n}\sum _{i=1}^n(x_i-\overline{x}{)}^3}{s^3}$$

where $\overline{x}$ is the mean, $s$ is the standard deviation, and $n$ is the number of data points.

The Fisher-Pearson coefficient of skewness is the most commonly used measure of skewness.

Interpreting Fisher-Pearson coefficient of skewness

The rule of thumb:

• A skewness between -0.5 and 0.5 means that the data are pretty symmetrical
• A skewness between -1 and -0.5 (negatively skewed) or between 0.5 and 1 (positively skewed) means that the data are moderately skewed.
• A skewness smaller than -1 (negatively skewed) or bigger than 1 (positively skewed) means that the data are highly skewed.

Pearson Mode Skewness

The Pearson mode skewness is used when a strong mode is exhibited by the sample data.

For univariate data $x_1,x_2,...,x_n$ the formula for Pearson mode Skewness is:

$${Sk}_1=\frac{\overline{x}-m_o}{s}$$

where $\overline{x}$ is the mean, $s$ is the standard deviation, and $m_o$ is the mode of data points.

Interpretation:

• The direction of skewness is given by the sign.
• The coefficient compares the sample distribution with a normal distribution. The larger the value, the larger the distribution differs from a normal distribution.
• A value of zero means no skewness at all.
• A large negative value means the distribution is negatively skewed.
• A large positive value means the distribution is positively skewed.

Pearson's Second Coefficient (Pearson Median Skewness)

Pearson's second coefficient is used when the data includes multiple modes or a weak mode.

For univariate data $x_1,x_2,...,x_n$ the formula for Pearson mode Skewness is:

$${Sk}_2=\frac{3(\overline{x}-m_d)}{s}$$

where $\overline{x}$ is the mean, $s$ is the standard deviation, and $m_d$ is the median of data points.

It has the sam interpretation as the Pearson mode skewness.

Remedies for Skewewness

You generally have three choices if your statistical procedure requires a normal distribution and your data is skewed:

Do nothing. Many statistical tests, including t tests, ANOVAs, and linear regressions, aren’t very sensitive to skewed data. Especially if the skew is mild or moderate, it may be best to ignore it.

Use a different model. You may want to choose a model that doesn’t assume a normal distribution. Non-parametric tests or generalized linear models could be more appropriate for your data.

Transform the variable. Another option is to transform a skewed variable so that it’s less skewed. “Transform” means to apply the same function to all the observations of a variable.

Transformations Based on the Type of Skewness

*In this context, “reflect” means to take the largest observation, $x_l$, then subtract each observation from $x_l+1$. Keep in mind that the reflection reverses the direction of the variable and its relationships with other variables (i.e., positive relationships become negative).

Kurtosis

Measuring Kurtosis

For univariate data $x_1,x_2,\dots ,x_n$ the formula for kurtosis is:

$$k=\frac{\frac{1}{n}\sum _{i=1}^n(x_i-\overline{x}{)}^4}{s^4}$$

If there is a high kurtosis, then you may want to investigate why there are so many outliers.

Low kurtosis in a data set is an indication that data has light tails or lacks outliers. If we get low kurtosis, then also we need to investigate and trim the dataset of unwanted results.

Excess kurtosis

In practic, excess kurtosis, which is defined as Pearson's kurtosis minus 3, to provide a simple comparison to the normal distribution.

$$k_e=k-3=\frac{\frac{1}{n}\sum _{i=1}^n(x_i-\overline{x}{)}^3}{s^3}-3$$

Types of Kurtosis

Types of Kurtosis