Skewness Kurtosis

The measure of skewness, Negative skewed, positive skewed, Karl Pearson Measures of Relative Skewness, Bowley Measures of Skewness, Moment Coefficient of Skewness, Measure of kurtosis, formula, interpretation, application of skewness kurtosis

What is the Measure of Kurtosis (2012)

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Introduction to Kurtosis

In statistics, a measure of kurtosis is a measure of the “tailedness” of the probability distribution of a real-valued random variable. The standard measure of kurtosis is based on a scaled version of the fourth moment of the data or population. Therefore, the measure of kurtosis is related to the tails of the distribution, not its peak.

Measure of Kurtosis

Sometimes, the Measure of Kurtosis is characterized as a measure of peakedness that is mistaken. A distribution having a relatively high peak is called leptokurtic. A distribution that is flat-topped is called platykurtic. The normal distribution which is neither very peaked nor very flat-topped is also called mesokurtic.  The histogram in some cases can be used as an effective graphical technique for showing the skewness and kurtosis of the data set.

Measure of Kurtosis

Data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak.

Moment ratio and Percentile Coefficient of kurtosis are used to measure the kurtosis

Moment Coefficient of Kurtosis= $b_2 = \frac{m_4}{S^2} = \frac{m_4}{m^{2}_{2}}$

Percentile Coefficient of Kurtosis = $k=\frac{Q.D}{P_{90}-P_{10}}$
where Q.D = $\frac{1}{2}(Q_3 – Q_1)$ is the semi-interquartile range. For normal distribution, this has a value of 0.263.

Dr. Wheeler defines kurtosis as:

The kurtosis parameter is a measure of the combined weight of the tails relative to the rest of the distribution.

So, kurtosis is all about the tails of the distribution – not the peakedness or flatness.

A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is considered Leptokurtic. If its kurtosis is less than 3, it is considered Platykurtic.

A large value of kurtosis indicates a more serious outlier issue and hence may lead the researcher to choose alternative statistical methods.

Measure of Kurtosis

Some Examples of Kurtosis

  • In finance, risk and insurance are examples of needing to focus on the tail of the distribution and not assuming normality.
  • Kurtosis helps in determining whether the resource used within an ecological guild is truly neutral or which it differs among species.
  • The accuracy of the variance as an estimate of the population $\sigma^2$ depends heavily on kurtosis.

For further reading see Moments in Statistics

FAQs about Kurtosis

  1. Define Kurtosis.
  2. What is the moment coefficient of Kurtosis?
  3. What is the definition of kurtosis by Dr. Wheeler?
  4. Give examples of kurtosis from real life.

R Frequently Asked Language

Rules for Skewed Data Free Guide

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Introduction to Skewed Data: Lack of Symmetry

Skewness is the lack of symmetry (lack of normality) in a probability distribution. The skewness is usually quantified by the index as given below

$$s = \frac{\mu_3}{\mu_2^{3/2}}$$

where $\mu_2$ and $\mu_3$ are the second and third moments about the mean.

The index formula described above takes the value zero for a symmetrical distribution. A distribution is positively skewed when it has a longer and thin tail to the right. A distribution is negatively skewed when it has a longer thin tail to the left.

Any distribution is said to be skewed when the data points cluster more toward one side of the scale than the other. Creating such a curve that is not symmetrical.

Skewed Data

Skewed Data

The two general rules for Skewed Data are

  1. If the mean is less than the median, the data are skewed to the left, and
  2. If the mean is greater than the median, the data are skewed to the right.

Therefore, if the mean is much greater than the median the data are probably skewed to the right.

Misinterpretation of Mean and Median: The mean can be sensitive to outliers in skewed distributions and might not accurately represent the “typical” value. The median, which is the middle value when the data is ordered, can be a more robust measure of the central tendency for skewed data.

Statistical Tests: Some statistical tests assume normality (zero skewness). If the data is skewed, alternative tests or transformations might be necessary for reliable results.

Identifying Skewed Data

There are a couple of ways to identify skewness in data:

  • Visual Inspection: Histograms and box plots are useful tools for visualizing the distribution of the data. Skewed distributions will show an asymmetry in the plots.
  • Skewness Coefficient: This statistic measures the direction and magnitude of the skew in the distribution. A positive value indicates a positive skew, a negative value indicates a negative skew, and zero indicates a symmetrical distribution.

FAQs about Skewed Data

  1. What is the skewness of data?
  2. What is the lack of symmetry?
  3. What is a positive skewed distribution?
  4. What is a negative skewed distribution?
  5. How a skewness in data be identified?
  6. What is the assumption of different statistical tests?
  7. What is the visual inspection of data skewness?
  8. What is the use of the skewness coefficient?
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