Basic Statistics

The post outlines key skewness formulas providing essential tools for analyzing data distribution asymmetry. The skewness formulas help quantify the direction and degree of skewness, aiding in data analysis and decision-making.

What is Skewness?

Skewness is a statistical measure that describes the asymmetry of a probability distribution around its mean. It indicates whether the data is skewed to the left (negative skew), the right (positive skew), or symmetrically distributed (zero skew). In short, Skewness is the degree of asymmetry or departure from the symmetry of the distribution of a real-valued random variable. The post describes some important skewness formulas.

Positive Skewed

If the frequency curve of distribution has a longer tail to the right of the central maximum than to the left, the distribution is said to be skewed to the right or to have positively skewed. In a positively skewed distribution, the mean is greater than the median and the median is greater than the mode i.e. $$Mean > Median > Mode$$

Negative Skewed

If the frequency curve has a longer tail to the left of the central maximum than to the right, the distribution is said to be skewed to the left or to be negatively skewed. In a negatively skewed distribution, the mode is greater than the median and the median is greater than the mean i.e. $$Mode > Median > Mean$$

Zero Skewness

For zero skewness, the data is symmetrically distributed, as in a normal distribution.

In a symmetrical distribution, the mean, median, and mode coincide. In a skewed distribution, these values are pulled apart.

Pearson’s Coefficient of Skewness Formula

Karl Pearson, (1857-1936) introduced a coefficient to measure the degree of skewness of distribution or curve, which is denoted by $S_k$ and defined by

\begin{eqnarray*}
S_k &=& \frac{Mean – Mode}{Standard Deviation}\\
S_k &=& \frac{3(Mean – Median)}{Standard Deviation}\\
\end{eqnarray*}
Usually, this coefficient varies between –3 (for negative) to +3 (for positive) and the sign indicates the direction of skewness.

Bowley’s Coefficient of Skewness Formula (Quartile Coefficient)

Arthur Lyon Bowley (1869-1957) proposed a measure of skewness based on the median and the two quartiles.

\[S_k=\frac{Q_1+Q_3-2Median}{Q_3 – Q_1}\]
Its values lie between 0 and ±1.

Moment Coefficient of Skewness Formula

This measure of skewness is the third moment expressed in standard units (or the moment ratio) thus given by

\[S_k=\frac{\mu_3}{\sigma^3} \]
Its values lie between -2 and +2.

If $S_k$ is greater than zero, the distribution or curve is said to be positively skewed. If $S_k$ is less than zero the distribution or curve is said to be negatively skewed. If $S_k$ is zero the distribution or curve is said to be symmetrical.

The skewness of the distribution of a real-valued random variable can easily be seen by drawing a histogram or frequency curve.

The skewness may be very extreme and in such a case these are called J-shaped distributions.

Skewness helps identify deviations from normality, which is crucial for selecting appropriate statistical methods and interpreting data accurately. It is commonly used in finance, economics, and data analysis to understand the shape and behavior of datasets

FAQs about Skewness

1. What is the degree of asymmetry called?
2. What is a departure from symmetry?
3. If a distribution is negatively skewed then what is the relation between mean, median, and mode?
4. If a distribution is positively skewed then what is the relation between mean, median, and mode?
5. What is the relation between mean, median, and mode for a symmetrical distribution?
6. What is the range of the moment coefficient of skewness?

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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.

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.

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.

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