Skewness Formula

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?

Learn R Frequently Asked Questions