Basic Statistics and Data Analysis

Lecture notes, MCQS of Statistics

Tag: Measure of Dispersion

Skewness Introduction, formula, Interpretation

Skewness

Skewness is the degree of asymmetry or departure from symmetry of the distribution of a real valued random variable.

Positive Skewed
If the frequency curve of a 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 positive skewed. In a positive skewed distribution, the mean is greater than the media and 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 have negative skewed. In a negatively skewed distribution, mode is greater than median and median is greater than mean i.e. Mode > Median > Mean.

Skewness

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

Pearson’s Coefficient of Skewness
Karl Pearson, (1857-1936) introduced a coefficient of skewness to measure the degree of skewness of a distribution or curve, which is denote by $S_k$ and define 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 or Quartile Coefficient of Skewness
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
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 positive skewed. If $S_k$ is less than zero the distribution or curve is said to be negative 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 histogram or frequency curve.

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

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Kurtosis Normality Measure, Introduction, Interpretation

Measure of Kurtosis

Kurtosis is a measure of peakedness of a distribution relative to the normal distribution. A distribution having a relatively high peak is called leptokurtic. A distribution which is flat topped is called platykurtic. The normal distribution which is neither very peaked nor very flat-topped is also called mesokurtic.  The histogram is an effective graphical technique for showing both the skewness and kurtosis of data set.

Kurtosis Pict

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 the value 0.263.

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 said to be Leptokurtic. If its kurtosis is less than 3, it is said to be Platykurtic.

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