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Maths Exercices

Central Tendency Measures
Measures of Dispersion
Measures of skewness and kurtosis


Skewness is the degree of asymmetry, or departure from symmetry, of a distribution.
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 skewness. If the reverse is true, it is said to be skewed to the left, or to have negative skewness.


Negative skew
Elongated tail at the left.
More data in the left tail than would be expected in a normal distribution.

Positive skew
Elongated tail at the right.
More data in the right tail than would be expected in a normal distribution.

For skewness distribution, the mean tends to lie on the same side of the mode as the longer tail.

In many books measure of skewness is denoted by , J or Sk.

Different formulae for measuring skewness are:

i) Bowley's formula for measuring skewness in terms of quartiles is:

ii) Kelley gave the formula in terms of percentiles and deciles. Kelley's absolute measures os skewnes are:

Sk=P90+P10-2P50= D9+D1-2D5

These formulae are not practically used. Instead, it is measured as coefficient of skewness which is given as:

Kelly's formulae are seldom used.

iii) Karl Pearson's measure of skewness:

iv) Karl pearson's formulae for a wide class of frequency distributions in terms of moments is

gives only the measure of skewness but not the direction of skewness. So another measure  is defined as:

How to interpret the value of measure of skewness:

  • Measure of skewness = 0 means that the frequency curve is symmetrical.
  • Measure of skewness > 0 leads to positive skewness.
  • Measure of skewness < 0 leads to negative skewness.

Calculate the coefficient of skewness of the following data by using Karl Pearson's method.

2 3 3 4 4 6 6

Step 1. Find the mean:

Step 2. Find the standard deviation:


Step 3. Find the coefficient of skeness: (negative skewness)


  • Skewness is dimensionless.
  • Skewness is independent of change of scale.