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Skewness Formula

Skewness formula is called so because the graph plotted is displayed in skewed manner. Skewness is a measure used in statistics that helps reveal the asymmetry of a probability distribution. It can either be positive or negative, irrespective of signs. To calculate the skewness, we have to first find the mean and variance of the given data.

The formula is:

\[\large g=\sqrt{\frac{\sum_{i-1}^{n}\left(x-x_{i}\right)^{3}}{\left(n-1\right)s^{3}}}\]

Where,
x is the observations
$x_{i}$ is the mean
n is the total number of observations
s is the variance

Solved example

Question. Find the skewness in the following data.  

 

Height (inches) Class Marks Frequency
59.5 – 62.5 61 5
62.5 – 65.5 64 18
65.5 – 68.5 67 42
68.5 – 71.5 70 27
71.5 – 74.5 73 8

To know how skewed these data are as compared to other data sets, we have to  compute the skewness.

Sample size and sample mean should be found out.

N = 5 + 18 + 42 + 27 + 8 = 100

$\overline{x}=\frac{\left(61\times 5\right)+\left(64\times 18\right)+\left(67\times 43\right)+\left(70\times 27\right)+\left(73\times 8\right)}{100}$

$\overline{x}=\frac{6745}{100}=67.45$

Now with the mean we can compute the skewness.

 

Class Mark, x Frequency, f xf $\left(x-\overline{x}\right)$  $\left(x-\overline{x}\right)^{2}\times f$ $\left(x-\overline{x}\right)^{3}\times f$ 
61 5 305 -6.45 208.01 -1341.68
64 18 1152 -3.45 214.25 -739.15
67 42 2814 -0.45 8.51 -3.83
70 27 1890 2.55 175.57 447.70
73 8 584 5.55 246.42 1367.63
6745 n/a 852.75 -269.33
67.45 n/a 8.5275 -2.6933

Now, the skewness is

$g_{i}=\sqrt{\frac{\sum_{i=1}^{n}\left(x-x_{i}\right)^{3}}{\left(n-1 \right)s^{3}}}=-\frac{2.6937}{8.5275^{\frac{3}{2}}}=-0.1082$

For interpreting we have the folowing rules as per Bulmer in the year 1979:

  • If the skewness comes to less than -1 or greater than +1, the data distribution is highly skewed
  • If the skewness comes to between -1 and $-\frac{1}{2}$ or between $+\frac{1}{2}$ and +1, the data distribution is moderately skewed.
  • If the skewness is between $-\frac{1}{2}$ and $+\frac{1}{2}$,the distribution is approximately symmetric