Computing skewness and kurtosis by hand can often be better organized using a table. Below, notice that the \(x_k\) column would be the given data values but the other columns you could again easily compute.

Table 1.7.5. Computing data statistics by hand

\(x_k\)	\(x_k^2\)	\(x_k^3\)	\(x_k^4\)
1	1	1	1
-1	1	-1	1
0	0	0	0
2	4	8	16
2	4	8	16
5	25	125	625

So, \(\Sigma x_k = 9\) and \(\Sigma x_k^2 = 35\) as before and so

\begin{equation*} \overline{x} = \frac{9}{6} = \frac{3}{2} \end{equation*}

and

\begin{gather*} v = \frac{25}{6} - \left ( \frac{3}{2} \right )^2 = \frac{43}{12} \approx 3.58 ,\\ s^2 = \frac{6}{5} \times v = \frac{43}{10} = 4.3,\\ \text{and so } s = \sqrt{4.3} \approx 2.07\text{.} \end{gather*}

But also, \(\Sigma x_k^3 = 141\) and \(\Sigma x_k^4 = 659\text{.}\) Use these in the Alternate Skewness and Kurtosis formulas 1.7.4 to obtain skewness of

\begin{equation*} g_1 = \frac{6}{5} \left [ \frac{141}{6} - 3 \cdot \frac{26}{6} \cdot \frac{3}{2} + 2 \left ( \frac{3}{2} \right )^3 \right ] / s^3 \end{equation*}

and kurtosis of

\begin{equation*} g_2 = \frac{6}{5} \left [ \frac{659}{6} - 4 \cdot \frac{3}{2} \cdot \frac{141}{6} + 6 \left ( \frac{3}{2} \right )^2 \cdot \frac{26}{6} - 3 \cdot \left ( \frac{3}{2} \right )^4 \right ] / s^4. \end{equation*}

Of course, these expanded formulas are much more useful when the data set is significantly larger.

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