Section 1.7 Other Statistical Point Measures
Above, we have investigated statistical measures that help determine the middle and the spread of a given data set. There are however other metrics available that help describe the distribution of that data. Skewness is one of those metrics and describes any lack of symmetry of the data set’s distribution and whether data is stretched out to one side or the other.
Consider a standard die (a cube with six sides often used in games) and pick a desired side. Continue to roll the die until that side appears the first time and report the number of rolls that it took. Repeat this experiment 12 times and collect the various responses.
- Collect the response numbers on a number line.
- What would be the set of "possible" response numbers?
- Discuss whether you would ever get a response number of (say) 50 rolls.
- Would you expect any symmetries in general if this experiment were repeated a large number of times?
Definition 1.7.2. Skewness.
For population data, the Skewness of \(x_1, x_2, ..., x_n\) is given by
\begin{equation*}
\gamma_1 = \frac{1}{\sigma^3} \frac{\sum_{k=1}^n ( x_k-\mu )^3}{n}.
\end{equation*}
For sample data, the Skewness of \(x_1, x_2, ..., x_n\) is given by
\begin{equation*}
g_1 = \frac{1}{s^3} \frac{\sum_{k=1}^n ( x_k-\overline{x} )^3}{n-1}.
\end{equation*}
A positive skewness indicates that the positive \((x_k - \mu)^3\) terms (likewise \((x_k - \overline{x})^3\) terms) overwhelm the negative terms. So, a positive skewness indicates that the data set is strung out to the right. Likewise, a negative skewness indicates a data set that is strung out to the left.
Definition 1.7.3. Kurtosis.
For population data, the Kurtosis of \(x_1, x_2, ..., x_n\) is given by
\begin{equation*}
\gamma_2 = \frac{1}{\sigma^4} \frac{\sum_{k=1}^n ( x_k-\mu )^4}{n}.
\end{equation*}
For sample data, the Kurtosis of \(x_1, x_2, ..., x_n\) is given by
\begin{equation*}
g_2 = \frac{1}{s^4} \frac{\sum_{k=1}^n ( x_k-\overline{x} )^4}{n-1}.
\end{equation*}
Data might tend to be clustered around the mean. The "kurtosis" can be used to measure how closely data resembles a "bell-shaped" collection. However, note that data values that are far from the mean will generate relatively larger contributions when the difference in the formula is raised to the fourth power. So, the kurtosis that is relatively large can indicate the presence of "outliers"...that is, a significant number of data values far from the mean. In this case, notice that that would also preclude the clumping of data in the middle (since lots are far from the middle) and so a possibly flatter distribution would be indicated by a larger kurtosis.
A kurtosis of 3 indicates that the data is perfectly bell shaped (a "normal" distribution).
Theorem 1.7.4. Alternate Formulas for Skewness and Kurtosis.
Skewness =
\begin{equation*}
\gamma_1 = \frac{1}{\sigma^3}
\left [ \frac{\sum_{k=1}^n x_k^3}{n} - 3 v \mu - \mu^3 \right ]
\\ = \frac{1}{\sigma^3}
\left [ \frac{\sum_{k=1}^n x_k^3}{n} - 3 \mu \sum_{k=1}^n \frac{x_k^2}{n} + 2 \mu^3 \right ]
\end{equation*}
and Kurtosis =
\begin{equation*}
\gamma_2 = \frac{1}{\sigma^4} \left [ \frac{\sum_{k=1}^n x_k^4}{n} - 4 \mu \frac{\sum_{k=1}^n x_k^3 }{n} + 6 \mu^2 v + 3 \mu^4
\right ]
\\ = \frac{1}{\sigma^4}
\left [ \frac{\sum_{k=1}^n x_k^4}{n} - 4 \mu \frac{\sum_{k=1}^n x_k^3 }{n} + 6 \mu^2 \frac{\sum_{k=1}^n x_k^2}{n} - 3 \mu^4 \right ]
\end{equation*}
Similar formulas can be used for sample data with \(\mu\) replaced with \(\overline{x}\) and then by adjusting at the end with the factor \(\frac{n}{n-1}\) in the same way that was used when computing the sample variance. Proof.
For skewness, expand the cubic and break up the sum. Factoring out constants (such as \(\mu\)) gives
\begin{align*}
& \frac{\sum_{k=1}^n ( x_k-\mu )^3}{n}\\
& = \frac{\sum_{k=1}^n x_k^3}{n} - 3 \mu \frac{\sum_{k=1}^n x_k^2 }{n} + 3 \mu^2 \frac{\sum_{k=1}^n x_k}{n} - \frac{\sum_{k=1}^n \mu^3}{n}\\
& = \frac{\sum_{k=1}^n x_k^3}{n} - 3 \mu (v + \mu^2) + 3 \mu^3 - \mu^3\\
& = \frac{\sum_{k=1}^n x_k^3}{n} - 3 \mu v - \mu^3\\
& = \frac{\sum_{k=1}^n x_k^3}{n} - 3 \mu \left ( \sum_{k=1}^n \frac{x_k^2}{n} - \mu^2 \right ) - \mu^3\\
& = \frac{\sum_{k=1}^n x_k^3}{n} - 3 \mu \sum_{k=1}^n \frac{x_k^2}{n} + 2\mu^2.
\end{align*}
and divide by the cube of the standard deviation to finish. Note that the first expansion in the derivation above can be used quickly if the data is collected in a table and powers easily computed.
For kurtosis, similarly expand the quartic and break up the sum as before. Then,
\begin{align*}
& \frac{\sum_{k=1}^n ( x_k-\mu )^4}{n}\\
& = \frac{\sum_{k=1}^n x_k^4}{n} - 4 \mu \frac{\sum_{k=1}^n x_k^3 }{n} + 6 \mu^2 \frac{\sum_{k=1}^n x_k^2}{n} - 4 \mu^3 \frac{\sum_{k=1}^n x_k}{n} + \frac{\sum_{k=1}^n \mu^4}{n}\\
& = \frac{\sum_{k=1}^n x_k^4}{n} - 4 \mu \frac{\sum_{k=1}^n x_k^3 }{n} + 6 \mu^2 (v + \mu^2) - 4 \mu^4 + \mu^4\\
& = \frac{\sum_{k=1}^n x_k^4}{n} - 4 \mu \frac{\sum_{k=1}^n x_k^3 }{n} + 6 \mu^2 v - 3 \mu^4
\end{align*}
and then divide by the fourth power of the standard deviation. Note again that the first expansion in the derivation above might also be a useful shortcut.
The interactive cell below creates the various sample measures using the original formulas and then recreates them using the alternate formulas.
It should be noted that there are other possible ways to come up with interesting alternate ways to compute the kurtosis, etc. based on previous values. The ones presented above relate back to the mean and variance whose values might be more readily obtained from calculators for example.
Going back to a previous example...
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
So, \(\Sigma x_k = 9\) and \(\Sigma x_k^2 = 35\) as before and so
| \(x_k\) | \(x_k^2\) | \(x_k^3\) | \(x_k^4\) |
| 1 | 1 | 1 | 1 |
| -1 | 1 | -1 | 1 |
| 0 | 0 | 0 | 0 |
| 2 | 4 | 8 | 16 |
| 2 | 4 | 8 | 16 |
| 5 | 25 | 125 | 625 |
\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.
