Skewness
Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A data set is symmetric if it looks the same to the left and right of the center point. The data set is said to be skewed if it is not symmetric. The data set is said to be positively (negatively) skewed if it has a longer tail towards right (left). The degree of skewness is measured by its coefficient.
Various measures of skewness are
- Karl Pearson's measure of skewness
- Bowley's measure of skewness
- Kelly's measure of skewness
- Moments measure of skewness
Moment coefficient of skewness for ungrouped data
Let $x_1, x_2,\cdots, x_n$ be $n$ observations. The sample mean of $X$ is denoted by $\overline{x}$ and is given by
$$ \begin{eqnarray*} \overline{x}& =\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*} $$
The moment coefficient of skewness is denoted by $\beta_1$ and is defined as
$$\beta_1=\dfrac{m_3^2}{m_2^3}$$
The drawback of $\beta_1$ coefficient of skewness is that, it is always positive.
The moment coefficient of skewness is denoted by $\gamma_1$ and is defined as
$$\gamma_1=\sqrt{\beta_1}=\dfrac{m_3}{m_2^{3/2}}$$
where
$n$total number of observations$\overline{x}$sample mean$m_2 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2$is second sample central moment$m_3 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3$is third sample central moment
Interpretation of coefficient of skewness
- If $\gamma_1 > 0$, then the data is positively skewed.
- If $\gamma_1 = 0$, then the data is symmetric (i.e., absence of skewness).
- If $\gamma_1 < 0$, then the data is negatively skewed.
Moment Coefficient of Skewness Calculator for ungrouped data
Use this calculator to find the Coefficient of Skewness based on moments for ungrouped (raw) data.
| Moment coeff. of Skewness | |
|---|---|
| Enter the X Values (Separated by comma,) | |
| Results | |
| Number of Obs. (n): | |
| Mean of X values: | |
| First Central Moment :($m_1$) | |
| Second Central Moment :($m_2$) | |
| Third Central Moment :($m_3$) | |
| Fourth Central Moment :($m_4$) | |
| Coeff. of Skewness :($\beta_1$) | |
| Coeff. of Skewness :($\gamma_1$) | |
How to calculate Moment Coefficient of Skewness for ungrouped data?
Step 1 - Enter the $x$ values separated by commas
Step 2 - Click on "Calculate" button to get moment coefficient of skewness for ungrouped data
Step 3 - Gives the output as number of observations $n$
Step 4 - Gives the mean, $m_1$,$m_2$,$m_3$,$m_4$, $\beta_1$ and $\gamma_1$.
Step 5 - Gives output as Moment Coefficient of Skewness
Moment Coefficient of Skewness Example 1
The hourly earning (in dollars) of sample of 7 workers are : $27,27,24,26,25,24,22$.
Compute coefficient of skewness based on moments and interpret it.
Solution
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{175}{7}\\ &=25 \text{ dollars} \end{aligned} $$
| $x$ | $(x-xb)$ | $(x-xb)^2$ | $(x-xb)^3$ | |
|---|---|---|---|---|
| 27 | 2 | 4 | 8 | |
| 27 | 2 | 4 | 8 | |
| 24 | -1 | 1 | -1 | |
| 26 | 1 | 1 | 1 | |
| 25 | 0 | 0 | 0 | |
| 24 | -1 | 1 | -1 | |
| 22 | -3 | 9 | -27 | |
| Total | 175 | 0 | 20 | -12 |
Second sample central moment
The second sample central moment is
$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{20}{7}\\ &=2.8571 \end{aligned} $$
Third sample central moment
The third sample central moment is
$$ \begin{aligned} m_3 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3\\ &=\frac{-12}{7}\\ &=-1.7143 \end{aligned} $$
Coefficient of Skewness
The coefficient of skewness based on moments ($\beta_1$) is
$$ \begin{aligned} \beta_1 &=\frac{m_3^2}{m_2^3}\\ &=\frac{(-1.7143)^2}{(2.8571)^3}\\ &=\frac{2.9388}{23.3226}\\ &=0.126 \end{aligned} $$
The coefficient of skewness based on moments ($\gamma_1$) is
$$ \begin{aligned} \gamma_1 &=\frac{m_3}{m_2^{3/2}}\\ &=\frac{-1.7143}{(2.8571)^{3/2}}\\ &=\frac{-1.7143}{4.8293}\\ &=-0.355 \end{aligned} $$
As the value of $\gamma_1 < 0$, the data is $\text{negatively skewed}$.
Moments Coefficient of Skewness Example 2
Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:
75, 80, 72, 78, 82, 85, 73, 75, 97, 87,
84, 76, 73, 79, 99, 86, 83, 76, 78, 73.Compute coefficient of skewness based on moments and interpret it.
Solution
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1611}{20}\\ &=80.55 \text{ mg/dl} \end{aligned} $$
| $x$ | $(x-xb)$ | $(x-xb)^2$ | $(x-xb)^3$ | |
|---|---|---|---|---|
| 75 | -5.55 | 30.8025 | -170.9539 | |
| 80 | -0.55 | 0.3025 | -0.1664 | |
| 72 | -8.55 | 73.1025 | -625.0264 | |
| 78 | -2.55 | 6.5025 | -16.5814 | |
| 82 | 1.45 | 2.1025 | 3.0486 | |
| 85 | 4.45 | 19.8025 | 88.1211 | |
| 73 | -7.55 | 57.0025 | -430.3689 | |
| 75 | -5.55 | 30.8025 | -170.9539 | |
| 97 | 16.45 | 270.6025 | 4451.4111 | |
| 87 | 6.45 | 41.6025 | 268.3361 | |
| 84 | 3.45 | 11.9025 | 41.0636 | |
| 76 | -4.55 | 20.7025 | -94.1964 | |
| 73 | -7.55 | 57.0025 | -430.3689 | |
| 79 | -1.55 | 2.4025 | -3.7239 | |
| 99 | 18.45 | 340.4025 | 6280.4261 | |
| 86 | 5.45 | 29.7025 | 161.8786 | |
| 83 | 2.45 | 6.0025 | 14.7061 | |
| 76 | -4.55 | 20.7025 | -94.1964 | |
| 78 | -2.55 | 6.5025 | -16.5814 | |
| 73 | -7.55 | 57.0025 | -430.3689 | |
| Total | 1611 | 0.00 | 1084.9500 | 8825.5045 |
Second sample central moment
The second sample central moment is
$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{1084.95}{20}\\ &=54.2475 \end{aligned} $$
Third sample central moment
The third sample central moment is
$$ \begin{aligned} m_3 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3\\ &=\frac{8825.5045}{20}\\ &=441.2752 \end{aligned} $$
Coefficient of Skewness
The coefficient of skewness based on moments ($\beta_1$) is
$$ \begin{aligned} \beta_1 &=\frac{m_3^2}{m_2^3}\\ &=\frac{(441.2752)^2}{(54.2475)^3}\\ &=\frac{194723.8021}{159639.0687}\\ &=1.2198 \end{aligned} $$
The coefficient of skewness based on moments ($\gamma_1$) is
$$ \begin{aligned} \gamma_1 &=\frac{m_3}{m_2^{3/2}}\\ &=\frac{441.2752}{(54.2475)^{3/2}}\\ &=\frac{441.2752}{399.5486}\\ &=1.1044 \end{aligned} $$
As the value of $\gamma_1 > 0$, the data is $\text{positively skewed}$.
Moments Coefficient of Skewness Example 3
The following data gives the hourly wage rates (in dollars) of 25 employees of a company.
20, 28, 30, 18, 27, 19, 22, 21, 24, 25,
18, 25, 20, 27, 24, 20, 23, 32, 20, 35,
22, 26, 25, 28, 31.
Compute coefficient of skewness based on moments and interpret it.
Solution
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{610}{25}\\ &=24.4 \text{ dollars} \end{aligned} $$
| $x$ | $(x-xb)$ | $(x-xb)^2$ | $(x-xb)^3$ | |
|---|---|---|---|---|
| 20 | -4.4 | 19.36 | -85.184 | |
| 28 | 3.6 | 12.96 | 46.656 | |
| 30 | 5.6 | 31.36 | 175.616 | |
| 18 | -6.4 | 40.96 | -262.144 | |
| 27 | 2.6 | 6.76 | 17.576 | |
| 19 | -5.4 | 29.16 | -157.464 | |
| 22 | -2.4 | 5.76 | -13.824 | |
| 21 | -3.4 | 11.56 | -39.304 | |
| 24 | -0.4 | 0.16 | -0.064 | |
| 25 | 0.6 | 0.36 | 0.216 | |
| 18 | -6.4 | 40.96 | -262.144 | |
| 25 | 0.6 | 0.36 | 0.216 | |
| 20 | -4.4 | 19.36 | -85.184 | |
| 27 | 2.6 | 6.76 | 17.576 | |
| 24 | -0.4 | 0.16 | -0.064 | |
| 20 | -4.4 | 19.36 | -85.184 | |
| 23 | -1.4 | 1.96 | -2.744 | |
| 32 | 7.6 | 57.76 | 438.976 | |
| 20 | -4.4 | 19.36 | -85.184 | |
| 35 | 10.6 | 112.36 | 1191.016 | |
| 22 | -2.4 | 5.76 | -13.824 | |
| 26 | 1.6 | 2.56 | 4.096 | |
| 25 | 0.6 | 0.36 | 0.216 | |
| 28 | 3.6 | 12.96 | 46.656 | |
| 31 | 6.6 | 43.56 | 287.496 | |
| Total | 610 | 0.0 | 502.00 | 1134.000 |
Second sample central moment
The second sample central moment is
$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{502}{25}\\ &=20.08 \end{aligned} $$
Third sample central moment
The third sample central moment is
$$ \begin{aligned} m_3 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3\\ &=\frac{1134}{25}\\ &=45.36 \end{aligned} $$
Coefficient of Skewness
The coefficient of skewness based on moments ($\beta_1$) is
$$ \begin{aligned} \beta_1 &=\frac{m_3^2}{m_2^3}\\ &=\frac{(45.36)^2}{(20.08)^3}\\ &=\frac{2057.5296}{8096.3845}\\ &=0.2541 \end{aligned} $$
The coefficient of skewness based on moments ($\gamma_1$) is
$$ \begin{aligned} \gamma_1 &=\frac{m_3}{m_2^{3/2}}\\ &=\frac{45.36}{(20.08)^{3/2}}\\ &=\frac{45.36}{89.9799}\\ &=0.5041 \end{aligned} $$
As the value of $\gamma_1 > 0$, the data is $\text{positively skewed}$.
Moments Coefficient of Skewness Example 4
Diastolic blood pressure (in mmHg) of a sample of 18 patients admitted to the hospitals are as follows:
65,76,64,73,74,80,71,68,66,
81,79,75,70,62,83,63,77,78.Compute coefficient of skewness based on moments and interpret it.
Solution
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1305}{18}\\ &=72.5 \text{ mmHg} \end{aligned} $$
| $x$ | $(x-xb)$ | $(x-xb)^2$ | $(x-xb)^3$ | |
|---|---|---|---|---|
| 65 | -7.5 | 56.25 | -421.875 | |
| 76 | 3.5 | 12.25 | 42.875 | |
| 64 | -8.5 | 72.25 | -614.125 | |
| 73 | 0.5 | 0.25 | 0.125 | |
| 74 | 1.5 | 2.25 | 3.375 | |
| 80 | 7.5 | 56.25 | 421.875 | |
| 71 | -1.5 | 2.25 | -3.375 | |
| 68 | -4.5 | 20.25 | -91.125 | |
| 66 | -6.5 | 42.25 | -274.625 | |
| 81 | 8.5 | 72.25 | 614.125 | |
| 79 | 6.5 | 42.25 | 274.625 | |
| 75 | 2.5 | 6.25 | 15.625 | |
| 70 | -2.5 | 6.25 | -15.625 | |
| 62 | -10.5 | 110.25 | -1157.625 | |
| 83 | 10.5 | 110.25 | 1157.625 | |
| 63 | -9.5 | 90.25 | -857.375 | |
| 77 | 4.5 | 20.25 | 91.125 | |
| 78 | 5.5 | 30.25 | 166.375 | |
| Total | 1305 | 0.0 | 752.50 | -648.000 |
Second sample central moment
The second sample central moment is
$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{752.5}{18}\\ &=41.8056 \end{aligned} $$
Third sample central moment
The third sample central moment is
$$ \begin{aligned} m_3 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3\\ &=\frac{-648}{18}\\ &=-36 \end{aligned} $$
Coefficient of Skewness
The coefficient of skewness based on moments ($\beta_1$) is
$$ \begin{aligned} \beta_1 &=\frac{m_3^2}{m_2^3}\\ &=\frac{(-36)^2}{(41.8056)^3}\\ &=\frac{1296}{73063.9896}\\ &=0.0177 \end{aligned} $$
The coefficient of skewness based on moments ($\gamma_1$) is
$$ \begin{aligned} \gamma_1 &=\frac{m_3}{m_2^{3/2}}\\ &=\frac{-36}{(41.8056)^{3/2}}\\ &=\frac{-36}{270.3035}\\ &=-0.1332 \end{aligned} $$
As the value of $\gamma_1 < 0$, the data is $\text{negatively skewed}$.
Moments Coefficient of Skewness Example 5
The following data are the heights, correct to the nearest centimeters, for a group of children:
126, 129, 129, 132, 132, 133, 133, 135, 136, 137,
137, 138, 141, 143, 144, 146, 147, 152, 154, 161 Compute coefficient of skewness based on moments and interpret it.
Solution
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{2785}{20}\\ &=139.25 \text{ cm} \end{aligned} $$
| $x$ | $(x-xb)$ | $(x-xb)^2$ | $(x-xb)^3$ | |
|---|---|---|---|---|
| 126 | -13.25 | 175.5625 | -2326.2031 | |
| 129 | -10.25 | 105.0625 | -1076.8906 | |
| 129 | -10.25 | 105.0625 | -1076.8906 | |
| 132 | -7.25 | 52.5625 | -381.0781 | |
| 132 | -7.25 | 52.5625 | -381.0781 | |
| 133 | -6.25 | 39.0625 | -244.1406 | |
| 133 | -6.25 | 39.0625 | -244.1406 | |
| 135 | -4.25 | 18.0625 | -76.7656 | |
| 136 | -3.25 | 10.5625 | -34.3281 | |
| 137 | -2.25 | 5.0625 | -11.3906 | |
| 137 | -2.25 | 5.0625 | -11.3906 | |
| 138 | -1.25 | 1.5625 | -1.9531 | |
| 141 | 1.75 | 3.0625 | 5.3594 | |
| 143 | 3.75 | 14.0625 | 52.7344 | |
| 144 | 4.75 | 22.5625 | 107.1719 | |
| 146 | 6.75 | 45.5625 | 307.5469 | |
| 147 | 7.75 | 60.0625 | 465.4844 | |
| 152 | 12.75 | 162.5625 | 2072.6719 | |
| 154 | 14.75 | 217.5625 | 3209.0469 | |
| 161 | 21.75 | 473.0625 | 10289.1094 | |
| Total | 2785 | 0.00 | 1607.7500 | 10642.8755 |
Second sample central moment
The second sample central moment is
$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{1607.75}{20}\\ &=80.3875 \end{aligned} $$
Third sample central moment
The third sample central moment is
$$ \begin{aligned} m_3 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3\\ &=\frac{10642.8755}{20}\\ &=532.1438 \end{aligned} $$
Coefficient of Skewness
The coefficient of skewness based on moments ($\beta_1$) is
$$ \begin{aligned} \beta_1 &=\frac{m_3^2}{m_2^3}\\ &=\frac{(532.1438)^2}{(80.3875)^3}\\ &=\frac{283177.0239}{519476.0957}\\ &=0.5451 \end{aligned} $$
The coefficient of skewness based on moments ($\gamma_1$) is
$$ \begin{aligned} \gamma_1 &=\frac{m_3}{m_2^{3/2}}\\ &=\frac{532.1438}{(80.3875)^{3/2}}\\ &=\frac{532.1438}{720.7469}\\ &=0.7383 \end{aligned} $$
As the value of $\gamma_1 > 0$, the data is $\text{positively skewed}$.
Conclusion
In this tutorial, you learned about how to calculate moment coefficient of skewness. You also learned about how to solve numerical problems based on moment coefficient of skewness for ungrouped data.
To learn more about other descriptive statistics, please refer to the following tutorial:
Let me know in the comments if you have any questions on Moment measure of Skewness calculator for ungrouped data with examples and your thought on this article.
