# Moment coefficient of skewness Calculator for ungrouped data

## 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

## 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.