Moment coefficient of kurtosis calculator for ungrouped data

Kurtosis

Kurtosis is the peakedness of a frequency curve. Even if two curves have the same average, dispersion and skewness, one may have higher (or lower) concentration of values near the mode, and in this case, its frequency curve will show a sharper peak (or flatter peak) than the other. This characteristics of a frequency distribution is known as kurtosis.

The literal meaning of kurtosis is peakedness or flatness of a frequency curve.

  • A frequency curve is said to be leptokurtic, if it has a higher peak than the normal curve.
  • A frequency curve is said to be platykurtic, if it has a lower peak than the normal curve.
  • A frequency curve is said to be mesokurtic, if it is neither peaked nor flatted.

Moment coefficient of kurtosis for ungrouped data

Let $x_1, x_2,\cdots, x_n$ be $n$ observations. The 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 kurtosis (also known as Pearson's moment coefficient of kurtosis) is denoted by $\beta_2$ and is defined as

$\beta_2=\dfrac{m_4}{m_2^2}$

The moment coefficient of kurtosis $\gamma_2$ is defined as

$\gamma_2=\beta_2-3$

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
  • $m_4 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4$ is fourth sample central moment

Interpretation of moment coefficient of kurtosis

  • If $\gamma_2 > 0$ or $\beta_2 > 3$, then the data is leptokurtic.
  • If $\gamma_2 = 0$ or $\beta_2 = 3$, then the data is mesokurtic.
  • If $\gamma_2 < 0$, or $\beta_2 < 3$ then the data is platykurtic.

Moment Coefficient of Kurtosis Calculator for ungrouped data

Use this calculator to find the Coefficient of Kurtosis based on moments for ungrouped (raw) data.

Moment coeff. of kurtosis
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 Kurtosis :($\beta_2$)
Coeff. of Kurtosis :($\gamma_2$)

How to calculate Moment Coefficient of kurtosis for ungrouped data?

Step 1 - Enter the $x$ values separated by commas

Step 2 - Click on "Calculate" button to get moment coefficient of kurtosis 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 kurtosis

Moment coefficient of kurtosis Example 1

The hourly earning (in dollars) of sample of 7 workers are : 27,27,24,26,25,24,22.

Compute coefficient of kurtosis based on moments.

Solution

The mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{175}{7}\\ &=25 \text{ dollars} \end{aligned} $$

$x$ $(x-xb)$ $(x-xb)^2$ $(x-xb)^4$
27 2 4 16
27 2 4 16
24 -1 1 1
26 1 1 1
25 0 0 0
24 -1 1 1
22 -3 9 81
Total 175 0 20 116

Second sample central moment

The second sample central moment is

$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{20}{7}\\ &=2.8571 \end{aligned} $$

Fourth sample central moment

The fourth sample central moment is

$$ \begin{aligned} m_4 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4\\ &=\frac{116}{7}\\ &=16.5714 \end{aligned} $$

Coefficient of Kurtosis

The coefficient of kurtosis based on moments ($\beta_2$) is

$$ \begin{aligned} \beta_2 &=\frac{m_4}{m_2^2}\\ &=\frac{(16.5714)}{(2.8571)^2}\\ &=\frac{16.5714}{8.163}\\ &=2.0301 \end{aligned} $$

The coefficient of kurtosis based on moments ($\gamma_2$) is

$$ \begin{aligned} \gamma_2 &=\beta_2-3\\ &=2.0301 -3\\ &=-0.9699 \end{aligned} $$

As the value of $\gamma_2 < 0$, the data is $\text{platy-kurtic}$.

Moment coefficient of kurtosis 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 moment coefficient of kurtosis and interpret.

Solution

The mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1611}{20}\\ &=80.55 \text{ mg/dl} \end{aligned} $$

$x$ $(x-xb)$ $(x-xb)^2$ $(x-xb)^4$
75 -5.55 30.8025 948.7940
80 -0.55 0.3025 0.0915
72 -8.55 73.1025 5343.9755
78 -2.55 6.5025 42.2825
82 1.45 2.1025 4.4205
85 4.45 19.8025 392.1390
73 -7.55 57.0025 3249.2850
75 -5.55 30.8025 948.7940
97 16.45 270.6025 73225.7130
87 6.45 41.6025 1730.7680
84 3.45 11.9025 141.6695
76 -4.55 20.7025 428.5935
73 -7.55 57.0025 3249.2850
79 -1.55 2.4025 5.7720
99 18.45 340.4025 115873.8620
86 5.45 29.7025 882.2385
83 2.45 6.0025 36.0300
76 -4.55 20.7025 428.5935
78 -2.55 6.5025 42.2825
73 -7.55 57.0025 3249.2850
Total 1611 0.00 1084.9500 210223.8745

Second sample central moment

The second sample central moment is

$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{1084.95}{20}\\ &=54.2475 \end{aligned} $$

Fourth sample central moment

The fourth sample central moment is

$$ \begin{aligned} m_4 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4\\ &=\frac{210223.8745}{20}\\ &=10511.1937 \end{aligned} $$

Coefficient of Kurtosis

The coefficient of kurtosis based on moments ($\beta_2$) is

$$ \begin{aligned} \beta_2 &=\frac{m_4}{m_2^2}\\ &=\frac{(10511.1937)}{(54.2475)^2}\\ &=\frac{10511.1937}{2942.7913}\\ &=3.5718 \end{aligned} $$

The coefficient of kurtosis based on moments ($\gamma_2$) is

$$ \begin{aligned} \gamma_2 &=\beta_2-3\\ &=3.5718 -3\\ &=0.5718 \end{aligned} $$

As the value of $\gamma_2 > 0$, the data is $\text{lepto-kurtic}$.

Moment coefficient of kurtosis 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 moment coefficient of kurtosis and interpret.

Solution

The mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{610}{25}\\ &=24.4 \text{ dollars} \end{aligned} $$

$x$ $(x-xb)$ $(x-xb)^2$ $(x-xb)^4$
20 -4.4 19.36 374.8096
28 3.6 12.96 167.9616
30 5.6 31.36 983.4496
18 -6.4 40.96 1677.7216
27 2.6 6.76 45.6976
19 -5.4 29.16 850.3056
22 -2.4 5.76 33.1776
21 -3.4 11.56 133.6336
24 -0.4 0.16 0.0256
25 0.6 0.36 0.1296
18 -6.4 40.96 1677.7216
25 0.6 0.36 0.1296
20 -4.4 19.36 374.8096
27 2.6 6.76 45.6976
24 -0.4 0.16 0.0256
20 -4.4 19.36 374.8096
23 -1.4 1.96 3.8416
32 7.6 57.76 3336.2176
20 -4.4 19.36 374.8096
35 10.6 112.36 12624.7696
22 -2.4 5.76 33.1776
26 1.6 2.56 6.5536
25 0.6 0.36 0.1296
28 3.6 12.96 167.9616
31 6.6 43.56 1897.4736
Total 610 0.0 502.00 25185.0400

Second sample central moment

The second sample central moment is

$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{502}{25}\\ &=20.08 \end{aligned} $$

Fourth sample central moment

The fourth sample central moment is

$$ \begin{aligned} m_4 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4\\ &=\frac{25185.04}{25}\\ &=1007.4016 \end{aligned} $$

Coefficient of Kurtosis

The coefficient of kurtosis based on moments ($\beta_2$) is

$$ \begin{aligned} \beta_2 &=\frac{m_4}{m_2^2}\\ &=\frac{(1007.4016)}{(20.08)^2}\\ &=\frac{1007.4016}{403.2064}\\ &=2.4985 \end{aligned} $$

The coefficient of kurtosis based on moments ($\gamma_2$) is

$$ \begin{aligned} \gamma_2 &=\beta_2-3\\ &=2.4985 -3\\ &=-0.5015 \end{aligned} $$

As the value of $\gamma_2 < 0$, the data is $\text{platy-kurtic}$.

Moment coefficient of kurtosis 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 moment coefficient of kurtosis and interpret.

Solution

The mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1305}{18}\\ &=72.5 \text{ mmHg} \end{aligned} $$

$x$ $(x-xb)$ $(x-xb)^2$ $(x-xb)^4$
65 -7.5 56.25 3164.0625
76 3.5 12.25 150.0625
64 -8.5 72.25 5220.0625
73 0.5 0.25 0.0625
74 1.5 2.25 5.0625
80 7.5 56.25 3164.0625
71 -1.5 2.25 5.0625
68 -4.5 20.25 410.0625
66 -6.5 42.25 1785.0625
81 8.5 72.25 5220.0625
79 6.5 42.25 1785.0625
75 2.5 6.25 39.0625
70 -2.5 6.25 39.0625
62 -10.5 110.25 12155.0625
83 10.5 110.25 12155.0625
63 -9.5 90.25 8145.0625
77 4.5 20.25 410.0625
78 5.5 30.25 915.0625
Total 1305 0.0 752.50 54767.1250

Second sample central moment

The second sample central moment is

$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{752.5}{18}\\ &=41.8056 \end{aligned} $$

Fourth sample central moment

The fourth sample central moment is

$$ \begin{aligned} m_4 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4\\ &=\frac{54767.125}{18}\\ &=3042.6181 \end{aligned} $$

Coefficient of Kurtosis

The coefficient of kurtosis based on moments ($\beta_2$) is

$$ \begin{aligned} \beta_2 &=\frac{m_4}{m_2^2}\\ &=\frac{(3042.6181)}{(41.8056)^2}\\ &=\frac{3042.6181}{1747.7082}\\ &=1.7409 \end{aligned} $$

The coefficient of kurtosis based on moments ($\gamma_2$) is

$$ \begin{aligned} \gamma_2 &=\beta_2-3\\ &=1.7409 -3\\ &=-1.2591 \end{aligned} $$

As the value of $\gamma_2 < 0$, the data is $\text{platy-kurtic}$.

Moment coefficient of kurtosis 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 moment coefficient of kurtosis and interpret.

Solution

The mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{2785}{20}\\ &=139.25 \text{ cm} \end{aligned} $$

$x$ $(x-xb)$ $(x-xb)^2$ $(x-xb)^4$
126 -13.25 175.5625 30822.1914
129 -10.25 105.0625 11038.1289
129 -10.25 105.0625 11038.1289
132 -7.25 52.5625 2762.8164
132 -7.25 52.5625 2762.8164
133 -6.25 39.0625 1525.8789
133 -6.25 39.0625 1525.8789
135 -4.25 18.0625 326.2539
136 -3.25 10.5625 111.5664
137 -2.25 5.0625 25.6289
137 -2.25 5.0625 25.6289
138 -1.25 1.5625 2.4414
141 1.75 3.0625 9.3789
143 3.75 14.0625 197.7539
144 4.75 22.5625 509.0664
146 6.75 45.5625 2075.9414
147 7.75 60.0625 3607.5039
152 12.75 162.5625 26426.5664
154 14.75 217.5625 47333.4414
161 21.75 473.0625 223788.1289
Total 2785 0.00 1607.7500 365915.1405

Second sample central moment

The second sample central moment is

$$ \begin{aligned} m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\ &=\frac{1607.75}{20}\\ &=80.3875 \end{aligned} $$

Fourth sample central moment

The fourth sample central moment is

$$ \begin{aligned} m_4 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4\\ &=\frac{365915.1405}{20}\\ &=18295.757 \end{aligned} $$

Coefficient of Kurtosis

The coefficient of kurtosis based on moments ($\beta_2$) is

$$ \begin{aligned} \beta_2 &=\frac{m_4}{m_2^2}\\ &=\frac{(18295.757)}{(80.3875)^2}\\ &=\frac{18295.757}{6462.1502}\\ &=2.8312 \end{aligned} $$

The coefficient of kurtosis based on moments ($\gamma_2$) is

$$ \begin{aligned} \gamma_2 &=\beta_2-3\\ &=2.8312 -3\\ &=-0.1688 \end{aligned} $$

As the value of $\gamma_2 < 0$, the data is $\text{platy-kurtic}$.

Conclusion

In this tutorial, you learned about how to calculate moment coefficient of kurtosis. You also learned about how to solve numerical problems based on moment coefficient of kurtosis for ungrouped data.

To learn more about other descriptive statistics, please refer to the following tutorial:

Descriptive Statistics

Let me know in the comments if you have any questions on Moment measure of kurtosis calculator for ungrouped data with examples and your thought on this article.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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