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:
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.
