The kurtosis (also known as a measure of aiming) is a statistical measure, which determines the degree of concentration presented by the values of a variable around the central area of the frequency distribution.
When we measure a random variable, in general, the results that have a higher frequency are those that are around the average of the distribution. Imagine the height of the students in a class. If the average height of the class is 1.72, the most normal is that the heights of the rest of the students are around this value (with a certain degree of variability, but without being too large). If this happens, it is considered that the distribution of the random variable is distributed normally. But given the infinity of variables that can be measured, this does not always happen.
There are some variables that have a higher degree of concentration (less dispersion) of the values around their average and others, on the contrary, have a lower degree of concentration (greater dispersion) of their values around their central value. Therefore, the kurtosis informs us of what is targeted (higher concentration) or the flat (lower concentration) that is a distribution.
Types of kurtosis
Depending on the degree of kurtosis, we have three types of distributions:
- Leptocurtic:There is a large concentration of values around their average (g 2> 3)
- Mesocúrtica:There is a normal concentration of the values around its mean (g 2= 3).
- Platicúrtica:There is a low concentration of the values around its average (g 2<3).
Kurtosis measures according to data
Depending on the grouping or not of the data, one formula or another is used.
Ungrouped data:
Data grouped in frequency tables:
Data grouped in intervals:
Example of kurtosis calculation for ungrouped data
Suppose we want to calculate the kurtosis of the following distribution:
8,5,9,10,12,7,2,6,8,9,10,7.7.
First we calculate the arithmetic mean (µ), which would be 7.69.
Next, we calculate the standard deviation, which would be 2.43.
After having this data and for convenience in the calculation, a table can be made to calculate the part of the numerator (fourth moment of distribution). For the first calculation it would be: (Xi-µ) ^ 4 = (8-7.69) ^ 4 = 0.009.
| Data | (Xi-µ) ^ 4 |
| 8 | 0.0090 |
| 5 | 52.5411 |
| 9 | 2.9243 |
| 10 | 28,3604 |
| 12 | 344.3330 |
| 7 | 0.2297 |
| two | 1049.9134 |
| 6 | 8,2020 |
| 8 | 0.0090 |
| 9 | 2.9243 |
| 10 | 28,3604 |
| 7 | 0.2297 |
| 7 | 0.2297 |
| N = 13 | ∑ = 1,518.27 |
Once we have this table made, we would simply have to apply the formula described above to have the kurtosis.
g 2 = 1,518.27 / 13 * (2.43) ^ 4 = 3.34
In this case given that g 2 is greater than 3, the distribution would be leptocurtic, presenting a greater aim than the normal distribution.
Excess kurtosis
In some manuals, kurtosis is presented as excess kurtosis. In this case, it compares directly with that of the normal distribution. Since the normal distribution has kurtosis 3, to obtain the excess, we would only have to subtract 3 from our result.
Excess kurtosis = g 2 -3 = 3.34-3 = 0.34.
The interpretation of the result in this case would be as follows:
g 2 -3> 0 -> leptocortic distribution.
g 2 -3 = 0 -> meso-curt distribution (or normal).
g 2 -3 <0 -> platicuric distribution.
