kurtosis (Statistics Toolbox)

Sample kurtosis

Syntax

```
k = kurtosis(X)
k = kurtosis(X,flag)
```

Description

k = kurtosis(X) returns the sample kurtosis of X. For vectors, kurtosis(x) is the kurtosis of the elements in the vector x. For matrices kurtosis(X) returns the sample kurtosis for each column of X.

Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 3; distributions that are less outlier-prone have kurtosis less than 3.

The kurtosis of a distribution is defined as

where is the mean of x, is the standard deviation of x, and E(t) represents the expected value of the quantity t.

Note Some definitions of kurtosis subtract 3 from the computed value, so that the normal distribution has kurtosis of 0. The kurtosis function does not use this convention.

k = kurtosis(X,flag) specifies whether to correct for bias (flag = 0) or not (flag = 1, the default). When X represents a sample from a population, the kurtosis of X is biased, that is, it will tend to differ from the population kurtosis by a systematic amount that depends on the size of the sample. You can set flag = 0 to correct for this systematic bias.

Example

X = randn([5 4])
X =
    1.1650    1.6961   -1.4462   -0.3600
    0.6268    0.0591   -0.7012   -0.1356
    0.0751    1.7971    1.2460   -1.3493
    0.3516    0.2641   -0.6390   -1.2704
   -0.6965    0.8717    0.5774    0.9846
k = kurtosis(X)
k =
    2.1658    1.2967    1.6378    1.9589

See Also
mean, moment, skewness, std, var

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