# Distribution shape

The symmetrical level of the probability distribution (or asymmetrical level)

### Symmetrical dustribution

For a perfect symmetrical distribution the skweness is zero. In real data the skewness level will be around the zero. With a smaller sample size the absolute skweness number may be higher for a symmetrical distribution.

In a symmetrical distribution Mean = Median = Mode

### Positive skewness (right)

A positive skewness represent asymmetrical distribution with long right tail, like chi-squred distribution, binomial distribution.

In a positive distribution Mean ≥ Median ≥ Mode

### Negative skewness (left)

A negative skewness represent asymmetrical distribution with long left tail.

In a negative distribution Mean ≤ Median ≤ Mode

The Kurtosis measured the level of the tails. $$population\;kurtosis=n\frac{\sum_{i=1}^{n}(x_i-\bar{x})^4}{(\sum_{i=1}^{n}(x_i-\bar{x})^2)^2}= \frac{\sum_{i=1}^{n}(x_i-\bar{x})^4}{n\sigma^2}$$ Since the "stronger" part in the formula is:

$$(x_i-\bar{x})^4$$ Values that are

**far from the average** will mostly determine the Kurtosis value, independently of the value sign. For example if the average is 10 two values of 10 and 100 will increase the kurtosis more tnan 2 values of 80: $$(100-10)^4>2(80-10)^4$$ ~65M>~48M But 2 values of 90 will increase the kurtosis more than [10,100], ~82M>65M. So it is mainly the distance from the average but also the amount if values far from the average.

## Excess Kurtosis

The kurtosis value of the normal distribution is 3. most statistical software shift the measurement to be 0 for the normal distribution. This website uses the following sample excess kurtosis formula to calculate the kurtosis. $$sample\;excess\; kurtosis=\frac{n(n+1)(n-1)*\sum_{i=1}^{n}(x_i-\bar{x})^4}{(n-2)(n-3)*\sum_{i=1}^{n}(x_i-\bar{x})^2}-\frac{3(n-1)^2}{(n-2)(n-3)}$$

### Mesokurtic - normal tail

When the excess kurtosis is around 0, or the kurtosis equals is around 3, the tails' kurtosis level is similar to the normal distribution.

### Leptokurtic - potitive excess kurtosis, **long heavy tails**

When excess kurtosis possitive, the balance is shifted toward the tails, so **usually** the peak will be **low**, but a high peak with some values far from the average may also have a potitive kurtosis!

### Platykurtic - negative excess kurtosis, **short thin tails**

When excess kurtosis possitive, the balance is shifted out of the tails, so **usually** the peak will be **high**, but a low-medium peak with no values far from the average may also have negative kurtosis!