# TEST_SKEW - Skewness Test

Calculates the p-value of the statistical test for the population skew (i.e. 3rd moment).

## Syntax

TEST_SKEW(X, Return_type, Alpha)

X is the input data sample (one/two dimensional array of cells (e.g. rows or columns))

Return_type is a switch to select the return output (1 = P-Value (default), 2 = Test Stats, 3 = Critical Value.

Method Description
1 P-Value
2 Test Statistics (e.g. Z-score)
3 Critical Value

Alpha is the statistical significance of the test (i.e. alpha). If missing or omitted, an alpha value of 5% is assumed.

## Remarks

1. The data sample may include missing values (e.g. #N/A).
2. The test hypothesis for the population distribution skewness:
$$H_{o}: S=0$$
$$H_{1}: S\neq 0$$
Where:
• $H_{o}$ is the null hypothesis.
• $H_{1}$ is the alternate hypothesis.
• $S$ is the population skew.
3. For the case in which the underlying population distribution is normal, the sample skew also has a normal sampling distribution:
$$\hat S \sim N(0,\frac{6}{T})$$
Where:
• $\hat S$ is the sample skew (i.e. 3rd moment).
• $T$ is the number of non-missing values in the data sample.
• $N(.)$ is the normal (i.e. gaussian) probability distribution function.
4. The sample data skew is calculated as:
$$\hat S(x)= \frac{\sum_{t=1}^T(x_t-\bar x)^3}{(T-1)\times \hat \sigma^3}$$
Where:
• $\hat S$ is the sample skew (i.e. 3rd moment).
• $x_i$ is the i-th non-missing value in the data sample.
• $T$ is the number of non-missing values in the data sample.
• $\hat \sigma$ is the data sample standard deviation.
5. In the case where the population skew is not zero, the mean is farther out than the median in the long tail. The underlying distribution is referred to as skewed, unbalanced, or lopsided.
6. The underlying population distribution is assumed normal (gaussian).
7. This is a two-sides (i.e. two-tails) test, so the computed p-value should be compared with half of the significance level ($\alpha/2$).

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
A B
Date Data
1/1/2008 #N/A
1/2/2008 -2.83
1/3/2008 -0.95
1/4/2008 -0.88
1/5/2008 1.21
1/6/2008 -1.67
1/7/2008 0.83
1/8/2008 -0.27
1/9/2008 1.36
1/10/2008 -0.34
1/11/2008 0.48
1/12/2008 -2.83
1/13/2008 -0.95
1/14/2008 -0.88
1/15/2008 1.21
1/16/2008 -1.67

Formula Description (Result)
=TEST_SKEW($B$2:$B$17) p-value of the test when skewness = 0 (0.4025)
=SKEW($B$3:$B$17) Sample skewness (-0.1740)