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 Federal Highway Administration > Publications > Research Publications > LTPP Publications > 03041 > Appb.Cfm Evaluation and Analysis of LTPP Pavement Layer Thickness Data
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# Evaluation and Analysis of LTPP Pavement Layer Thickness Data

### Statistical Formulations Used in the Skewness and Kurtosis Test

The following formulations for the combined skewness and kurtosis test were developed based on the reference [41].

For the skewness, we have:

Figure 76: Equation. Skewness definition.

For kurtosis, we have:

Figure 77: Equation. Kurtosis definition.

To evaluate the skewness and kurtosis tests results, the non-dimensional skewness and kurtosis coefficients are computed, as following:

Figure 78: Equation. Non-dimensional skewness coefficient definition.

Figure 79: Equation. Non-dimensional kurtosis coefficient definition.

Based on the g1 and g2 values, the statistics and b2 are found next:

Figure 80: Equation. Definition of statistic.

Figure 81: Equation. Definition of b2 statistic.

To find z1 value, the following parameters are computed using and b2 statistics:

Figure 82: Equation. Definition of intermediate parameter A.

Figure 83: Equation. Definition of intermediate parameter B.

Figure 84: Equation. Definition of intermediate parameter C.

Figure 85: Equation. Definition of intermediate parameter D.

Figure 86: Equation. Definition of intermediate parameter E.

The corresponding z1 value used as a skewness test statistic is the following:

Figure 87: Equation. Definition of skewness test statistic z1.

To find z2 value, the following intermediate parameters are computed next:

Figure 88: Equation. Definition of the mean of intermediate parameter meanb2.

Figure 89: Equation. Definition of the variance of intermediate parameter varb2.

Figure 90: Equation. Definition of intermediate parameter F.

Figure 91: Equation. Definition of intermediate parameter G.

Figure 92: Equation. Definition of intermediate parameter H.

The corresponding z2 value used as a kurtosis test statistic is the following:

Figure 93: Equation. Definition of kurtosis test statistic z2.

The z1 and z2 statistics are used to obtain the p-values (the probability that values of the standard normal distribution are more extreme than the computed z1 and z2 statistics).