Statistical Hydrology 1 Dr. Muhammad Ajmal Lecturer, Agri. Engg. Dept. UET Peshawar DATA TRANSFORMATION

What is a data transformation?  It is a mathematical function that is applied to all the observations of a given variable Y represents the original variable, Y* is the transformed variable, and f is a mathematical function that is applied to the data Transformation of a variable can change its distribution from a skewed distribution to a normal distribution (bell-shaped, symmetric about its center 04. Data Transformation

3 ▣ Data Transformation Transformation are used for three purposes ① To make data more symmetric ② To make data more linear, and ③ To make data more constant in variance In order to make an asymmetric distribution become more symmetric, the data can be transformed or re-expressed into new units. The new nits alter the distances between observations on a line plot. The effect is to either expand or contract the distances to extreme observations on one side of the median making it look more like the other side.

04. Data Transformation 4 ▣ Data Transformation The most commonly used transformation in water resources is the logarithm. Logs of water resources data for example, stream discharge, hydraulic conductivity and sediment concentration are often taken before statistical analyses are performed.

UP Bigger Impact Bigger Impact.. Middle rung: No transformation ( = 1) Middle rung: No transformation ( = 1) DOWN Here V represents our variable of interest. We are going to consider this variable raised to a power, i.e. V We go up the ladder to remove left skewness and down the ladder to remove right skewness. Right skewed Left skewed Tukey’s Ladder of Powers for Transformation

To remove right skewness we typically take the square root, cube root, logarithm, or reciprocal of a the variable etc., i.e. V.5, V.333, log 10 (V) (think of V 0 ), V -1, etc. To remove left skewness we raise the variable to a power greater than 1, such as squaring or cubing the values, i.e. V 2, V 3, etc. Tukey’s Ladder of Powers for Transformation

Tukey’s Ladder of Power

To remove right skewness we typically take the square root, cube root, logarithm, or reciprocal of a the variable etc., i.e. V.5, V.333, log 10 (V) (think of V 0 ), V -1, etc. To remove left skewness we raise the variable to a power greater than 1, such as squaring or cubing the values, i.e. V 2, V 3, etc. Tukey’s Ladder of Power

How can we determine if observations are normally distributed? Graphical examination Histogram or boxplot Normal quantile-quantile plot (QQ-plot) Goodness of fit tests Kolmogorov-Smirnov test Shapiro-Wilks test Anderson Darling test Transformations to Achieve Normality

How can we determine if observations are normally distributed? Transformations to Achieve Normality

Questions? 11