Insights into concepts of Hypothesis Testing (Part 10)

Assumption of Normality of Parent Population

The population from which a sample is drawn is called a parent population. This parent population is usually unknown. Parametric tests under testing of hypothesis are based on the assumption of normality of parent population. Non parametric tests are not based on this assumption. Today’s discussion delves deeper into this assumption of normal population. We look at various transformations that convert a skewed (non normal) population to a normal population.

For the sake of simplicity let’s consider the following population. It is unknown under normal conditions.

Population = {1, 1, 1, 1, 1, 1, 2, 2, 2, 5, 5, 20, 25}

This is a non normal or a positively skewed data. It is represented by following frequency curve on the left side of the image. Here. Hence Mode<Median<Mean for a positively skewed data.

If the population was normal it’s simplified version will be the following.

Population = {3, 4, 5, 5, 5, 5, 5, 5, 6, 7}

It is represented by the frequency curve in the right side. Here Mean = Mode = Median = 5. For a normal data, Mean = Mode = Median.

For conducting parametric tests the parent population should be normal where mean, mode and median are very close to one another.

Some transformations like logarithmic, square root and power (1/4) can transform a skewed data by bringing the mean, mode and median closer to each other. But these transformations don’t change the shape of the frequency curve.

After we conduct logarithmic transformation to the non normal population that is {1, 1, 1, 1, 1, 1, 2, 2, 2, 5, 5, 20, 25}, Mean = 5.15, Median = 2 and Mode = 1 changes to Mean = 0.385, Median = 0.301 and Mode = 0.  After doing the square root of the original data Mean = 1.86, Median = 1.414 and Mode = 1. And if power (1/4) is done for the original data, Mean = 1.3007, Mode = 1 and Median = 1.189. But these transformations don’t change the shape of the frequency curve. This is illustrated by the image below giving the screen shot of the excel worksheet comparing different transformations.