Question

Does it make sense?

8. Incomes. The distribution of annual incomes of U.S. adults is a normal distribution.

Answer 1

Now it is said that the distribution of annual incomes of U.S. adults is a normal distribution.

Now what is a normal distribution?

A random variable X is said to have a normal distribution it its PDF(Probability Density Function) is given by,

Properties of Normal Distribution

• The normal distribution is symmetric
• It is an unimodal distribution
• The mean, median and mode are all equal

Now in fact in real life no data exactly follows a specific distribution.

In this case it is said the distribution of annual incomes of U.S. adults is a normal distribution which does not implies that the distribution exactly follows a normal distribution but what we can say about this is that the distribution of income is approximately symmetric and that the income distribution is unimodal and the mean, median and the mode of the income distribution are approximately equal so that is the reason why we can that the income distribution approximately follows a normal distribution.

In the exact sense offcourse it is not correct because no distribution exactly follows a normal distribution but if we consider it as a sense of approximation we can say that yes the income distribution follows a normal distribution provided the properties of normal distribution are approximately met by the income distribution.

Now why do we need a distribution like in this case income distribution to follow a distribution approximately like in this case normal distribution. Now there are many parametric tests in statistics which needs distributions to follow a normal distribution which in return helps us calculating test statistics, confidence intervals etc.