Solved – How to calculate quantiles

descriptive statisticsquantilesself-study

Could someone please show(!) and explain step-by-step how to calculate the theoretical quantiles for the following normally distributed dataset:

{1,1,2,2,2,3,3,5,8,14,24,40}

Given this table:

http://imgur.com/5oHgSVu

EDIT: I am looking for the numbers at the bottom. The original "question": http://imgur.com/TZ4V1RG

Best Answer

These (indeed - see comments) are theoretical quantiles of the normal distribution for different $\alpha_j$. These are the values such that $\alpha_j$ of the probability mass of a standard normally distributed random variable is to the left of that value.

The reported quantiles are following:

> round(pnorm(c(-1.73,-1.15,-.81,-.55,-.32,-.1,.1,.32,.55,.81,1.15,1.73)),3)
 [1] 0.042 0.125 0.209 0.291 0.374 0.460 0.540 0.626 0.709 0.791 0.875 0.958

So, this for example means that $-1.73$ is the 4.2%-quantile of the standard normal distribution, the value such that smaller realizations from a standard normal distribution will occur with probability 4.2%.

They are "equally spaced" (evenredig verdeeld) in the sense that there is a roughly constant amount of probability mass between each pair of neighboring quantiles:

> round(diff(x),3)
 [1] 0.083 0.084 0.082 0.083 0.086 0.080 0.086 0.083 0.082 0.084 0.083

As the cdf of the standard normal is not available in closed form, calculating these values by hand is not possible and numerical approximations are used.

Related Question