# How to calculate quantiles?

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

• These are the theoretical quantiles. Dec 31, 2015 at 16:05
• Why do you need the data for the theoretical quantiles? Also, I would accept any bet that these data are not from a normal distribution. At the very least, they are rounded. Dec 31, 2015 at 16:12
• @ChristophHanck Those numbers are from slides provided by my university, stating those are normally distributed Dec 31, 2015 at 16:13
• Which book? Can you state the complete question? Also, please add the self study tag Dec 31, 2015 at 16:15
• Are you certain this exercise is not about computing sample quantiles? Dec 31, 2015 at 16:19

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.