How can I find percentile $P_{10}$ and $P_{90}$ for Normal Distribution with Mean as $100$ and Standard Deviation as $3$?

Question

How can I find percentile $P_{10}$ and $P_{90}$ for Normal Distribution with Mean as $100$ and Standard Deviation as $3$?

Example

I don't understand from where the $-1.28$ come. Any help would be appreciated.

Answer 1

-1.28 is the 10th percentile of the standard normal distribution.

If you're working with the distribution $\mathsf{Norm}(\mu=100,\sigma=3)$ then you can you can find its percentiles by starting with the percentiles of a standard normal distribution.

The formula for the 10th percentile of $\mathsf{Norm}(\mu=100,\sigma=3)$ is $$\mu + (-1.28)\sigma = 100 - 1.28(3) = 96.16.$$

From R statistical software, here are percentiles 10, 20, ..., 90, first of standard normal $\mathsf{Norm}(\mu=0, \sigma=1),$ which you can get (to 2 decimal places) from a printed table of the CDF of standard normal, and second of $\mathsf{Norm}(\mu=100,\sigma=3),$ which you can get from the former by using the formula above:

p = seq(10, 90, by=10);  p
[1] 10 20 30 40 50 60 70 80 90

qnorm(p/100, 0, 1)
[1] -1.2815516 -0.8416212 -0.5244005 -0.2533471
[5]  0.0000000  0.2533471  0.5244005  0.8416212
[9]  1.2815516

qnorm(p/100, 100, 3)
[1]  96.15535  97.47514  98.42680  99.23996
[5] 100.00000 100.76004 101.57320 102.52486
[9] 103.84465

Notes: In R, qnorm is the 'inverse Cumulative Distribution Function' (inverse CDF), which is sometimes called the 'quantile function'.

If you have a statistical calculator, maybe you can figure out how to get percentiles from it.

If you are using a printed normal table, try this: Look for the closest number in the body of the table to .50, .60, ..., .90. Then find the percentile value from the margin of the table. For percentiles with negative values, you may have to use the symmetry of the normal distribution.

'Percentiles' are a particular kind of quantiles (based on 1/100). Other particular kinds of quantile that you may encounter later are quartiles (based on 1/4), quintiles (based on 1/5), and deciles (based on 1/10). The lower quartile is the 25th percentile, the median is the 50th percentile.