Why Confidence Level 95% is -1.65? I'm learning about Value at Risk for Investing which got me into statistical questions.
Here it's said that
For a confidence level of 95%, # of Standard Deviations (σ) = -1.65 x σ
And for a confidence level of 99%, # of Standard Deviations (σ) = -2.33 x σ
My question is, where do we get the -1.65 from? I've googled and found that for investing, the 95% confidence level always uses -1.65 in here, here, and this youtube video.
However when from the answers on this post, you can see that people are sure that -1.65 belongs to a 90% confidence level. Also see this answer.
So which one is correct? And how does it get the number from anyway? (Sorry I did not learn about statistics, please share a source to learn if you have)
 A: The sources you mention assume normal distribution, or use the normal distribution to approximate the underlying distribution. You are probably familiar with the 68-95-99.7 rule, mean $\pm 1.65$ standard deviations covers the middle $90\%$, with $\pm 2.58$ it's the $99\%$. In the links, probably somebody mixed both.
A: As a bit of an expansion to the answer by @Tim :
First, the sources are assuming that the observed values follow a normal distribution.
Second, the sources are using "confidence level" in a particular sense.  By "95% confidence", they are saying to report the 5th percentile of observed values. Which is essentially the highest value in the bottom 5% of the observed values.
Assuming a normal distribution, this is approximately mean - 1.65 * sd.
Practically speaking, if you have the actual observed values, looking at the 5th percentile has the advantage of not needing to assume a normal distribution of values.
As you might imagine, mean + 1.65 * sd (plus !) approximates the 95th percentile, if this were also of interest.
So, mean ± 1.65 * sd captures the middle 90% of the observations.
You can run the following R code at https://rdrr.io/snippets/ .  It will generate random numbers that are approximately normal, show you a histogram, and calculate the "95% confidence" value as suggested by your sources, by two methods.  You can change the values for NumberOfObs, Mean, and StdDev.
NumberOfObs = 100
Mean = 0
StdDev = 10

A = rnorm(NumberOfObs, Mean, StdDev)

SD = sd(A)

mean(A) - 1.645 * SD

quantile(A, 0.05)

hist(A)

A: Tim is correct.  If you want to know how the 1.65 is calculated.
It is the inverse of the standard normal cumulative distribution.  You can use the excel formula normsinv to calculate it. 50% will give a value of zero.  The probability can be from 0 to 1, although these values return an error in excel.
