# 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)

• 95% confidence level or 95th percentile?
– Dave
Commented Jul 1, 2022 at 17:51

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.

• And then the top end of the middle 95% is the 95th percentile, which is probably the source of the mistake.
– Dave
Commented Jul 1, 2022 at 18:03

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)

• "Confidence" and "5th percentile of observed values" are two different things! The former is an indicator of the precision of an estimate while the latter is a statement of the data distribution.
– whuber
Commented Apr 24 at 20:04
• @whuber, right. I've seen this use of "confidence" a few times; I think always in the subject of investing. Commented Apr 25 at 14:32

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.

I imagine this comes from z-score table. If you check the z-table for z=1.65 you will see that P(Z<=z) is precisely 5% (1-0.95 = 5). The same for z=2.33, P(Z<=z) = 1%.

One sided confidence intervals use different boundary levels than two sides confidence intervals with the same confidence.

The 95% one sided interval boundary is instead the same as the 90% two sided interval boundary.

That different test (one-sided vs two-sided), or a mistake with it, can explain the differences.