# How to estimate 100% confidence interval aka. what is the Z value of standard normal distribution at probability of 100%?

Thinking of the various tests and parameter estimates we perform with 99% confindence interval based on assumption of "normal distribution of errors" I asked myself a question what would be the 100% confidence interval for these estimated parameters ? This leads to formulation of normal distribution value Z at 0% or 100%. Is this virtually infinity or some specific number "big enough" ?

• since the normal distribution has support on the entire real line, it would need to be infinity – Christoph Hanck Aug 19 '15 at 8:30

1. The normal distribution covers the entire real line from $-\infty$ to $\infty$. To include 100% of the probability under the normal distribution would involve having an infinite range.

2. A "normal distribution of errors" does not imply that the distribution you use for a parameter estimate is itself normal. For example, an interval for the mean of a $N(\mu,\sigma^2)$ (where both parameters are unknown) will be based on the $t$-distribution (also infinite in range), while an interval for $\sigma^2$ would be based on the $\chi^2$ distribution (which is semi-infinite). Other parameters might be on a finite range. For example correlations lie in $(-1,1)$.

However, in the limit as simple sizes go to infinity, in each of those three examples there will be a convergence of all the corresponding pivotal quantities to normality.

More generally, 100% intervals will typically cover the entire possible range of the parameter. The benefit of taking a smaller-than-100% interval is that you can take advantage of the way the intervals become narrower with larger sample size. A 99.9% interval for the mean of a normally distributed population will tend to shrink as $n$ becomes larger, but a 100% interval won't.

The normal distribution extends to +/- infinity, therefore to cover the 100% confidence interval you need to cover this range.

You can test this in Excel or Python using the inverse T value (i'm assuming 100 degrees of freedom. Use T.INV in Excel):

In [1]: import scipy.stats
In [2]: scipy.stats.t.ppf(1, 100)
Out[2]: inf
In [3]: scipy.stats.t.ppf(0, 100)
Out[3]: -inf


Your question about a "big enough" number will depend on your population. For example, for a distribution of the height of people, it makes no sense to have a +/- infinity range, so in practice you might want to truncate your range to say the height of the tallest man in the world.