Reconstructing normal distribution according to data ranges I have a temperature data and I believe it follows the normal distribution. The problem is that I know just values for few ranges, but I need to have the results for finer temperature classes.
So, as input I have:
    T<23°C --> 21.50%
23°<T<65°C --> 75.27%
65°<T<85°C -->  2.15%
85°<T<95°C -->  1.08%

and I know that it follow a normal distribution.
and then I need to estimate the temperature for the following ranges:
X°<T<X+5°C -->  % ; where X=-40,-35, ..., 85, 90, 95, 100,...

Since I do not have a solid mathematical question, I appreciate your help on this topic.
 A: Given any two quantiles of a normal distribution, you can find its mean $\mu$ and SD $\sigma.$ The method is to standardize and then to solve two equations in the two
unknowns $\mu$ and $\sigma.$ (This assumes that exact quantiles are known. If quantiles are approximated from data, then results will be only approximate.)
First, you are given 
$$.215 = P(X \le 23) = P\left(Z = \frac{X-\mu}{\sigma} \le \frac{23-\mu}{\sigma}\right),$$
where $Z$ is standard normal, so $\frac{23-\mu}{\sigma} = -0.7892$ (approximately
from standard normal tables or exactly using R statistical software in which the 'quantile fundtion' qnorm is the inverse of a normal CDF).
qnorm(.215)
[1] -0.7891917

Second, you have 
$$ .215 + .7527 + .0215 = 0.9892 = P(X \le 85) = P(Z \le (85 - \mu)/\sigma),$$
so $(85 - \mu)/\sigma) = 2.2974.$
qnorm(.9892)
[1] 2.297329

Roughly, I get $\mu = 38.853,\, \sigma = 20.087.$ Once you have
$\mu$ and $\sigma,$ you can use standardization and normal tables
or you can use software to find the probabilities of you intervals
of interest (listed in your Question).
