I'm trying to figure out the formula for a bell curve knowing only the y-value of the cusp and the standard deviation.
Is it supposed to be the mean? Because I tried putting it into a calculator and the y-value of the cusp increased when I increased the standard deviation.
Best Answer
By the 'cusp' I suppose you mean the mode, which is also the same as the mean and the median for a normal distribution. In general, these are values on the x-axis (which coincide for a normal distrobution).
As in the Comment, the 'y-value' must mean the value of the pdf $y = f(\mu)$ at the mean $\mu$. That is $\frac{1}{\sqrt{2\pi}\sigma}.$ So, given that y-value, you can solve for the standard deviation $\sigma.$ Notice that the y-value at the mode should decrease as the standard deviation increases.
The general formula for a normal density curve (PDF) is $$f(x|\mu,\sigma) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right].$$ When $\mu = x$ the argument of the exponential factor is $0$ so the exponential factor is $1$. If you know $x = \mu$ and the corresponding $y$, then you can write the density function with no unknown constants.
Here is the graph of a normal distribution with $\mu = 10$ (mean, median, and mode), and standard deviation $\sigma = 2.$
The vertical purple line is at $x = \mu = 10$ and the horizontal purple line is at $y = f(\mu) = \frac{1}{2\sqrt{2\pi}} = 0.1994711.$