Why does normal distribution have the same shape regardless of its parameters?

Question

The formula for normal distribution is quite complicated, it has $\sigma$ in the exponent and in denominator. And no matter what $\sigma$ is, the shape of its pdf is the same (i.e. for example 3 standard deviations lie on the same point of the graph, no matter what $\sigma$ we choose). Could anyone explain how is that possible?

I guess that's why we can use things such as Z-score - the shape is universal, and thus probability as well. We only calculate the distance from the mean expressed in standard deviations.

http://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule

http://en.wikipedia.org/wiki/Normal_distribution

Answer 1

Note if $X\sim N[\mu,\sigma]$, then $\mathbb{P}[\beta\leq X\leq \alpha]=\mathbb{P}[z_\beta\leq z\leq z_\alpha]$ where $z\sim N[0,1]$ and $$z_{\alpha}=\frac{\alpha-\mu}{\sigma}$$ and similarly for $z_\beta$.

Well by definition, we have

$$\mathbb{P}[\beta\leq X\leq \alpha]=\int_{\beta}^\alpha f(x)\,dx=\int_\beta^\alpha \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}\,dx.$$

Now make the $u$-substitution $u=\frac{x-\mu}{\sigma}\Rightarrow \frac{du}{dx}=\frac{1}{\sigma}\Rightarrow dx=\sigma\cdot du$.

Also the limits change $\alpha\rightarrow z_\alpha$ and $\beta\rightarrow z_\beta$ so we have

$$\mathbb{P}[\beta\leq Z\leq \alpha]=\int_{z_\beta}^{z_\alpha}\frac{1}{\sigma \sqrt{2\pi}}e^{-u^2/2}\sigma\,du=\int_{z_\beta}^{z_\alpha}\frac{1}{ \sqrt{2\pi}}e^{-u^2/2}\,du=\mathbb{P}[z_\beta\leq z\leq z_\alpha].$$

Answer 2

Normal distribution two parameters Mean and Standard Deviation do effect Normal distribution.

Mean Mean will shift the Normal distribution forward or backward on it's axis. Mean will not distort shape of Normal Distribution

Standard Deviation Standard Deviation does effect shape A large Standard Deviation mean that normal distribution will approach to less peakedness and more tailedness and a low standard deviation will clutter values around Mean and normal distribution will have large peak. Theoretically speaking Mount Everest the world tallest Mountain of the world if assumed to be like Normal distribution has low standard deviation.