distribution question

Question

anonymous · Answer

Which factor does the width of the peak of a normal curve depend on?

anonymous · Answer

Here's the normal distribution function: $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}~\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$ What happens when you change $\mu$ and $\sigma$? A change in $\mu$ shifts the function. A change in $\sigma$ alters the distribution: a higher standard deviation spreads out the data, while a lower one groups the data closer together. That means a lower $\sigma$ "pushes" the data toward the mean value, which means $\sigma$ also determines the peak of the curve. Also note the fraction coefficient containing $\sigma$: as $\sigma$ increases, the value of $f(x)$ decreases. Play around with this app to see the effects: http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html