Question

distribution question

Answer 1

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

Answer 2

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