91Ó°ÊÓ

Normal distribution with mean 0 and variance 1 is called the standard normal distribution.

Pdf of standard normal distribution:

\(\begin{array}{l}\phi \left( x \right) = f\left( {x|0,1} \right)\\\phi \left( x \right) = \frac{1}{{{{\left( {2\pi } \right)}^{\frac{1}{2}}}}}\exp \left( { - \frac{1}{2}{x^2}} \right)\,for\, - \infty < x < \infty \end{array}\)

The value of 0.5 quantile of standard normal distribution:.

By the symmetry of the standard normal distribution around 0, the 0.5 quantile must be 0

The value of 0.25 quantile of standard normal distribution:

By symmetry the 0.25 quantile is -0.6745

The value of 0.75 quantile of standard normal distribution:

The 0.75 quantile is found by locating 0.75 in\(\phi \left( x \right)\)column of the standard normal table and interpolating in x column, we find\(\phi \left( {0.67} \right) = 0.7486\)and\(\phi \left( {0.68} \right) = 0.7517\)

Interpolating gives the 0.75 quantile as 0.6745

The value of 0.1 quantile of standard normal distribution:

By symmetry we found the 0.1 quantile is\(\left( { - 1.282} \right)\)

The value of 0.9 quantile of standard normal distribution:

We find the 0.9 quantile by interpolation using

\(\phi \left( {1.28} \right) = 0.8997\)and\(\phi \left( {1.29} \right) = 0.9015\)

The 0.9 quantile is then 1.282