What is the value of the IQR for the standard Normal distrib

What is the value of the IQR for the standard Normal distribution

Solution

The IQR range is the the width of an interval which contains the middle 50% of the data set. It is normally computed by subtracting the first quartile from the third quartile.

As the standard normal distribution is symmetric, so let us assume that the 50% of the data is contained in the interval (-a, a).

So, we can write -

P(-a < Z < a) = 0.5

=> P(Z < a) - P(Z < -a) = 0.5---------------------(i)

And due to symmetry of standard normal distribution about its mean 0, we can write -

P(Z < -a) = P(Z > a)

=> P(Z < -a) = 1 - P(Z < a)

=> P(Z < a) = 1 - P(Z < -a)-------------------- (ii)

So, from (i) and (ii), we conclude -

1 - P(Z < -a) - P(Z < -a) = 0.5

=> 2 P(Z < -a) = 0.5

=> P(Z < -a) = 0.25

From the standard Z-table, we can get the value of Z for which the cumulative probability = 0.25 as -

P(Z < -0.675) = 0.25.

So, a = 0.675.

Hence, the IQR = 0.675 - (-0.675) = 1.35.