Question

If the inter-quartile range is the distance between the first and third quartiles, then the inter-decile range is the distance between the first and ninth decile. (Deciles divide a distribution into ten equal parts.) If IQ is normally distributed with a mean of 100 and a standard deviation of 16, what is the inter-decile range of IQ

Answer 1

The inter-decile range of IQ is 40.96.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

`\mu = 100, \sigma = 16`

First decile:

100/10 = 10th percentile, which is X when Z has a pvalue of 0.1. So it is X when Z = -1.28.

`Z = (X - \mu)/(\sigma)`

`-1.28 = (X - 100)/(16)`

`X - 100 = -1.28*16`

`X = 79.52`

Ninth decile:

9*(100/10) = 90th percentile, which is X when Z has a pvalue of 0.9. So it is X when Z = 1.28.

`Z = (X - \mu)/(\sigma)`

`1.28 = (X - 100)/(16)`

`X - 100 = 1.28*16`

`X = 120.48`

Interdecile range:

120.48 - 79.52 = 40.96

The inter-decile range of IQ is 40.96.