Question

Deciles The deciles of any distribution are the values at the $10th,20th,\dots ,$and $90th$ percentiles. The first and last deciles are the $10th$ and the $90th$ percentiles, respectively. What are the first and last deciles of the standard Normal distribution?

Short answer

First decile:

$z=-1.28$

Last decile:

$z=1.28$

Step-by-step solution

Given information

The deciles of any distribution are the values at the${10}^{th},{20}^{th}\dots {90}^{th}$percentiles.

Concept

The percentile of an individual is the percentage of the distribution that is less than the data value of the individual.

Calculation

First decile:

The first decile has the attribute of having $10\%$of the data values below it.

Now,

In the normal probability table of the appendix, find the $z-$score that corresponds to a probability of $10\%$(or $0.10$).

Note that

The probability that comes closest is $0.1003$, which is found in row $-1.2$and column$.08$of the normal probability table.

Then

The corresponding $z-$score,

$z=-1.28$

Last decile:

The last decile has the attribute of having 90% of the data values below it.

Similarly, $10\%$of the data values are above it.

Now,

In the normal probability table of the appendix, find the $z-$score that corresponds to a probability of $90\%$(or$0.90$).

Note that

The probability that comes closest is $0.8997$ which is found in row $1.2$ and column$.08$ of the normal probability table.

Then

The corresponding $z-$score,

$z=1.28$