91Ó°ÊÓ

Measurements of head circumference (in inches):

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

Let

$n,$the number of data values that are fewer than the data value of a single individual.

$N,$Total number of data values

Now,

Sort the values of all the data in ascending order:

$$\begin{gathered} 20.8,20.8,21.0,21.5,21.6,21.7,21.9,22.0,22.2,22.3,22.4,22.5, \\ 22.5,22.6,22.6,22.7,22.7,22.7,23.0,23.0,23.1,23.3,23.4,23.5,23.5, \\ 23.9,23.9,24.0,24.2,25.6. \end{gathered}$$

Note that

$10$of the $30$data, values are smaller than $22.4$

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

$$\begin{gathered} Percentile=\frac{n}{N}×100\% \\ =\frac{10}{30}×100\% \\ =0.33×100\% \\ =33\% \end{gathered}$$

Thus,

The percentile that corresponds with a head circumference of $22.4$ inches is the $33rd$ percentile.

Let

$n,$The number of data values that are less than the data value of an individual.

$N,$Total number of data values

Now,

Sort the values of all the data in ascending order:

$$\begin{gathered} 20.8,20.8,21.0,21.5,21.6,21.7,21.9,22.0,22.2,22.3,22.4,22.5, \\ 22.5,22.6,22.6,22.7,22.7,22.7,23.0,23.0,23.1,23.3,23.4,23.5,23.5, \\ 23.9,23.9,24.0,24.2,25.6 \end{gathered}$$

Note that

The data set consists of $30$data values.

Such that

$N=30$

The $xth$ percentile represents a data value with $x$ percent of the data values below it.

Then

$90th$ percentile has $90\%$ of the $30$ data values below it.

Calculate $90\%$ of $30$ data values:

$n=90\%×N=\frac{90}{100}×30=\frac{9}{10}×30=27$

We have

$27$ out of $30$ data values are smaller than the $90th$ percentile.

$90th$ percentile needs to be the $28th$ data value in the ascending order, and that is $24.0$

Thus,

Head circumference of the player at the $90th$ percentile of the distribution is $24.0$ inches.