Percentile no. $$$75$$$ of $$$6$$$, $$$6$$$, $$$-8$$$, $$$8$$$, $$$-1$$$, $$$5$$$, $$$9$$$, $$$2$$$, $$$-4$$$, $$$6$$$, $$$-3$$$, $$$1$$$

Find the percentile no. $$$75$$$ of $$$6$$$, $$$6$$$, $$$-8$$$, $$$8$$$, $$$-1$$$, $$$5$$$, $$$9$$$, $$$2$$$, $$$-4$$$, $$$6$$$, $$$-3$$$, $$$1$$$.

The percentile no. $$$p$$$ is a value such that at least $$$p$$$ percent of the observations is less than or equal to this value and at least $$$100 - p$$$ percent of the observations is greater than or equal to this value.

The first step is to sort the values.

The sorted values are $$$-8$$$, $$$-4$$$, $$$-3$$$, $$$-1$$$, $$$1$$$, $$$2$$$, $$$5$$$, $$$6$$$, $$$6$$$, $$$6$$$, $$$8$$$, $$$9$$$.

Since there are $$$12$$$ values, then $$$n = 12$$$.

Now, calculate the index: $$$i = \frac{p}{100} n = \frac{75}{100} \cdot 12 = 9$$$.

Since the index $$$i$$$ is an integer, the percentile no. $$$75$$$ is the average of the values at the positions $$$i$$$ and $$$i + 1$$$.

The value at the position $$$i = 9$$$ is $$$6$$$; the value at the position $$$i + 1 = 10$$$ is $$$6$$$.

Their average is the percentile: $$$\frac{6 + 6}{2} = 6$$$.

The percentile no. $$$75$$$A is $$$6$$$A.