Why do we need to add one to the number of cases when calculating the percentile?

To calculate the value for pth percentile for the following marks:


73, 58, 67, 93, 33, 18, 147





We first need to sort them, right?


So it becomes:


18, 33, 58, 67, 73, 93, 147





Since there are n=7 cases and suppose that I want to calculate the value for the 30th percentile.





By using this formula: p*(n+1)





0.30 * (7+1) = 2.4





So the value for the 30th percentile is 33, am I right?





But how come we need to add one to n?


What is the purpose of doing that?


Thanks a lot.Why do we need to add one to the number of cases when calculating the percentile?
Actually, the 30%-tile is about 35.14.





The first number (18) is at the 7.1%-tile, the second number (33) is at the 21.4%, and the third number (58) is at the 35.7%-tile. The last number is at the 92.9%-tile.





I use the formula p = (x - 0.5) / n, which is the same as


index x = p*n +0.5