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Basically, my math textbook just gave me the formula without any explanation on about the "why"/intuition of the formula. Here's the text from the book:

• Order the data from smallest to largest.

• x = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile.

• y = the number of data values equal to the data value for which you want to find the percentile.

• n = the total number of data.

• Calculate $\frac{x + 0.5y} {n} (100)$. Then round to the nearest integer.

Joe
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2 Answers2

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The percentile is another representation of the 'ranking' of a person. So if there are 100 people with unique ranks, a person at the 25th rank will be at the 75th percentile because there are 75 people after him, i.e., 75% of the candidates are below him. Hence the percentile for people with unique ranks will be expressed as a percentage by $$\frac{x}{n} \cdot 100$$

Now if there are $y$ people having the same rank, I believe the book is placing the candidate's rank in the middle of the range. So it says that there are $(x + 0.5)$ people less than this rank. Hence the ranking expressed as a percentage becomes: $$\frac{x+0.5y}{n}\cdot 100$$

user1952500
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  • What do you mean about the book placing the candidate's rank in the middle of the range? Why does it do that? – Joe Apr 27 '19 at 17:21
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    @user65141 Suppose there are 100 candidates and 5 ranks. Suppose the number of candidates per rank are 1:15, 2:20, 3:15, 4:30, 5:20. Now if someone gets the second rank, what is the percentile? There are 15+30+20=65 people in ranks 3,4,5, so it’s atleast 65. But there are only 15 people better, and we don’t know the relative position of the candidate within rank 2. So we can place him in the middle and say there are about 10 people in rank 2 who are after him. Hence his percentile will become 10+65=75. – user1952500 Apr 27 '19 at 17:50
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    Oh, I see now! Thanks so much for answering! – Joe Apr 28 '19 at 17:57
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The percentile $p \times 100\%$ is the value $v$ below which we have close to $p \times 100\%$ of the values of the data (I used the term "close" because if data is discrete we may not be able to get exactly $p \times 100\%$ of the values).

If all the data values are different, the value at position $x+1$ in ascending order will be an estimate of the percentile $$ \frac{x+0.5}{n} \times 100\% $$

If you have repetitions you have to take that into account, and the value halfway through the repeated values will be an estimate of the percentile $$ \frac{x+0.5y}{n} \times 100\% $$

On a side note, the formulas you presented are one of the possible ways to estimate the percentiles.

Ertxiem - reinstate Monica
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