# How do percentiles work?

If I have 5 students (A-E) that score 80%, 70%, 70%, 60% and 50% on a test what percentiles do they fall in?

• A - 20th percentile (80%)
• B - 40th percentile (70%)
• C - 40th percentile (70%)
• D - 80th percentile (60%)
• E - 100th percentile (50%)

Is this correct?

Let me expand the question as I wasn't clear earlier: If A is in the top 20th percentile, are B and C in the top 40th percentile or top 60th percentile?

• @Arturo - Perhaps this should be an answer. – Justin L. Oct 26 '10 at 19:54
• when you ask questions like this, you should try to consult other sources, e.g. the Wikipedia article, first. If they confuse you, ask about the thing that's confusing you. – Qiaochu Yuan Oct 26 '10 at 20:02
• You are misusing "percentile" as a synonym for "percent". They are not the same thing! – Arturo Magidin Oct 27 '10 at 2:46

You've got them backwards. The 20th percentile would be the level below which only 20% of the observations fall. Since 80% of the scores fall below A's score (four out of 5), then A is in the 80th percentile. Since 20% of the scores fall below D's score, then D is in the 20th percentile, not the 80th. See Wikipedia's page on percentiles (it's rather rough, but the lead will do it).

Edit: You edited the question, so here goes the answer to that: It seems that you are not really asking about percentiles, but about percents. They are not the same thing. A is in the top 20 percent, but not "the top 20 percentile". By definition, percentile refers to how many quantities are below. It makes no sense to talk about "top 20 percentile". Your intended question is about percents, not 'percentiles'.

To answer that: You have 5 people; A is the top, B and C are tied in second, D is in fourth, and E in fifth. A is certainly in the top 20% of the group, E in the bottom 20%; D and E are in the bottom 40%. A, B, and C are in the top 60% (not the top 40%); B, C, D, and E are in the bottom 80%. But you are using "percentile" wrong. If X is in the 40th percentile, then that means that 40% of the scores are below him; that would mean he is at the bottom of the top 60%. A is in the 80 percentile, B and C are in the 40th percentile, D is in the 20th percentile.

If you use a z table, the percentiles would make more sense. For example, mean IQ is 100 and the standard deviation is 15. Someone in the 84th percentile would have an IQ of 115.

The z score is 1 found by

z = x - mean IQ/standard deviation

So, z = 115 - 100/15 = 1

Depending on the type of z table you have (half the bell curve or not), you will see a series of percentages that correspond to any given z score. If you have a z table that is based on half the bell curve, then the percentage (not percentile) is .3413 To get the percentile, you add .50 + .3413 given you 84th percentile. Lind, Marchal and Wathen statistics textbooks are a great resource for getting a solid basic understanding of statistics.