# Understanding The Algorithms Behind Quantile() in R

So I was trying to compute the quantiles from a dataset in R:

c(3,5,7,8,12,13,14,18,21)


using the quantile() function, and I realised that it was returning unexpected quartile values for Q1 and Q3 (7 and 14, respectively, should be 6 and 16) on it's own. So I went into the documentation and saw you can assign a type to the function 1-9 that correspond to algorithms. By process of elimination type 6 yields the expected quantiles for that specific dataset, but due to my weak maths I don't actually understand the algorithms being used, and why type 6 specifically worked on my data type (more on those). Does a good explanation for when to use each type exist for someone really new to stats?

• What is unclear for you?
– Tim
Commented Jan 9, 2016 at 20:32
• Commented Jan 9, 2016 at 20:35
• As you found, there are many ways to calculate quantiles. Commented Jan 9, 2016 at 20:39
• @Kendall why do you consider this result to be "wrong"?
– Tim
Commented Jan 9, 2016 at 22:20
• Some useful threads can be found by searching our site. The one at stats.stackexchange.com/questions/13399 seems like it might be particularly helpful.
– whuber
Commented Jan 9, 2016 at 23:40

Why should the answers be $$6$$ and $$16$$? The $$p^\text{th}$$ quantile $$q_p$$ is the smallest value for which the proportion of the sample lesser than or equal to $$q_p$$ is at least $$p$$, and the proportion greater than or equal to $$q_p$$ is at most $$1 - p$$. If we take $$p = 1/4$$ and look at your sample $$\{3, 5, 7, 8, 12, 13, 14, 18, 21 \}$$, $$7$$ matches this definition but $$6$$ doesn't. The same is true for the $$p = 3/4$$ quantile.